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* (c) 2018-present Yuxi (Evan) You and Vue contributors
* @license MIT
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* @license MIT
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<div class="callout" data-callout="example">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-list"><line x1="8" x2="21" y1="6" y2="6"/><line x1="8" x2="21" y1="12" y2="12"/><line x1="8" x2="21" y1="18" y2="18"/><line x1="3" x2="3.01" y1="6" y2="6"/><line x1="3" x2="3.01" y1="12" y2="12"/><line x1="3" x2="3.01" y1="18" y2="18"/></svg>
</div>
<div class="callout-title-inner"><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6-.html" target="_blank">总目录</a></div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.83,words:250},title:"",type:"article",s:"Home"}}],["/",{loader:()=>B(()=>import("./index.html--uItrwRC.js"),[]),meta:{date:1775154041e3,excerpt:`<h2>简介</h2>
<p>基于Obsidian建立公开共享的数学知识库！</p>
<p>传统教材是线性的, 而数学概念是错综复杂的网络. 本库除了提供<strong>线性目录</strong>外, 还利用 Obsidian 的双向链接, 将知识点互相连接, 构建动态互联的<strong>数学知识图谱</strong>, 让学习者能够顺着思路学习, 而不仅仅是顺着章节学习. 内容上致力于以通俗易懂的语言梳理高等数学的主要知识点. <strong>最终目的: 取代防自学教材</strong>.</p>
<p><strong>本仓库的一切都优先建立在本地Obsidian仓库使用</strong>, 文档遵循 <a href="proxy.php?url=https%3A%2F%2Fhelp.obsidian.md%2Fobsidian-flavored-markdown" target="_blank" rel="noopener noreferrer">Obsidian Flavored Markdown</a> 规范, 充分利用 Obsidian 特色语法.<br>
同时感谢<a href="proxy.php?url=https%3A%2F%2Fgithub.com%2FLincZero" target="_blank" rel="noopener noreferrer">LincZero</a>提供的网页部署: <a href="proxy.php?url=https%3A%2F%2Fpkm-er.github.io%2FPkmer-Math%2F" target="_blank" rel="noopener noreferrer">https://pkm-er.github.io/Pkmer-Math/</a></p>`,readingTime:{minutes:4.2,words:1259},title:"",type:"article",s:"/"}}],["/Other/--%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95--.html",{loader:()=>B(()=>import("./--微分几何--.html-D7aaiETQ.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学","目录"],excerpt:`
<div class="callout" data-callout="question">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-help-circle"><circle cx="12" cy="12" r="10"/><path d="M9.09 9a3 3 0 0 1 5.83 1c0 2-3 3-3 3"/><path d="M12 17h.01"/></svg>
</div>
<div class="callout-title-inner">Question</div>
</div>
<div class="callout-content"><p>微分几何的核心问题: <strong>当空间不再是平直的欧几里得空间时, 如何定义&quot;导数&quot; , &quot;长度&quot; , &quot;角度&quot; 和&quot;曲率&quot; ？</strong></p>
</div></div>`,readingTime:{minutes:.91,words:274},title:"",type:"article",s:"--微分几何--"}}],["/Other/--%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90--.html",{loader:()=>B(()=>import("./--泛函分析--.html-BkM4HRNL.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li>赋范空间
<ul>
<li>向量空间</li>
<li>范数</li>
<li>赋范空间的定义</li>
</ul>
</li>
<li>巴拿赫空间
<ul>
<li>巴拿赫空间的定义</li>
<li>完备性</li>
<li>巴拿赫空间的例子</li>
</ul>
</li>
<li>内积空间
<ul>
<li>内积的定义</li>
<li>诱导范数</li>
<li>欧几里得空间</li>
</ul>
</li>
<li>希尔伯特空间
<ul>
<li>希尔伯特空间的定义</li>
<li>正交性</li>
<li>正交分解定理</li>
</ul>
</li>
<li>有界线性算子
<ul>
<li>有界线性算子的定义</li>
<li>有界性与连续性</li>
<li>线性算子的范数</li>
</ul>
</li>
<li>对偶空间
<ul>
<li>对偶空间的定义</li>
<li>典范嵌入</li>
<li>Hahn-Banach定理</li>
</ul>
</li>
<li>弱拓扑
<ul>
<li>弱收敛</li>
<li>弱拓扑的定义</li>
<li>Alaoglu定理</li>
</ul>
</li>
<li>紧算子
<ul>
<li>紧算子的定义</li>
<li>紧性与列紧性</li>
<li>谱定理</li>
</ul>
</li>
<li>巴拿赫代数
<ul>
<li>巴拿赫代数的定义</li>
<li>Gelfand表示</li>
<li>光滑性问题</li>
</ul>
</li>
<li>自伴算子
<ul>
<li>自伴算子的定义</li>
<li>谱性质</li>
<li>作用在希尔伯特空间上的算子</li>
</ul>
</li>
<li>泛函分析中的主要定理
<ul>
<li>Hahn-Banach定理</li>
<li>开映射定理</li>
<li>闭图像定理</li>
<li>Riesz表示定理</li>
</ul>
</li>
<li>Sobolev空间
<ul>
<li>Sobolev空间的定义</li>
<li>函数空间的嵌入</li>
<li>Sobolev不等式</li>
</ul>
</li>
<li>泛函分析的应用
<ul>
<li>偏微分方程的弱解</li>
<li>谱理论在物理中的应用</li>
<li>变分法</li>
</ul>
</li>
<li>补充
<ul>
<li>半范数</li>
<li>Fréchet空间</li>
<li>局部凸空间</li>
</ul>
</li>
</ul>`,readingTime:{minutes:1.06,words:319},title:"目录",type:"article"}}],["/Other/-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6-.html",{loader:()=>B(()=>import("./-高等数学-.html-Csow_2OO.js"),[]),meta:{author:["Cyletix"],tag:["总览","数学","目录"],excerpt:`
<p>相比微积分是一个更广泛的概念, 不仅包括微积分,还包括线性代数、微分方程、复变函数、实变函数、泛函分析等</p>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F--%E5%BE%AE%E7%A7%AF%E5%88%86--.html" target="_blank">--微积分--</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F--%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0--.html" target="_blank">--线性代数--</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F--%E6%A6%82%E7%8E%87%E8%AE%BA--.html" target="_blank">--概率论--</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6%E8%BF%9B%E9%98%B6-.html" target="_blank">-高等数学进阶-</a></li>
</ul>`,readingTime:{minutes:.56,words:167},title:"广义的高等数学",type:"article"}}],["/Other/-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6%E8%BF%9B%E9%98%B6-.html",{loader:()=>B(()=>import("./-高等数学进阶-.html-CVrUfInD.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">实分析</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F--%E5%A4%8D%E5%88%86%E6%9E%90--.html" target="_blank">--复分析--</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F--%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90--.html" target="_blank">--泛函分析--</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F--%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B--.html" target="_blank">--偏微分方程--</a></li>
</ul>`,readingTime:{minutes:.5,words:149},title:"分析",type:"article"}}],["/Other/Lp%E8%8C%83%E6%95%B0.html",{loader:()=>B(()=>import("./Lp范数.html-C9CoplAe.js"),[]),meta:{author:["Cyletix"],tag:["线性代数","向量空间","范数"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">核心思想</div>
</div>
<div class="callout-content"><p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">L_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 范数度量了向量的“长度”，其不同取值对应不同的几何度量方式：从菱形到圆形，再到方形。</p>
</div></div>`,readingTime:{minutes:1.08,words:324},title:"内容大纲",type:"article"}}],["/Other/%E5%8F%AF%E5%8A%A0%E6%80%A7.html",{loader:()=>B(()=>import("./可加性.html-CoF21g_H.js"),[]),meta:{author:["Cyletix","DeepSeekV3"],tag:["数学"],excerpt:`
<h2>简介</h2>
<p>可加性是<a class="route-link nolebase-route-link-invalid" href="#" target="_target">泛函分析</a>和<a class="route-link nolebase-route-link-invalid" href="#" target="_target">算子理论</a>中的核心概念，描述算子是否保持加法结构，是研究<a class="route-link nolebase-route-link-invalid" href="#" target="_target">线性算子</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84.html" target="_blank">线性映射</a>的基础。</p>`,readingTime:{minutes:.84,words:252},title:"可加性",type:"article"}}],["/Other/%E5%BE%AE%E5%88%86%E6%B5%81%E5%BD%A2.html",{loader:()=>B(()=>import("./微分流形.html-DMayZWkv.js"),[]),meta:{author:["Cyletix","GPT-5"],tag:["数学","微分几何"],excerpt:`<p>微分流形是一个在局部与欧几里得空间同胚、并能在其上定义光滑函数的集合。<br>
它提供了“在一般空间上进行微积分”的最小结构要求。</p>
<hr>
<h2>定义</h2>
<p>一个拓扑空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> 称为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 维<strong>微分流形</strong>，若：</p>`,readingTime:{minutes:1.1,words:331},title:"",type:"article",s:"微分流形"}}],["/Other/%E6%8B%93%E6%89%91%E5%AD%A6.html",{loader:()=>B(()=>import("./拓扑学.html-mMxwZ-9O.js"),[]),meta:{tag:["数学"],excerpt:`
<h3>1. 点集拓扑</h3>
<ul>
<li>拓扑空间的定义
<ul>
<li>开集与闭集的基本性质</li>
<li>拓扑基与子基</li>
<li>子空间拓扑</li>
</ul>
</li>
<li>连续函数与同胚
<ul>
<li>连续性的拓扑定义</li>
<li>同胚与拓扑等价</li>
</ul>
</li>
<li>分离性与紧致性
<ul>
<li>T0、T1、T2（Hausdorff）分离公理</li>
<li>紧致性与极限点紧性</li>
<li>紧性与有限覆盖</li>
</ul>
</li>
<li>连通性
<ul>
<li>连通空间与路径连通</li>
<li>局部连通性</li>
</ul>
</li>
<li>度量化与嵌入
<ul>
<li>度量空间与拓扑的关系</li>
<li>Urysohn嵌入定理</li>
</ul>
</li>
</ul>`,readingTime:{minutes:1.43,words:428},title:"目录",type:"article"}}],["/Other/%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80%E6%A6%82%E5%BF%B5%E6%A8%A1%E5%9E%8B.html",{loader:()=>B(()=>import("./数学基础概念模型.html-B775HCLa.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],excerpt:`<ul>
<li><strong>数学世界的事物</strong>
<ul>
<li>实体
<ul>
<li>数学对象：数、图形、结构……，被研究的东西；
<ul>
<li><strong>除了属性，还有属性的持有者——实体。</strong> 实体与属性相互依存，就像硬币的两面，共同构成了我们认知中的“事物”。</li>
</ul>
</li>
<li><mark>概念</mark>：反映对象在空间形式和数量关系等方面本质属性的思维形式”，通常通过“内涵（本质属性）+ 外延（所指对象的全体）”来刻画。是我们理解、命名、区分各种数学对象的“词汇/术语”。</li>
</ul>
</li>
<li>属性
<ul>
<li><strong><mark>定义</mark></strong>
<ul>
<li><strong><mark>性质</mark></strong>：是“由内向外”的推导。对象⟹特征
<ul>
<li>【数学对象（数、集合、函数、图形、代数结构等）或数学概念在其定义下所具有的、经证明成立的某种稳定的特征或规律，且这种特征或规律不依赖于具体表示方式或应用场景的改变而改变】</li>
<li>【因为它是某个东西，所以它具备什么特征。】</li>
</ul>
</li>
<li><mark>判定</mark>：是“由外向内”的确认。特征⟹对象
<ul>
<li>【依据特定的一组充分条件或充要条件，通过逻辑推理确认数学对象是否归属于特定集合或类别，或是否具备特定性质的命题与过程。】</li>
<li>【因为它具备某些特征，所以它是某个东西。】</li>
</ul>
</li>
<li><mark>命题</mark>
<ul>
<li><mark>公理</mark>
<ul>
<li>【是指依据人类理性的不证自明的基本事实。】</li>
<li>【世界api【自然资源），相当于python提供的基本api】</li>
</ul>
</li>
<li><mark>猜想</mark>
<ul>
<li>【证明、证伪、不可被证】</li>
<li><mark>假设</mark></li>
</ul>
</li>
<li><mark>引理</mark>
<ul>
<li>【证明过程中的辅助命题】</li>
<li>【中间产物，相当复杂】</li>
<li>【相当于python的函数】</li>
</ul>
</li>
<li><mark>定理</mark>
<ul>
<li>【是经过受逻辑限制的证明为真的陈述。】</li>
<li>【相当复杂的推理【推理，基本作用）】</li>
<li>【从公理或其他已被证明的定理出发，经过受逻辑限制的演绎推导，得到的另一个真命题。】</li>
<li>【相当于python程序，或者一个打包好的程序级别的API【利用自然资源创造的事物）】</li>
<li>【各种语言软件包的第三方库】</li>
<li><mark>推论</mark></li>
</ul>
</li>
</ul>
</li>
<li><mark>原理</mark>
<ul>
<li>【有无实践】</li>
<li>【通过实践或是试验等，总结出来的，规则或概念。】</li>
<li><mark>定律</mark>
<ul>
<li>【<strong>定律是一种理论模型</strong>，是描述客观世界变化规律的表达式或者文字。】</li>
<li>【通过观察和试验发现的一种<strong>规律性</strong>，它描述了自然界中一种<strong>恒定关系</strong>或模式。】</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li>关系与运算</li>
<li>结构与系统</li>
</ul>
</li>
</ul>`,readingTime:{minutes:2.09,words:627},title:"",type:"article",s:"数学基础概念模型"}}],["/Other/%E6%95%B0%E8%AE%BA.html",{loader:()=>B(()=>import("./数论.html-BtSA21Eq.js"),[]),meta:{tag:["数学"],excerpt:`
<h2>初等数论</h2>
<ul>
<li>整数的基本性质
<ul>
<li>整除性与最大公因数</li>
<li>欧几里得算法</li>
<li>同余与模运算</li>
</ul>
</li>
<li>素数
<ul>
<li>素数分布与质数定理</li>
<li>素性测试</li>
<li>素数生成</li>
</ul>
</li>
<li>数论函数
<ul>
<li>除数函数与σ函数</li>
<li>欧拉函数</li>
<li>Möbius函数与莫比乌斯反演公式</li>
</ul>
</li>
<li>同余方程
<ul>
<li>一次同余方程</li>
<li>中国剩余定理</li>
<li>二次剩余与平方同余</li>
</ul>
</li>
</ul>`,readingTime:{minutes:1.49,words:446},title:"目录",type:"article"}}],["/Other/%E8%8C%83%E7%95%B4%E8%AE%BA.html",{loader:()=>B(()=>import("./范畴论.html-D7CG9t9L.js"),[]),meta:{tag:["数学"],excerpt:`
<ul>
<li>范畴的定义
<ul>
<li>对象</li>
<li>态射</li>
<li>组合</li>
</ul>
</li>
<li>范畴的例子
<ul>
<li>集合范畴</li>
<li>拓扑范畴</li>
<li>群范畴</li>
<li>模范畴</li>
</ul>
</li>
<li>态射的分类
<ul>
<li>单态射</li>
<li>满态射</li>
<li>同构</li>
<li>等价</li>
</ul>
</li>
<li>函子
<ul>
<li>函子</li>
<li>协变函子</li>
<li>逆变函子</li>
</ul>
</li>
<li>自然变换
<ul>
<li>自然变换</li>
<li>自然同构</li>
</ul>
</li>
<li>极限与余极限
<ul>
<li>锥</li>
<li>极限</li>
<li>余锥</li>
<li>余极限</li>
</ul>
</li>
<li>加法范畴
<ul>
<li>加法范畴</li>
<li>零对象</li>
<li>二元积</li>
<li>二元余积</li>
</ul>
</li>
<li>阿贝尔范畴
<ul>
<li>阿贝尔范畴</li>
<li>核与上核</li>
<li>余核与下核</li>
</ul>
</li>
<li>导出函子
<ul>
<li>导出函子</li>
<li>Tor函子</li>
<li>Ext函子</li>
</ul>
</li>
<li>拓扑中的范畴论
<ul>
<li>拓扑空间的范畴</li>
<li>同伦范畴</li>
<li>层范畴</li>
</ul>
</li>
<li>几何中的范畴论
<ul>
<li>Sheaf</li>
<li>函子性解释</li>
<li>模空间</li>
</ul>
</li>
<li>范畴论在代数中的应用
<ul>
<li>同调代数的范畴视角</li>
<li>局部化</li>
<li>导出范畴</li>
</ul>
</li>
<li>范畴论在计算机科学中的应用
<ul>
<li>λ-演算与范畴论</li>
<li>类型论中的范畴理论</li>
<li>函子性设计模式</li>
</ul>
</li>
<li>补充
<ul>
<li>扇形范畴</li>
<li>层</li>
<li>Grothendieck范畴</li>
<li>双范畴</li>
<li>2-范畴</li>
</ul>
</li>
</ul>`,readingTime:{minutes:.87,words:260},title:"目录",type:"article"}}],["/assets_/WolframAlpha.html",{loader:()=>B(()=>import("./WolframAlpha.html-DK5UdoVU.js"),[]),meta:{tag:["数学","工具"],excerpt:`<div style="height: 600px; margin: -100px 0 -40px -40px; overflow: hidden; position: relative;">
    <iframe id="carbon-iframe" src="proxy.php?url=https%3A%2F%2Fwww.wolframalpha.com%2F" style="width: 110%; height: 600px; border: none; position: absolute; top: 0; left: -5%; transform: scale(0.9); " frameborder="no" border="0" marginwidth="0" marginheight="0" scrolling="no" allowtransparency="yes">
    </iframe>
</div>`,readingTime:{minutes:.19,words:56},title:"",type:"article",s:"WolframAlpha"}}],["/assets_/%E5%85%AC%E5%BC%8F%E7%BC%96%E8%BE%91%E5%99%A8.html",{loader:()=>B(()=>import("./公式编辑器.html-CrAOHGbB.js"),[]),meta:{tag:["数学","工具"],excerpt:`<div style="height:800px; margin: -120px -60px -60px -60px; overflow: hidden; position: relative;">
    <iframe id="carbon-iframe" src="proxy.php?url=https%3A%2F%2Fwww.2weima.com%2Fgongshi.html%3Fgongshi%3Dpower" style="width: 100%; height: 800px; border: none; position: absolute; top: 0; left: 0; transform: scale(0.9); filter: invert(1);" frameborder="no" border="0" marginwidth="0" marginheight="0" scrolling="no" allowtransparency="yes"> </iframe> </div>`,readingTime:{minutes:.2,words:61},title:"",type:"article",s:"公式编辑器"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/--%E5%BE%AE%E7%A7%AF%E5%88%86--.html",{loader:()=>B(()=>import("./--微积分--.html-ChDKb6oO.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">核心思想</div>
</div>
<div class="callout-content"><p>微积分研究<strong>瞬时变化</strong>和<strong>累积效应</strong>, 用极限理解连续世界的动态结构</p>
</div></div>`,readingTime:{minutes:1.87,words:561},title:"目录",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/--%E6%A6%82%E7%8E%87%E8%AE%BA--.html",{loader:()=>B(()=>import("./--概率论--.html-CHyhK5Kg.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">核心思想</div>
</div>
<div class="callout-content"><p>概率论研究<strong>可能性的规律</strong>, 以逻辑的方式理解<strong>偶然中的必然</strong></p>
</div></div>`,readingTime:{minutes:.47,words:141},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/--%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0--.html",{loader:()=>B(()=>import("./--线性代数--.html-DtZAKn65.js"),[]),meta:{author:"Cyletix",tag:["数学","目录"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">核心思想</div>
</div>
<div class="callout-content"><p>线性代数是研究<strong>线性结构与变换</strong>的数学。用结构化的方式理解空间的骨架。</p>
</div></div>`,readingTime:{minutes:.4,words:119},title:"目录-同济版",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/Gauss-Jordan%20Elimination%20solve%20Inverse%20Matrix.html",{loader:()=>B(()=>import("./Gauss-Jordan Elimination solve Inverse Matrix.html-BJz9jKFW.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>To find the inverse of the matrix <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">A = \\begin{pmatrix} 1 &amp; 1 \\\\ 1 &amp; -1 \\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span> using Gauss-Jordan elimination, we can follow these steps:</p>',readingTime:{minutes:.9,words:270},title:"",type:"article",s:"Gauss-Jordan Elimination solve Inverse Matrix"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B8%89%E7%BB%B4%E7%A9%BA%E9%97%B4%E5%A0%B5%E8%BD%A6%E6%A6%82%E7%8E%87%E8%AF%84%E4%BC%B0.html",{loader:()=>B(()=>import("./三维空间堵车概率评估.html-BUIcyPOk.js"),[]),meta:{tag:["数学"],excerpt:`<p>计算二维和三维空间中拥堵概率的具体数值涉及到复杂的概率和统计模型。以下是一个简化的示例，用于计算拥堵概率的基础概念和步骤：</p>
<ol>
<li>
<p><strong>定义参数</strong>：</p>
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span>：总车辆或无人机数量</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>：二维空间的面积</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>：三维空间的体积</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>2</mn><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\\lambda_{2D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：二维空间中的车辆密度，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>2</mn><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mi>N</mi><mi>A</mi></mfrac></mrow><annotation encoding="application/x-tex">\\lambda_{2D} = \\frac{N}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>3</mn><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\\lambda_{3D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：三维空间中的无人机密度，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>3</mn><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mi>N</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">\\lambda_{3D} = \\frac{N}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.2222em;">V</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li>
</ul>
</li>
<li>
<p><strong>泊松过程模型</strong>：</p>
<ul>
<li>假设车辆或无人机的分布符合泊松过程。</li>
<li>二维空间中某一区域拥堵的概率：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>2</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mrow><mn>2</mn><mi>D</mi></mrow></msub><mo>⋅</mo><msub><mi>A</mi><mtext>note</mtext></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{2D} = 1 - e^{-\\lambda_{2D} \\cdot A_{\\text{note}}} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mbin mtight">⋅</span><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2963em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
其中，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mtext>note</mtext></msub></mrow><annotation encoding="application/x-tex">A_{\\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是考虑的区域面积。</li>
<li>三维空间中某一区域拥堵的概率：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>3</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>λ</mi><mrow><mn>3</mn><mi>D</mi></mrow></msub><mo>⋅</mo><msub><mi>V</mi><mtext>note</mtext></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{3D} = 1 - e^{-\\lambda_{3D} \\cdot V_{\\text{note}}} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mbin mtight">⋅</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2963em;"><span style="top:-2.357em;margin-left:-0.2222em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
其中，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mtext>note</mtext></msub></mrow><annotation encoding="application/x-tex">V_{\\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是考虑的区域体积。</li>
</ul>
</li>
<li>
<p><strong>计算示例</strong>：</p>
<ul>
<li>假设二维空间的面积 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mn>1000</mn><msup><mtext>m</mtext><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">A = 1000 \\text{m}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1000</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，总车辆数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">N = 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">100</span></span></span></span>。</li>
<li>假设三维空间的体积 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mn>10000</mn><msup><mtext>m</mtext><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">V = 10000 \\text{m}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">10000</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>，总无人机数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">N = 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">100</span></span></span></span>。</li>
</ul>
<p>二维和三维的密度分别为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>λ</mi><mrow><mn>2</mn><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mn>100</mn><mn>1000</mn></mfrac><mo>=</mo><mn>0.1</mn><msup><mtext>vehicles/m</mtext><mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> \\lambda_{2D} = \\frac{100}{1000} = 0.1 \\text{vehicles/m}^2 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1000</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">100</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord">0.1</span><span class="mord"><span class="mord text"><span class="mord">vehicles/m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>λ</mi><mrow><mn>3</mn><mi>D</mi></mrow></msub><mo>=</mo><mfrac><mn>100</mn><mn>10000</mn></mfrac><mo>=</mo><mn>0.01</mn><msup><mtext>drones/m</mtext><mn>3</mn></msup></mrow><annotation encoding="application/x-tex"> \\lambda_{3D} = \\frac{100}{10000} = 0.01 \\text{drones/m}^3 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10000</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">100</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord">0.01</span><span class="mord"><span class="mord text"><span class="mord">drones/m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>考虑二维空间中一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><msup><mtext>m</mtext><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">10 \\text{m}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">10</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 的区域：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>2</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.1</mn><mo>⋅</mo><mn>10</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≈</mo><mn>0.632</mn></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{2D} = 1 - e^{-0.1 \\cdot 10} = 1 - e^{-1} \\approx 0.632 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.1</span><span class="mbin mtight">⋅</span><span class="mord mtight">10</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.632</span></span></span></span></span></p>
<p>考虑三维空间中一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><msup><mtext>m</mtext><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">100 \\text{m}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">100</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 的区域：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>3</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.01</mn><mo>⋅</mo><mn>100</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≈</mo><mn>0.632</mn></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{3D} = 1 - e^{-0.01 \\cdot 100} = 1 - e^{-1} \\approx 0.632 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.01</span><span class="mbin mtight">⋅</span><span class="mord mtight">100</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.632</span></span></span></span></span></p>
</li>
<li>
<p><strong>调整参数和区域大小</strong>：</p>
<ul>
<li>可以根据实际情况调整区域大小和密度，重新计算拥堵概率。</li>
<li>例如，如果二维空间的区域减小到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mtext>m</mtext><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1 \\text{m}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>2</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.1</mn><mo>⋅</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.1</mn></mrow></msup><mo>≈</mo><mn>0.095</mn></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{2D} = 1 - e^{-0.1 \\cdot 1} = 1 - e^{-0.1} \\approx 0.095 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.1</span><span class="mbin mtight">⋅</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.095</span></span></span></span></span></p>
</li>
</ul>
<p>而三维空间中的一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mtext>m</mtext><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">1 \\text{m}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 区域：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mtext>note</mtext><mrow><mn>3</mn><mi>D</mi></mrow></msubsup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.01</mn><mo>⋅</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mn>0.01</mn></mrow></msup><mo>≈</mo><mn>0.01</mn></mrow><annotation encoding="application/x-tex"> P_{\\text{note}}^{3D} = 1 - e^{-0.01 \\cdot 1} = 1 - e^{-0.01} \\approx 0.01 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">D</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.01</span><span class="mbin mtight">⋅</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">0.01</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.01</span></span></span></span></span></p>
</li>
</ol>`,readingTime:{minutes:1.77,words:531},title:"",type:"article",s:"三维空间堵车概率评估"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B8%89%E7%BB%B4%E7%A9%BA%E9%97%B4%E7%BB%93%E6%9E%84%E6%8A%95%E5%BD%B1%E5%88%B0%E6%8C%87%E5%AE%9A%E5%B9%B3%E9%9D%A2.html",{loader:()=>B(()=>import("./三维空间结构投影到指定平面.html-CYMCLQME.js"),[]),meta:{author:["GPT-4o"],tag:["数学","python"],excerpt:`<p>三维空间结构投影到指定平面，可以用计算机图形学中的<strong>投影变换</strong>来实现。以下是具体步骤：</p>
<h3>1. 定义投影的平面 (Camera 平面)</h3>
<ul>
<li>
<p>假设投影平面是二维平面，可以用一个平面方程表示：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">n</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo>−</mo><msub><mi mathvariant="bold">r</mi><mn mathvariant="bold">0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\mathbf{n} \\cdot (\\mathbf{r} - \\mathbf{r_0}) = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathbf">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathbf">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>其中：</p>
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">n</mi></mrow><annotation encoding="application/x-tex">\\mathbf{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">n</span></span></span></span> 是平面法向量</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">r</mi></mrow><annotation encoding="application/x-tex">\\mathbf{r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">r</span></span></span></span> 是空间点的坐标</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mn mathvariant="bold">0</mn></msub></mrow><annotation encoding="application/x-tex">\\mathbf{r_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是平面上一点（通常可以选为原点或Camera中心）</li>
</ul>
</li>
<li>
<p>如果投影平面是正交的，我们需要摄像机的视角参数，例如：</p>
<ul>
<li>摄像机位置 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">c</mi></mrow><annotation encoding="application/x-tex">\\mathbf{c}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">c</span></span></span></span></li>
<li>投影方向（光线方向）<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">d</mi></mrow><annotation encoding="application/x-tex">\\mathbf{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">d</span></span></span></span></li>
</ul>
</li>
</ul>`,readingTime:{minutes:2.5,words:750},title:"",type:"article",s:"三维空间结构投影到指定平面"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%BE%8B%E9%A2%98-%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95.html",{loader:()=>B(()=>import("./例题-分部积分法.html-Dr7YUiIJ.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<ol>
<li>换元, 将 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 化入dx后, 分部积分</li>
</ol>`,readingTime:{minutes:.66,words:198},title:"求解",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%BE%8B%E9%A2%98-%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%87%BD%E6%95%B0%E6%B1%82%E6%9E%81%E5%80%BC.html",{loader:()=>B(()=>import("./例题-拉格朗日函数求极值.html-CfWOjpmt.js"),[]),meta:{tag:["数学"],excerpt:'<p>我们考虑 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>+</mo><msub><mi>r</mi><mn>2</mn></msub><mo>+</mo><msub><mi>r</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r_1 + r_2 + r_3 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，并要求 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">r_1, r_2, r_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 构成向量范数的最小值。我们需要最小化 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mi>r</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\|r\\|^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r = (r_1, r_2, r_3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。假设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">r_1, r_2, r_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是复数（或实数）。向量的范数定义为：</p>',readingTime:{minutes:2.34,words:703},title:"",type:"article",s:"例题-拉格朗日函数求极值"}}],["/Other/%E4%BE%8B%E9%A2%98/%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0%E9%A1%B9%E8%AE%A1%E7%AE%97%E5%8C%96%E7%AE%80.html",{loader:()=>B(()=>import("./傅立叶级数项计算化简.html-CDLaSU7G.js"),[]),meta:{tag:["数学"],excerpt:`<p>计算高次三角函数积分时，有一些常用的技巧和方法可以简化计算过程。以下是一些技巧：</p>
<h3>1. 使用三角函数恒等式</h3>
<p>三角函数的幂可以用三角恒等式化简。例如：</p>
<ul>
<li>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin^2(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\cos^2(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 可以用半角公式表示：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mspace width="1em"/><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\\sin^2(t) = \\frac{1 - \\cos(2t)}{2}, \\quad \\cos^2(t) = \\frac{1 + \\cos(2t)}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
</li>
<li>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>4</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin^4(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 可以进一步化简为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>4</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo fence="true">(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></mfrac></mrow><annotation encoding="application/x-tex">\\sin^4(t) = \\left(\\frac{1 - \\cos(2t)}{2}\\right)^2 = \\frac{1 - 2\\cos(2t) + \\cos^2(2t)}{4}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.604em;vertical-align:-0.95em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.654em;"><span style="top:-3.9029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
</li>
</ul>`,readingTime:{minutes:2.17,words:651},title:"",type:"article",s:"傅立叶级数项计算化简"}}],["/Other/%E4%BE%8B%E9%A2%98/%E5%86%85%E5%AD%98%E6%9D%A1%E7%9A%84%E5%AF%BF%E5%91%BD.html",{loader:()=>B(()=>import("./内存条的寿命.html-CC8AqTYM.js"),[]),meta:{author:["Cyletix"],tag:["数学","chatgpt"],excerpt:`<p>根据寿命内存条的寿命和使用时间，能否计算第n天使用时坏掉的概率</p>
<h1>GPT-4</h1>
<p>要计算第 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 天内存条坏掉的概率，我们需要一些额外的信息或假设，比如内存条的预期寿命分布和是否存在使用时间对故障率的影响。通常，设备故障率可以用<strong>指数分布</strong>或<strong>韦伯分布</strong>（Weibull distribution）来建模。以下是如何使用这两种分布来估算的步骤：</p>`,readingTime:{minutes:2.67,words:801},title:"GPT-4",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E5%A4%8D%E7%BA%BF%E7%A7%AF%E5%88%861.html",{loader:()=>B(()=>import("./复线积分1.html-BTzHAJ6U.js"),[]),meta:{tag:["数学"],excerpt:'<p>当求解复函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mn>7</mn><mi>z</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>−</mo><mi>z</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{7z-1}{z^2-z-6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mbin mtight">−</span><span class="mord mtight">6</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 沿路径 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">|z|=5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 的线积分时, 需要首先确认所有奇点是否在该路径内. 我们已知函数的奇点为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">z = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">z = -2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">2</span></span></span></span>. 因为这两个奇点的模都小于5( <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mn>3</mn><mi mathvariant="normal">∣</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">|3| = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣3∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mo>−</mo><mn>2</mn><mi mathvariant="normal">∣</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">|-2| = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>) , 所以它们都在由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">|z| = 5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 定义的圆路径内部.</p>',readingTime:{minutes:1.21,words:363},title:"",type:"article",s:"复线积分1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E5%A4%8D%E7%BA%BF%E7%A7%AF%E5%88%862.html",{loader:()=>B(()=>import("./复线积分2.html-DMhvICVW.js"),[]),meta:{tag:["数学"],excerpt:'<p>复素积分中，当极点是重根时，使用留数定理需要计算高阶留数。对于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z = z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处具有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 重极点的情况，其留数的计算公式如下：</p>',readingTime:{minutes:1.63,words:489},title:"",type:"article",s:"复线积分2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%8A%BD%E5%8D%A1%E6%A6%82%E7%8E%87%E8%AE%A1%E7%AE%97.html",{loader:()=>B(()=>import("./抽卡概率计算.html-C60Wx3WN.js"),[]),meta:{tag:["数学"],excerpt:`<p>在明日方舟这个手游中，官方给出的数据为抽卡有2%概率出六星稀有度(最高稀有度)，6星干员中娜仁图雅的概率是35%，佩佩的概率是35%，剩下30％概率是其他干员。一个人抽了300次，7个佩佩，2个娜仁图雅，3个其他。这样的结果是否合理？使用假说检验来验证是否能够接受官方给出的概率</p>
<hr>
<p>要验证这样的结果是否合理，可以使用假设检验方法。我们可以采用卡方检验（Chi-square test）来检查实际抽卡结果是否与官方给出的概率分布有显著差异。</p>
<h3>1. 确定假设</h3>
<ul>
<li><strong>原假设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>：抽卡结果符合官方给出的概率分布。</li>
<li><strong>备择假设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">H_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>：抽卡结果与官方给出的概率分布不符。</li>
</ul>`,readingTime:{minutes:2.09,words:626},title:"",type:"article",s:"抽卡概率计算"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%A4%AD%E5%9C%86%E4%BA%8C%E9%87%8D%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./椭圆二重积分.html-XO60ekii.js"),[]),meta:{author:["GPT-4"],tag:["数学","例题"],excerpt:`<h3>椭圆的标准形式和参数化</h3>
<p>椭圆的标准方程为:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>`,readingTime:{minutes:12.69,words:3806},title:"为什么要用雅可比矩阵？",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E7%9F%A9%E9%98%B5%E5%A4%9A%E9%A1%B9%E5%BC%8F%E5%AF%B9%E8%A7%92%E5%8C%96.html",{loader:()=>B(()=>import("./矩阵多项式对角化.html-BhFMWJMD.js"),[]),meta:{author:["Cyletix"],tag:["例题"],excerpt:'<p>化简矩阵多项式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mn>5</mn></msup><mo>−</mo><mn>6</mn><msup><mi>A</mi><mn>4</mn></msup><mo>−</mo><mn>3</mn><msup><mi>A</mi><mn>3</mn></msup><mo>+</mo><mn>17</mn><msup><mi>A</mi><mn>2</mn></msup><mo>−</mo><mi>A</mi><mo>+</mo><mn>19</mn><mi>E</mi></mrow><annotation encoding="application/x-tex">A^5 - 6A^4 - 3A^3 + 17A^2 - A + 19E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">17</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">19</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是一个可以对角化的矩阵，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> 是单位矩阵。</p>',readingTime:{minutes:1.14,words:343},title:"",type:"article",s:"矩阵多项式对角化"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%BD%90%E6%AC%A1%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./齐次微分方程例题.html-5Rx6-pqS.js"),[]),meta:{tag:["数学"],excerpt:`<h3>齐次微分方程例题</h3>
<p><strong>例题1:</strong></p>
<p>求解齐次微分方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} = \\frac{x + 2y}{2x + 3y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3783em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8972em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">3</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>。</p>`,readingTime:{minutes:2.23,words:670},title:"",type:"article",s:"齐次微分方程例题"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/--%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B--.html",{loader:()=>B(()=>import("./--偏微分方程--.html-fjTHjo61.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学","目录"],excerpt:`
<ol>
<li><strong>基本概念</strong>
<ul>
<li>偏微分方程的定义</li>
<li>一阶和二阶偏微分方程的区别</li>
<li>初值问题和边值问题</li>
</ul>
</li>
<li><strong>典型偏微分方程</strong>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E6%96%B9%E7%A8%8B.html" target="_blank">拉普拉斯方程</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">热方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E6%B3%A2%E5%8A%A8%E6%96%B9%E7%A8%8B.html" target="_blank">波动方程</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">泊松方程</a></li>
</ul>
</li>
<li><strong>求解方法</strong>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%88%86%E7%A6%BB%E5%8F%98%E9%87%8F%E6%B3%95.html" target="_blank">分离变量法</a></li>
<li>特征线法</li>
<li>傅里叶级数法</li>
<li>拉普拉斯变换</li>
</ul>
</li>
<li><strong>边界值问题</strong>
<ul>
<li>狄利克雷边界条件</li>
<li>诺依曼边界条件</li>
<li>混合边界条件</li>
</ul>
</li>
<li><strong>傅里叶分析</strong>
<ul>
<li>傅里叶级数</li>
<li>傅里叶变换</li>
<li>傅里叶变换在PDE中的应用</li>
</ul>
</li>
<li><strong>格林函数</strong>
<ul>
<li>格林函数的定义和性质</li>
<li>利用格林函数求解边值问题</li>
</ul>
</li>
<li><strong>特征值问题</strong>
<ul>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">Sturm-Liouville理论</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0.html" target="_blank">正交函数</a></li>
</ul>
</li>
<li><strong>非线性偏微分方程</strong>
<ul>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">非线性波动方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2FKdV+%E6%96%B9%E7%A8%8B.html" target="_blank">KdV 方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E7%BA%B3%E7%BB%B4-%E6%96%AF%E6%89%98%E5%85%8B%E6%96%AF%E6%96%B9%E7%A8%8B.html" target="_blank">纳维-斯托克斯方程</a></li>
</ul>
</li>
<li><strong>数值方法</strong>
<ul>
<li>有限差分法</li>
<li>有限元法</li>
<li>谱方法</li>
</ul>
</li>
<li><strong>Sobolev空间</strong>
<ul>
<li>函数空间的基本概念</li>
<li>Sobolev 不等式</li>
<li>Sobolev 空间在PDE中的应用</li>
</ul>
</li>
<li><strong>弱解与变分法</strong>
<ul>
<li>弱解的定义</li>
<li>变分法</li>
<li>能量方法</li>
</ul>
</li>
<li><strong>分布理论</strong>
<ul>
<li>分布的基本概念</li>
<li>分布的微分</li>
<li>分布在PDE中的应用</li>
</ul>
</li>
</ol>`,readingTime:{minutes:4.18,words:1254},title:"目录",type:"article"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/KdV%20%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./KdV 方程.html-C-rGA1tw.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<p>Korteweg-de Vries (KdV) 方程是一个非线性偏微分方程，用于描述浅水波的传播，特别是孤立子（soliton）现象。KdV 方程在流体力学、光学和等离子体物理等领域具有重要的应用。下面详细介绍 KdV 方程。</p>
<h2>KdV 方程的定义</h2>
<p>KdV 方程的标准形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>u</mi><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>3</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\frac{\\partial u}{\\partial t} + 6u \\frac{\\partial u}{\\partial x} + \\frac{\\partial^3 u}{\\partial x^3} = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord">6</span><span class="mord mathnormal">u</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:5.78,words:1733},title:"为什么标准形式会出现常数6",type:"article"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/Sturm-Liouville%20%E7%90%86%E8%AE%BA.html",{loader:()=>B(()=>import("./Sturm-Liouville 理论.html-BupGhqIq.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>Sturm-Liouville 理论是微分方程领域中一个重要的分支，特别是在解决物理问题中具有广泛的应用，如量子力学、热传导、电磁波等。以下是关于这一理论的核心内容和概念的简述。</p>
<h3>核心概念</h3>
<p>Sturm-Liouville 理论涉及的主要是二阶线性微分方程的一类特殊形式，通常表示为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true">[</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo fence="true">]</mo></mrow><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>+</mo><mi>λ</mi><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\frac{d}{dx}\\left[p(x) \\frac{dy}{dx}\\right] - q(x)y + \\lambda w(x)y = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:1.93,words:578},title:"",type:"article",s:"Sturm-Liouville 理论"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%88%86%E7%A6%BB%E5%8F%98%E9%87%8F%E6%B3%95.html",{loader:()=>B(()=>import("./分离变量法.html-XRPl03mA.js"),[]),meta:{tag:["数学"],excerpt:`<p>分离变量法是一种用于求解偏微分方程（PDE）的经典方法。该方法假设解可以表示为多个独立变量的乘积形式，从而将一个偏微分方程转化为多个常微分方程（ODE）。下面详细介绍分离变量法的具体细节和应用条件。</p>
<h2>分离变量法的基本步骤</h2>
<h3>1. 假设解的形式</h3>
<p>假设偏微分方程的解可以表示为多个独立变量的乘积形式。例如，对于二维空间的波动方程，假设解为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(x,t) = X(x)T(t)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:3.78,words:1134},title:"",type:"article",s:"分离变量法"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2%E4%B8%8E%E5%81%8F%E5%AF%BC%E6%95%B0%E7%9A%84%E4%BA%A4%E6%8D%A2.html",{loader:()=>B(()=>import("./拉普拉斯变换与偏导数的交换.html-BTno89-2.js"),[]),meta:{tag:["数学"],excerpt:`
<p>对于拉普拉斯变换与偏导数交换的一般性理论，确实存在一些更广泛的数学原则和定理，这些理论提供了对于何时可以安全地交换积分与微分操作的更深刻理解。这些理论通常涉及函数的局部与整体性质，如可微性、收敛性以及更复杂的函数行为。以下是一些关键的理论和概念：</p>
<ol>
<li>
<p><strong>多变量微积分中的微积分基本定理</strong>：</p>
<ul>
<li>当函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 以及其对 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的偏导数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial f}{\\partial x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 在区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[a, b] \\times [0, \\infty)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mclose">)</span></span></span></span> 上连续时，且相关的积分收敛，可以交换积分和偏导数。</li>
</ul>
</li>
<li>
<p><strong>Leibniz 积分规则</strong>：</p>
<ul>
<li>这是一个关于如何处理参数化积分的导数的规则。如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(x,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial u}{\\partial x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 在闭区间上连续，并且积分 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\\int_a^b u(x,t) \\, dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3998em;vertical-align:-0.3558em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.044em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span> 对于所有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 是存在的，那么<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\\frac{d}{dx} \\int_a^b u(x,t) \\, dt = \\int_a^b \\frac{\\partial u}{\\partial x}(x,t) \\, dt
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></p>
</li>
<li>这在拉普拉斯变换的上下文中非常有用，因为它提供了在一定条件下交换积分和导数的合法性。</li>
</ul>
</li>
<li>
<p><strong>Dominated Convergence Theorem (DCT)</strong>：</p>
<ul>
<li>这个定理提供了一种交换极限（在这里是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">t \\to \\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> 的极限）和积分的条件。如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial u}{\\partial x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 几乎处处存在，并且存在一个积分函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 使得对于所有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>，都有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">|\\frac{\\partial u}{\\partial x}(x,t)| \\leq g(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord">∣</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>，而 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\\int_0^\\infty g(t) dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2151em;vertical-align:-0.3558em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span> 是有限的，那么可以交换极限和积分。</li>
</ul>
</li>
<li>
<p><strong>Laplace Transform Specific Theories</strong>：</p>
<ul>
<li>在拉普拉斯变换的框架中，如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(x,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 和其导数的增长不超过 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>c</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^{ct}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span> （对某个常数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>），则该函数及其导数的拉普拉斯变换存在并可以交换积分和偏导数。</li>
</ul>
</li>
</ol>`,readingTime:{minutes:5.29,words:1588},title:"一般性理论",type:"article"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%B3%A2%E5%8A%A8%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./波动方程.html-BQcR4CrI.js"),[]),meta:{tag:["数学"],excerpt:`<p>要求解波动方程的二阶偏微分方程，可以采用分离变量法、变分法或傅里叶变换等方法。可以首先从经典波动方程的求解步骤开始，然后再具体讨论带有特定项的偏微分方程的求解方法。</p>
<h2>波动方程的基本形式</h2>
<p>波动方程的一般形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:5.06,words:1519},title:"",type:"article",s:"波动方程"}}],["/Other/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E7%BA%B3%E7%BB%B4-%E6%96%AF%E6%89%98%E5%85%8B%E6%96%AF%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./纳维-斯托克斯方程.html-BlGel1AC.js"),[]),meta:{tag:["数学"],excerpt:`<h2>斯托克斯方程</h2>
<h3>定义</h3>
<p>斯托克斯方程是描述黏性流体运动的偏微分方程组，是纳维-斯托克斯方程在低雷诺数情况下的简化形式。斯托克斯方程主要用于研究粘性流体在缓慢运动下的行为，其数学形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>μ</mi><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi mathvariant="bold">u</mi><mo>−</mo><mi mathvariant="normal">∇</mi><mi>p</mi><mo>+</mo><mi mathvariant="bold">f</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\mu \\nabla^2 \\mathbf{u} - \\nabla p + \\mathbf{f} = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathbf">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">∇</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf" style="margin-right:0.109em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:.79,words:238},title:"",type:"article",s:"纳维-斯托克斯方程"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/-%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6-.html",{loader:()=>B(()=>import("./-初等数学-.html-DcoMTplb.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录","总览"],excerpt:`<p>整理部分必要的初等数学知识, 不会整理全部知识.</p>
<h1>目录</h1>
<ul>
<li>函数
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%87%BD%E6%95%B0.html" target="_blank">函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E6%98%A0%E5%B0%84.html" target="_blank">映射</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E6%95%B0%E9%9B%86.html" target="_blank">数集</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A8%8B%E6%B1%82%E6%A0%B9%E5%85%AC%E5%BC%8F.html" target="_blank">二次方程求根公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E6%89%8B%E7%AE%97%E5%BC%80%E6%A0%B9%E5%8F%B7.html" target="_blank">手算开根号</a></li>
</ul>
</li>
<li>三角函数
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0.html" target="_blank">三角函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%92%8C%E8%A7%92%E5%85%AC%E5%BC%8F.html" target="_blank">和角公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F.html" target="_blank">倍角公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%8D%8A%E8%A7%92%E5%85%AC%E5%BC%8F.html" target="_blank">半角公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%B8%89%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F%E6%8E%A8%E5%AF%BC.html" target="_blank">三倍角公式推导</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%92%8C%E5%B7%AE%E5%8C%96%E7%A7%AF.html" target="_blank">和差化积</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E7%A7%AF%E5%8C%96%E5%92%8C%E5%B7%AE.html" target="_blank">积化和差</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E6%8D%A2%E5%BA%95%E5%85%AC%E5%BC%8F.html" target="_blank">换底公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%B8%B8%E7%94%A8%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E5%80%BC%E8%A1%A8.html" target="_blank">常用三角函数值表</a></li>
</ul>
</li>
<li>代数
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86.html" target="_blank">二项式定理</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%BA%8C%E9%A1%B9%E5%BC%8F%E7%B3%BB%E6%95%B0.html" target="_blank">二项式系数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E7%AB%8B%E6%96%B9%E5%92%8C%E5%85%AC%E5%BC%8F.html" target="_blank">立方和公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E7%AB%8B%E6%96%B9%E5%B7%AE%E5%85%AC%E5%BC%8F.html" target="_blank">立方差公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97.html" target="_blank">等比数列</a></li>
</ul>
</li>
<li>几何
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%BB%BB%E6%84%8F%E5%A4%9A%E8%BE%B9%E5%BD%A2%E9%9D%A2%E7%A7%AF.html" target="_blank">任意多边形面积</a></li>
</ul>
</li>
</ul>`,readingTime:{minutes:.51,words:152},title:"目录",type:"article"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%B8%89%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F%E6%8E%A8%E5%AF%BC.html",{loader:()=>B(()=>import("./三倍角公式推导.html-D6m5-8c0.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>三倍角公式是用于表示三倍角的正弦和余弦函数的公式。具体来说，三倍角公式如下：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>3</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>4</mn><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin(3\\theta) = 3\\sin(\\theta) - 4\\sin^3(\\theta)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:1.55,words:465},title:"",type:"article",s:"三倍角公式推导"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./三角函数.html-BdD_5KAv.js"),[]),meta:{author:"Cyletix",tag:["数学","三角函数"],excerpt:`<p>在数学中，三角函数是将直角三角形的角度与两条边长之比联系起来的实函数。它们广泛应用于与几何相关的所有科学，例如航海、固体力学、天体力学、大地测量学等。它们是最简单的周期函数之一，因此也广泛用于通过傅里叶分析研究周期现象。</p>
<table>
<thead>
<tr>
<th>Function</th>
<th>Description</th>
<th>Equal</th>
</tr>
</thead>
<tbody>
<tr>
<td>sine</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>opposite</mtext><mtext>hypotenuse</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2253em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3449em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">hypotenuse</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">opposite</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td></td>
</tr>
<tr>
<td>cosine</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>adjacent</mtext><mtext>hypotenuse</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">hypotenuse</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">adjacent</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td></td>
</tr>
<tr>
<td>tangent</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>tan</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>opposite</mtext><mtext>adjacent</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\tan \\theta = \\frac{\\text{opposite}}{\\text{adjacent}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2253em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3449em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">adjacent</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">opposite</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\frac{\\sin(\\theta)}{\\cos(\\theta)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td>cosecant</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>csc</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>hypotenuse</mtext><mtext>opposite</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\csc \\theta = \\frac{\\text{hypotenuse}}{\\text{opposite}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">csc</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">opposite</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">hypotenuse</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\frac{1}{\\sin(\\theta)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td>secant</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>sec</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>hypotenuse</mtext><mtext>adjacent</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\sec \\theta = \\frac{\\text{hypotenuse}}{\\text{adjacent}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">adjacent</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">hypotenuse</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\frac{1}{cos(\\theta)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td>cotangent</td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>cot</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mtext>adjacent</mtext><mtext>opposite</mtext></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\cot \\theta = \\frac{\\text{adjacent}}{\\text{opposite}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cot</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">opposite</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">adjacent</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\frac{\\cos(\\theta)}{\\sin(\\theta)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
</tbody>
</table>`,readingTime:{minutes:.9,words:271},title:"三角恒等式",type:"article"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A8%8B%E6%B1%82%E6%A0%B9%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./二次方程求根公式.html-BLJv4gPf.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p>二次方程的一般形式为:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>a</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ax^2 + bx + c = 0 ,(a \\neq 0)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:1.46,words:437},title:"",type:"article",s:"二次方程求根公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./二项式定理.html-Djnv9pHZ.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><msup><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><msup><mi>y</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(x+y)^n=\\sum_{k=0}^n\\binom nk x^{n-k}y^k
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.13,words:39},title:"",type:"article",s:"二项式定理"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E7%B3%BB%E6%95%B0.html",{loader:()=>B(()=>import("./二项式系数.html-CSo8r4k7.js"),[]),meta:{author:["Cyletix"],tag:["数学","代数","公式"],excerpt:`
<p>组合数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mi>m</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">C_m^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9303em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>（有时记作<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">C_n^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0961em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>,或者<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow/><mi>n</mi></msub><msub><mi>C</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">_nC_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）, 表示从 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个不同的元素中选择 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个元素的方式的数量，而不考虑选择的顺序。计算组合数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mi>m</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">C_m^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9303em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 的公式如下：</p>`,readingTime:{minutes:2.35,words:705},title:"组合数",type:"article"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E4%BB%BB%E6%84%8F%E5%A4%9A%E8%BE%B9%E5%BD%A2%E9%9D%A2%E7%A7%AF.html",{loader:()=>B(()=>import("./任意多边形面积.html-CPuAXMOj.js"),[]),meta:{tag:["数学"],excerpt:`<h2>三角形面积</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>S</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">S = \\frac{1}{2} ab \\sin \\theta
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span></span></p>`,readingTime:{minutes:1.41,words:424},title:"",type:"article",s:"任意多边形面积"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./倍角公式.html-CB-06F1-.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin(2x)=2\\sin(x)\\cos(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.11,words:33},title:"",type:"article",s:"倍角公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./函数.html-BBff0-gd.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`
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<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>设数集<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">D \\in R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>,则称映射<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">f:D \\to R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>为定义在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>上的函数, 记为<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">y=f(x),x \\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span></p>
</div></div>`,readingTime:{minutes:2.92,words:875},title:"",type:"article",s:"函数"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%8D%8A%E8%A7%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./半角公式.html-CzFKahJA.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\\sin^2(\\frac{x}{2})=\\frac{1-\\cos(x)}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.11,words:34},title:"",type:"article",s:"半角公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%92%8C%E5%B7%AE%E5%8C%96%E7%A7%AF.html",{loader:()=>B(()=>import("./和差化积.html-DidhsUeO.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>sin</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mn>2</mn></mfrac><mi>cos</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mi>β</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\\sin(\\alpha)+\\sin(\\beta)=2\\sin\\frac{\\alpha+\\beta}{2}\\cos\\frac{\\alpha-\\beta}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.15,words:44},title:"",type:"article",s:"和差化积"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%92%8C%E8%A7%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./和角公式.html-C_dssH2m.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo>±</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>±</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin(x \\pm y)=\\sin(x) \\cdot \\cos(y) \\pm \\cos(x) \\sin(y)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.14,words:43},title:"",type:"article",s:"和角公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E5%B8%B8%E7%94%A8%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E5%80%BC%E8%A1%A8.html",{loader:()=>B(()=>import("./常用三角函数值表.html-IYHhenCi.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>写成矩阵形式</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mi>i</mi><mi>n</mi><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>0</mn><mo>∘</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>30</mn><mo>∘</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>45</mn><mo>∘</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>60</mn><mo>∘</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>90</mn><mo>∘</mo></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mfrac><msqrt><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></msqrt><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">sin \\left[ \\begin{matrix} 0^\\circ &amp; 30^\\circ &amp; 45^\\circ &amp; 60^\\circ &amp;90^\\circ \\end{matrix}\\right]
=
\\frac{\\sqrt{\\left[\\begin{matrix}0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\end{matrix}\\right]} }{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">in</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.916em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.23em;"><span style="top:-2.549em;"><span class="pstrut" style="height:3.235em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.465em;"><span class="pstrut" style="height:3.235em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.23em;"><span class="pstrut" style="height:3.235em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.235em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span><span style="top:-3.195em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
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s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744
c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
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M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.605em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.42,words:125},title:"",type:"article",s:"常用三角函数值表"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E6%89%8B%E7%AE%97%E5%BC%80%E6%A0%B9%E5%8F%B7.html",{loader:()=>B(()=>import("./手算开根号.html-u82C2K5m.js"),[]),meta:{tag:["数学"],excerpt:`<p>《数学手册》（高等教育出版社）中记录了一种清新脱俗的手算开平方的方法。摘录如下：</p>
<p>[开平方] 开平方的一般方法用下面的例子说明。</p>
<p><strong>例</strong> 求316.4841的平方根。</p>
<p><strong>解</strong> 第一步，先将被开方的数，从小数点位置向左右每隔两位用逗号“,”分段，如把316.4841分段成3,16.48,41。</p>
<p>第二步，找出第一段数字的初商，使初商的平方不超过第一段数字，而初商加1的平方则大于第一段数字，本例中第一段数字为3，初商为1，因为 12&lt;3，而 (1+1)2=4&gt;3 。</p>`,readingTime:{minutes:1.38,words:414},title:"",type:"article",s:"手算开根号"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E6%8D%A2%E5%BA%95%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./换底公式.html-_M_BQEeU.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>l</mi><mi>o</mi><msub><mi>g</mi><mi>b</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mi>a</mi></mrow><mrow><mi>ln</mi><mo>⁡</mo><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">log_b(a) = \\frac{\\ln a}{\\ln b}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0197em;">l</span><span class="mord mathnormal">o</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.04,words:12},title:"",type:"article",s:"换底公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E6%95%B0%E9%9B%86.html",{loader:()=>B(()=>import("./数集.html-9iJwrptG.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],excerpt:`<p>定义：设P是由一些复数组成的集合，其中包括0与1，如果P中任意两个数的和、差、积、商（除数不为0）仍是P中的数，则称P为一个数域。<br>
显然，全体有理数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Q</mi></mrow><annotation encoding="application/x-tex">\\mathbb{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8556em;vertical-align:-0.1667em;"></span><span class="mord mathbb">Q</span></span></span></span>，全体实数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> ，全体复数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> 分别组成的集合都是一个数域。</p>`,readingTime:{minutes:.86,words:257},title:"",type:"article",s:"数集"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E6%98%A0%E5%B0%84.html",{loader:()=>B(()=>import("./映射.html-C7aEzpaD.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>在集合论中，映射（或称<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%87%BD%E6%95%B0.html" target="_blank">函数</a>）是描述两个集合间元素对应关系的核心概念。它是现代数学语言的基础构件，广泛应用于<a class="route-link nolebase-route-link-invalid" href="#" target="_target">关系代数</a>、<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8B%93%E6%89%91%E5%AD%A6.html" target="_blank">拓扑学</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E8%8C%83%E7%95%B4%E8%AE%BA.html" target="_blank">范畴论</a>等领域。</p>`,readingTime:{minutes:2.58,words:774},title:"",type:"article",s:"映射"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E7%A7%AF%E5%8C%96%E5%92%8C%E5%B7%AE.html",{loader:()=>B(()=>import("./积化和差.html-Cx5lK5Mv.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo fence="true">[</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mo>=</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo fence="true">[</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>−</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\\sin(\\alpha)\\cos(\\beta)=\\dfrac{1}{2}\\left[\\sin(\\alpha+\\beta)+\\sin(\\alpha-\\beta)\\right]=-\\dfrac{1}{2}\\left[\\cos(\\alpha+\\beta)-\\sin(\\alpha-\\beta)\\right]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span></span></p>`,readingTime:{minutes:.22,words:66},title:"",type:"article",s:"积化和差"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E7%AB%8B%E6%96%B9%E5%92%8C%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./立方和公式.html-Cqn-UDju.js"),[]),meta:{author:["Cyletix"],tag:["数学","代数","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>a</mi><mn>3</mn></msup><mo>+</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^3+b^3=(a+b)(a^2-ab+b^2)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.08,words:25},title:"",type:"article",s:"立方和公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E7%AB%8B%E6%96%B9%E5%B7%AE%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./立方差公式.html-CaD3N0ew.js"),[]),meta:{author:["Cyletix"],tag:["数学","代数","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>a</mi><mn>3</mn></msup><mo>−</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^3-b^3=(a-b)(a^2+ab+b^2)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.08,words:25},title:"",type:"article",s:"立方差公式"}}],["/Other/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6/%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97.html",{loader:()=>B(()=>import("./等比数列.html-lzbjnvW-.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学","公式"],excerpt:`<p>等比数列每一项与其前一项之比为常数，这个常数称为公比。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>⋅</mo><msup><mi>r</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_{n+1}=a_{1}\\cdot r^{n}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5945em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.79,words:238},title:"性质",type:"article"}}],["/Other/%E5%AE%9E%E5%88%86%E6%9E%90/--%E5%AE%9E%E5%88%86%E6%9E%90--.html",{loader:()=>B(()=>import("./--实分析--.html-Cob6_Vbe.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li>集合与测度
<ul>
<li>集合的基本性质</li>
<li>σ-代数与可测集合</li>
<li>测度的定义与性质</li>
<li>外测度与可测性</li>
<li>测度空间与Lebesgue测度</li>
</ul>
</li>
<li>可测函数
<ul>
<li>可测函数的定义</li>
<li>简单函数与逐点极限</li>
<li>函数的可测性判别</li>
<li>函数的积分性</li>
</ul>
</li>
<li>Lebesgue积分
<ul>
<li>Lebesgue积分的定义</li>
<li>单调收敛定理</li>
<li>Fatou引理</li>
<li>Dominated Convergence定理</li>
<li>与Riemann积分的比较</li>
</ul>
</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup></mrow><annotation encoding="application/x-tex">L^p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span></span></span></span>空间
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup></mrow><annotation encoding="application/x-tex">L^p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span></span></span></span>空间的定义</li>
<li>Hölder不等式与Minkowski不等式</li>
<li>完备性与Banach空间结构</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup></mrow><annotation encoding="application/x-tex">L^p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span></span></span></span>空间的内在性质</li>
</ul>
</li>
<li>测度理论的应用
<ul>
<li>Fubini定理与Tonelli定理</li>
<li>乘积测度</li>
<li>变化测度（Radon-Nikodym定理）</li>
<li>有界变差函数与绝对连续函数</li>
</ul>
</li>
<li>广义积分与测度生成
<ul>
<li>广义积分的定义</li>
<li>测度生成与外测度构造</li>
<li>哈恩分解与Jordan分解</li>
<li>狄拉克测度与Dirac函数</li>
</ul>
</li>
<li>拓展主题
<ul>
<li>Sobolev空间简介（与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">泛函分析</a>交叉）</li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">调和分析</a>中的测度工具</li>
<li>奇异测度与Cantor集的构造</li>
<li>不同测度之间的等价与偏序关系</li>
</ul>
</li>
</ul>`,readingTime:{minutes:.97,words:291},title:"目录",type:"article"}}],["/Other/%E5%AE%9E%E5%88%86%E6%9E%90/%E9%9B%86%E5%90%88%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%80%A7%E8%B4%A8.html",{loader:()=>B(()=>import("./集合的基本性质.html-C5cH1gxI.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">交换律</div>
</div>
<div class="callout-content"><p></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mi>B</mi><mo>∪</mo><mi>A</mi><mspace width="2em"/><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mi>B</mi><mo>∩</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \\cup B=B \\cup A \\qquad A \\cap B=B \\cap A
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:2em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.65,words:195},title:"",type:"article",s:"集合的基本性质"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/--%E5%A4%8D%E5%88%86%E6%9E%90--.html",{loader:()=>B(()=>import("./--复分析--.html-3eMQMe5r.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
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<div class="callout-title-inner"><a href="proxy.php?url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E8%25A4%2587%25E5%2588%2586%25E6%259E%2590" target="_blank" rel="noopener noreferrer">复分析 - 维基百科，自由的百科全书</a></div>
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<div class="callout-content"><p><strong>复分析</strong>（英语：Complex analysis）是研究复变的函数，特别是亚纯函数和复变解析函数的数学理论。<br>
 <br>
研究中常用的理论、公式以及方法包括柯西积分定理、柯西积分公式、留数定理、洛朗级数展开等。复变分析的应用领域较为广泛，在其它数学分支和物理学中也起着重要的作用。包括数论、应用数学、流体力学、热力学和电动力学。</p>
</div></div>`,readingTime:{minutes:1.75,words:525},title:"简介",type:"article"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./傅里叶变换.html-Bg3ZfLtK.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>ω</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">F(\\omega)=\\int _{-\\infty}^{\\infty} f(t)\\cdot e^{ -i\\omega t } dt 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">ω</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8747em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">iω</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></p>`,readingTime:{minutes:2.09,words:628},title:"一维正变换",type:"article"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./全纯函数.html-8TT7GaLD.js"),[]),meta:{tag:["数学"],excerpt:`<p>全纯函数是复分析的核心对象，定义在复平面开子集上的复值函数，在每个点都复可微。在整个复平面上全纯的函数称为整函数。</p>
<h2>定义</h2>
<p>设开集 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>⊆</mo><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">U \\subseteq \\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>，函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">f: U \\rightarrow \\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>。若在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 中的任意一点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，极限</p>`,readingTime:{minutes:2.42,words:727},title:"",type:"article",s:"全纯函数"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%8F%8D%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF%E5%87%BD%E6%95%B0%E4%B8%8E%E5%AF%B9%E6%95%B0%E5%87%BD%E6%95%B0%E7%9A%84%E7%BB%9F%E4%B8%80.html",{loader:()=>B(()=>import("./反圆锥曲线函数与对数函数的统一.html-DbaPulZs.js"),[]),meta:{tag:["数学"],excerpt:`<p>反三角函数也可以通过对数函数在复数域的形式统一表达。这种统一依赖于反三角函数的对数化表达，以及它们与双曲函数之间的紧密联系。</p>
<h3>1. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcsin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\arcsin(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arcsin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arccos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\arccos(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arccos</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></h3>`,readingTime:{minutes:2.89,words:868},title:"",type:"article",s:"反圆锥曲线函数与对数函数的统一"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A4%8D%E5%AF%B9%E6%95%B0%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./复对数函数.html-DZqlviC1.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p>对于复数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>，满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>w</mi></msup><mo>=</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">e^w = z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0269em;">w</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 的所有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> 称为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 的对数，记作 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ln</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">\\operatorname{Ln} z = w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop"><span class="mord mathrm">Ln</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span>。<br>
可以验证，复对数函数的表达式为：</p>`,readingTime:{minutes:2.8,words:841},title:"",type:"article",s:"复对数函数"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A4%8D%E5%B9%B3%E9%9D%A2%E7%9A%84%E5%9D%90%E6%A0%87%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./复平面的坐标变换.html-CUwPbEU9.js"),[]),meta:{tag:["数学","例题"],excerpt:`<h2>问题描述</h2>
<p>考虑 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 平面上的三角形区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>，其边界由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y = 1 - x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 给出。对于以下变换，绘制 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> 平面上的影像区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">S&#x27;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>，并给出区域边界的方程：</p>`,readingTime:{minutes:5.36,words:1607},title:"小结",type:"article"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A4%8D%E6%95%B0%E5%9F%9F%E7%9A%84%E5%AE%8C%E5%A4%87%E6%80%A7.html",{loader:()=>B(()=>import("./复数域的完备性.html-DoCAvktc.js"),[]),meta:{tag:["数学"],excerpt:`<p>复数的完备性体现在以下几个方面：</p>
<ol>
<li>
<p><strong>代数的完备性</strong>：</p>
<ul>
<li>复数域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> 是一个 <strong>代数闭域</strong>，这意味着任何一个复系数的多项式都能在复数域内找到根。</li>
<li>例如，方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^2 + 1 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 无法在实数域中求解，但在复数域中解为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mo>±</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">x = \\pm i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord mathnormal">i</span></span></span></span>。</li>
</ul>
</li>
<li>
<p><strong>函数的解析性</strong>：</p>
<ul>
<li>复数域上定义的函数（解析函数）具有许多良好的性质，比如导数和积分的存在性、泰勒展开的唯一性等。</li>
<li>例如，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\ln(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\cos(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 等函数可以在复数域中扩展，从而统一实数和虚数的计算。</li>
</ul>
</li>
<li>
<p><strong>操作的完备性</strong>：</p>
<ul>
<li>复数支持加、减、乘、除以及开方等操作，而这些操作在实数域中有时会遇到限制（如负数开方）。</li>
</ul>
</li>
<li>
<p><strong>几何意义</strong>：</p>
<ul>
<li>复数不仅是代数工具，还能自然地表示平面几何中的旋转、缩放等操作。</li>
<li>例如，通过欧拉公式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">e^{i\\theta} = \\cos\\theta + i\\sin\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">θ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span>，我们可以将三角函数和指数函数关联起来。</li>
</ul>
</li>
</ol>`,readingTime:{minutes:2.47,words:741},title:"",type:"article",s:"复数域的完备性"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A4%8D%E6%95%B0%E8%A1%A8%E7%A4%BA%E4%B8%BA%E6%9E%81%E5%9D%90%E6%A0%87%E5%BD%A2%E5%BC%8F.html",{loader:()=>B(()=>import("./复数表示为极坐标形式.html-BHDQKY85.js"),[]),meta:{tag:["数学"],excerpt:'<p>要将任意复数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">z = x + iy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 转换为极坐标形式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>r</mi><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = r(\\cos \\theta + i\\sin \\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">(</span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>r</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">z = re^{i\\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">θ</span></span></span></span></span></span></span></span></span></span></span></span>, 需要进行以下步骤:</p>',readingTime:{minutes:1.47,words:441},title:"",type:"article",s:"复数表示为极坐标形式"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A4%8D%E6%B3%B0%E5%8B%92%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./复泰勒级数.html-DX15upSm.js"),[]),meta:{tag:["数学"],excerpt:'<p>复数泰勒级数用于将复变函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在某一点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的邻域内表示为幂级数形式。它是实数泰勒级数的推广，但在复数领域，级数展开的条件更为严格，因此适用的范围也更具约束力。</p>',readingTime:{minutes:2.01,words:604},title:"",type:"article",s:"复泰勒级数"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E5%A5%87%E7%82%B9.html",{loader:()=>B(()=>import("./奇点.html-B_m7bZ1P.js"),[]),meta:{tag:["数学"],excerpt:`<h3>奇点 (Singularity)</h3>
<p>在复分析中，奇点是指函数在其上不解析的点。更具体地说，如果一个函数 <em>f(z)</em> 在点 <em>z₀</em> 的任何邻域内都不是解析的，则称 <em>z₀</em> 是 <em>f(z)</em> 的一个奇点。</p>
<p><strong>分类：</strong></p>
<p>奇点可以分为以下几类：</p>
<ol>
<li>
<p><strong>孤立奇点 (Isolated Singularity):</strong> 如果存在一个以 <em>z₀</em> 为中心的去心邻域，使得 <em>f(z)</em> 在该邻域内解析，则称 <em>z₀</em> 是 <em>f(z)</em> 的一个孤立奇点。孤立奇点又可以分为以下三种：</p>
<ul>
<li><strong>可去奇点 (Removable Singularity):</strong> 如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>z</mi><mo>→</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\lim_{z \\to z_0} f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 存在且有限，则 <em>z₀</em> 是可去奇点。可以通过重新定义 <em>f(z₀)</em> 的值来消除这个奇点，使函数在 <em>z₀</em> 处也解析。</li>
<li><strong>极点 (Pole):</strong> 如上文所述。</li>
<li><strong>本性奇点 (Essential Singularity):</strong> 如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>z</mi><mo>→</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\lim_{z \\to z_0} f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 不存在，且也不是无穷大，则 <em>z₀</em> 是本性奇点。函数在本性奇点附近的性质非常复杂。例如，根据卡西乌斯-魏尔施特拉斯定理，函数在本性奇点的任何邻域内都无限接近任何复数。</li>
</ul>
</li>
<li>
<p><strong>非孤立奇点 (Non-isolated Singularity):</strong> 如果 <em>z₀</em> 的任何邻域内都包含 <em>f(z)</em> 的其他奇点，则称 <em>z₀</em> 是 <em>f(z)</em> 的一个非孤立奇点。例如，函数 <em>f(z) = 1/sin(1/z)</em> 在 <em>z = 0</em> 处有一个非孤立奇点，因为 <em>sin(1/z) = 0</em> 有无穷多个解趋近于 0。</p>
</li>
</ol>`,readingTime:{minutes:1.91,words:573},title:"",type:"article",s:"奇点"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9E%81%E7%82%B9.html",{loader:()=>B(()=>import("./极点.html-Ccm-0xCx.js"),[]),meta:{tag:["数学"],excerpt:`<p>在复分析中，极点是一种特殊的奇点，它比一般的奇点有更强的结构。</p>
<h2>定义</h2>

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<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z₀</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的某个去心邻域内解析（即在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z₀</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的某个邻域内除了 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z₀</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 以外都解析）。如果存在正整数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>，使得：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>z</mi><mo>→</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></munder><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mi>m</mi></msup><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>L</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim_{z \\to z_0} (z - z_0)^m f(z) = L \\neq 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5501em;vertical-align:-0.8001em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8001em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>其中 L 是一个有限的复数，则称 z₀ 是 f(z) 的一个 m 阶极点 或 m 级极点。当 m = 1 时，称为 简单极点。</p>
</div></div>`,readingTime:{minutes:.88,words:264},title:"",type:"article",s:"极点"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9E%81%E7%82%B9%E7%9A%84%E8%AE%A1%E7%AE%97%E6%96%B9%E6%B3%95.html",{loader:()=>B(()=>import("./极点的计算方法.html-DyvMXUl0.js"),[]),meta:{tag:["数学"],excerpt:`<h3>问题</h3>
<p>求解方程的极点</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mrow><msup><mi>z</mi><mn>4</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1}{z^4 + 1}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:1.16,words:349},title:"",type:"article",s:"极点的计算方法"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9E%81%E7%82%B9%E8%AE%A1%E7%AE%97%E7%A4%BA%E4%BE%8B.html",{loader:()=>B(()=>import("./极点计算示例.html-lcXYnFgo.js"),[]),meta:{tag:["数学"],excerpt:'<p>在处理极点为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><msup><mi>z</mi><mn>3</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">iz^3 - 1 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的情形时，我们需要首先找出 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 的具体值，这些值即为极点所在的位置。</p>',readingTime:{minutes:1.88,words:565},title:"",type:"article",s:"极点计算示例"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9F%AF%E8%A5%BF-%E9%BB%8E%E6%9B%BC%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./柯西-黎曼方程.html-DTTDi7eP.js"),[]),meta:{tag:["数学"],excerpt:`<h2>简介</h2>
<p>柯西-黎曼方程用于描述复变函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在某区域内解析的必要条件。柯西-黎曼方程给出了函数的实部和虚部之间的关系，确保函数在复平面上的导数处处存在且独立于方向。<br>
复分析中的柯西-黎曼方程是全纯函数的必要条件，导致复函数具有比实函数更强的对称性。</p>`,readingTime:{minutes:2.6,words:781},title:"",type:"article",s:"柯西-黎曼方程"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9F%AF%E8%A5%BF%E7%A7%AF%E5%88%86%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./柯西积分公式.html-D91UG1jM.js"),[]),meta:{tag:["数学","公式"],excerpt:`<h2>Wikipedia</h2>
<p>柯西积分公式说明了任何一个闭合区域上的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0.html" target="_blank">全纯函数</a>在区域内部的值完全取决于它在区域边界上的值，并且给出了区域内每一点的任意阶导数的积分计算方式。柯西积分公式是复分析中全纯函数“微分等同于积分”特性的表现。而在实分析中这样的结果是完全不可能达到的。</p>
<h2>定义</h2>
<p>假设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 是定义在单连通区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上的解析函数，并且曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 内的一条简单闭合正向曲线（逆时针方向，右手螺旋确定）。如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> 内的点，则柯西积分公式如下：</p>`,readingTime:{minutes:1.99,words:597},title:"",type:"article",s:"柯西积分公式"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E6%9F%AF%E8%A5%BF%E7%A7%AF%E5%88%86%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./柯西积分定理.html-QuDb_nx7.js"),[]),meta:{tag:["数学"],excerpt:`<h2>概述</h2>
<p>柯西积分定理说明，复平面上全纯函数的闭合路径积分为零。这意味着，任何尝试绕回起点的积分都将抵消，可以简化留数定理和复平面上全纯函数的积分的计算。从柯西积分定理可以推导出柯西积分公式和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E7%95%99%E6%95%B0%E5%AE%9A%E7%90%86.html" target="_blank">留数定理</a>。</p>
<h2>定义</h2>
<p>柯西积分定理有两个等价的描述：</p>
<ol>
<li><strong>路径独立性</strong>：在复平面上，如果函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 内全纯（即处处解析且导数存在），并且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 是单连通的（没有洞），那么从一点到另一点的积分值不依赖于路径的选择，只要这些路径完全位于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 内。</li>
<li><strong>闭合路径积分为零</strong>：如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在单连通区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 内全纯，那么对于任何完全包含在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 内的可求长闭合曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>，有：$$\\oint_C f(z) , dz = 0$$</li>
</ol>`,readingTime:{minutes:.86,words:258},title:"",type:"article",s:"柯西积分定理"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E7%95%99%E6%95%B0.html",{loader:()=>B(()=>import("./留数.html-BHV1WG5G.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>留数是函数在其奇点处的行为的量化表示</p>

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>极点处的留数可以通过以下公式计算：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>R</mi><mi>e</mi><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo separator="true">,</mo><msub><mi>z</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mfrac><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>z</mi><mo>→</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></munder><mfrac><msup><mi>d</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mi>d</mi><msup><mi>z</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mi>m</mi></msup><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Res(f, z_0) = \\frac{1}{(m-1)!} \\lim_{z \\to z_0} \\frac{d^{m-1}}{dz^{m-1}} [(z - z_0)^m f(z)]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mord mathnormal">es</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4271em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">[(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)]</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.47,words:140},title:"",type:"article",s:"留数"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E7%95%99%E6%95%B0%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./留数定理.html-BiQ_NHY7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>留数定理用来计算解析函数沿着闭曲线的路径积分，也可以用来计算实函数的积分。它是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E6%9F%AF%E8%A5%BF%E7%A7%AF%E5%88%86%E5%AE%9A%E7%90%86.html" target="_blank">柯西积分定理</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E6%9F%AF%E8%A5%BF%E7%A7%AF%E5%88%86%E5%85%AC%E5%BC%8F.html" target="_blank">柯西积分公式</a>的推论。留数定理用于计算闭合路径上的复积分。这一定理尤其适用于函数在其积分路径内有奇点的情况。</p>
<h2>定义</h2>
<p>如果复函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 在某闭合路径 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span></span> 内部的所有点上都解析，除了有限个奇点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>z</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">z_1, z_2, ..., z_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，那么沿这个闭合路径 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span></span> 的积分可以通过以下公式计算：</p>`,readingTime:{minutes:2.45,words:735},title:"",type:"article",s:"留数定理"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E7%BD%97%E5%85%B0%E5%B1%95%E5%BC%80.html",{loader:()=>B(()=>import("./罗兰展开.html-D2gOQHdt.js"),[]),meta:{tag:["数学"],excerpt:'<p><strong>Laurent 展开</strong>用于将复函数在某点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">z_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的邻域内展开成幂级数形式。特别是在函数具有孤立奇点时，这种方法非常有用，是对 <strong>泰勒展开</strong> 的一种扩展。</p>',readingTime:{minutes:2.1,words:629},title:"",type:"article",s:"罗兰展开"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E8%A7%A3%E6%9E%90%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./解析函数.html-CDjYfTCB.js"),[]),meta:{tag:["数学"],excerpt:`<p>解析函数是局部上由收敛幂级数表示的函数，分为实解析函数和复解析函数。两者均为无穷可导，但复解析函数（<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0.html" target="_blank">全纯函数</a>）具有一些实解析函数所不具备的性质。</p>
<h2>定义</h2>
<p>设开集 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">D \\subseteq \\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>，函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">f: D \\rightarrow \\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>。若对任意 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x_0 \\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>，存在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的开邻域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">U \\subseteq D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 内可表示为收敛幂级数：</p>`,readingTime:{minutes:2.57,words:770},title:"",type:"article",s:"解析函数"}}],["/Other/%E5%A4%8D%E5%88%86%E6%9E%90/%E8%A7%A3%E6%9E%90%E5%BB%B6%E6%8B%93.html",{loader:()=>B(()=>import("./解析延拓.html-BePgEimr.js"),[]),meta:{tag:["数学"],excerpt:`<p>解析延拓将<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F%E8%A7%A3%E6%9E%90%E5%87%BD%E6%95%B0.html" target="_blank">解析函数</a>从较小定义域拓展到更大定义域。透过此方法，一些原先发散的级数在新的定义域可具有迥异而有限的值。其中最知名的例子为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E4%BC%BD%E9%A9%AC%E5%87%BD%E6%95%B0.html" target="_blank">伽马函数</a>与黎曼ζ函数。</p>
<hr>
<h3>定义</h3>
<p>解析函数在定义域内由收敛的幂级数唯一确定，其值受解析性严格约束。如果一个解析函数在某区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 上通过幂级数展开已知，则可以尝试将其延拓到超出 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 的更大区域。</p>`,readingTime:{minutes:2.79,words:838},title:"",type:"article",s:"解析延拓"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/--%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B--.html",{loader:()=>B(()=>import("./--差分方程--.html-CN465_gk.js"),[]),meta:{author:["GPT-4o"],tag:["数学","目录"],excerpt:`<h2>概述</h2>
<p>差分方程（Difference Equations）是描述<strong>离散系统动态行为</strong>的数学工具，广泛应用于数列、递推关系、信号处理和动态系统中。差分方程的内容不仅包括基本的递推关系，还涉及动力系统的稳定性分析、矩阵表示、特征值分布以及混沌行为的描述。</p>
<hr>
<h2>🔖 目录</h2>
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%9F%BA%E7%A1%80.html" target="_blank">差分方程基础</a><br>
1.1. 什么是差分方程？<br>
1.2. 差分方程与递推关系的联系与区别<br>
1.3. 差分方程的分类<br>
- 一阶与高阶差分方程<br>
- 线性与非线性差分方程<br>
- 齐次与非齐次差分方程</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E7%BA%BF%E6%80%A7%E9%BD%90%E6%AC%A1%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">线性齐次差分方程</a><br>
2.1. 基本形式与定义<br>
2.2. 通解的结构与特征方程法<br>
2.3. 实特征根与复特征根的解法<br>
2.4. 高阶差分方程的降阶方法<br>
2.5. 与矩阵表示的关系（状态空间表示）</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E9%9D%9E%E9%BD%90%E6%AC%A1%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">非齐次差分方程</a><br>
3.1. 非齐次项的形式<br>
3.2. 通解的求解方法（齐次解 + 特解）<br>
3.3. 常数变易法与特解的构造<br>
3.4. 差分方程的叠加原理</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E7%B3%BB%E7%BB%9F%E8%A1%8C%E4%B8%BA%E5%88%86%E6%9E%90%E4%B8%8E%E7%A8%B3%E5%AE%9A%E6%80%A7.html" target="_blank">系统行为分析与稳定性</a><br>
4.1. 特征值与稳定性分析<br>
4.2. 实特征值与收敛、发散<br>
4.3. 复特征值与振荡行为<br>
4.4. 临界稳定与周期解</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E4%B8%8E%E5%8A%A8%E5%8A%9B%E7%B3%BB%E7%BB%9F.html" target="_blank">差分方程与动力系统</a><br>
5.1. 差分方程描述的离散时间动力系统<br>
5.2. 周期点与不动点分析<br>
5.3. 分岔与混沌行为（Logistic 映射）<br>
5.4. 相空间分析</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E7%A4%BA%E4%BE%8B.html" target="_blank">差分方程示例</a><br>
6.1. 经济模型与人口增长<br>
6.2. 斐波那契数列与黄金分割<br>
6.3. 信号处理中的差分方程<br>
6.4. 物理系统中的振荡与阻尼</li>
</ol>`,readingTime:{minutes:1.51,words:454},title:"",type:"article",s:"--差分方程--"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%88%87%E6%AF%94%E9%9B%AA%E5%A4%AB%E5%A4%9A%E9%A1%B9%E5%BC%8F.html",{loader:()=>B(()=>import("./切比雪夫多项式.html-CC19o5QX.js"),[]),meta:{tag:["数学"],excerpt:`<p>切比雪夫多项式（Chebyshev polynomials）是一类在数学和工程领域广泛应用的正交多项式。它们在近似理论、数值分析、工程设计等领域中特别重要，常用于最佳近似和最小化最大误差问题。</p>
<p>切比雪夫多项式分为两类：第一类切比雪夫多项式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 和第二类切比雪夫多项式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。它们都可以通过递归关系或者显式的三角函数表达式定义。</p>`,readingTime:{minutes:1.73,words:518},title:"",type:"article",s:"切比雪夫多项式"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E4%B8%8E%E5%8A%A8%E5%8A%9B%E7%B3%BB%E7%BB%9F.html",{loader:()=>B(()=>import("./差分方程与动力系统.html-BrcgdBeL.js"),[]),meta:{tag:["数学"],excerpt:`<h3><strong>5. 差分方程与动力系统</strong></h3>
<ul>
<li><strong>不动点</strong>：满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a_{n+1} = a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的点。</li>
<li><strong>周期点</strong>：经过若干次迭代回到自身的点。</li>
<li><strong>分岔</strong>：系统参数变化导致解的稳定性或周期性改变。</li>
</ul>`,readingTime:{minutes:.22,words:66},title:"",type:"article",s:"差分方程与动力系统"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%9F%BA%E7%A1%80.html",{loader:()=>B(()=>import("./差分方程基础.html-BSQ0FurZ.js"),[]),meta:{tag:["数学"],excerpt:`<h3>1. 差分方程基础</h3>
<ul>
<li><strong>定义</strong>：差分方程是包含自变量的离散函数关系式，描述相邻离散点之间的关系。</li>
<li><strong>递推关系</strong>：差分方程是递推关系的推广，可以表达为：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_n = f(a_{n-1}, a_{n-2}, \\dots, a_{n-k})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
</li>
<li><strong>与微分方程的联系</strong>：
<ul>
<li>微分方程描述连续系统，差分方程描述离散系统。</li>
<li>微分方程：描述函数的变化率。</li>
<li>差分方程：描述函数的变化量。</li>
<li>当步长趋近于 0 时，差分方程可以收敛到微分方程。</li>
</ul>
</li>
</ul>`,readingTime:{minutes:.51,words:153},title:"",type:"article",s:"差分方程基础"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E7%A4%BA%E4%BE%8B.html",{loader:()=>B(()=>import("./差分方程示例.html-BJq2A1Ai.js"),[]),meta:{tag:["数学"],excerpt:`<h3><strong>6. 实际应用与例子</strong></h3>
<ul>
<li><strong>斐波那契数列</strong>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_n = F_{n-1} + F_{n-2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p>
斐波那契矩阵：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">T = \\begin{bmatrix} 1 &amp; 1 \\\\ 1 &amp; 0 \\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p>
</li>
<li><strong>Logistic 映射</strong>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>r</mi><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{n+1} = r x_n (1 - x_n)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
出现混沌行为。</li>
</ul>`,readingTime:{minutes:.2,words:60},title:"",type:"article",s:"差分方程示例"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F.html",{loader:()=>B(()=>import("./斐波那契数列的矩阵形式.html-DV75kskE.js"),[]),meta:{tag:["数学"],excerpt:`<h3>1. 斐波那契数列的矩阵形式表示</h3>
<p>斐波那契数列的定义递推式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_n = F_{n-1} + F_{n-2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.75,words:824},title:"",type:"article",s:"斐波那契数列的矩阵形式"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E7%B3%BB%E7%BB%9F%E8%A1%8C%E4%B8%BA%E5%88%86%E6%9E%90%E4%B8%8E%E7%A8%B3%E5%AE%9A%E6%80%A7.html",{loader:()=>B(()=>import("./系统行为分析与稳定性.html-BJL6jgpe.js"),[]),meta:{tag:["数学"],excerpt:`<h3><strong>4. 系统行为分析与稳定性</strong></h3>
<ul>
<li><strong>稳定性定义</strong>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mi>a</mi><mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim_{n \\to \\infty} a_n \\to 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
稳定的充要条件是所有特征值的模小于 1。</li>
<li><strong>振荡与发散</strong>：
<ul>
<li>若特征值为复数或模大于 1，系统可能振荡或发散。</li>
</ul>
</li>
<li><strong>周期解</strong>：特征值模等于 1 的情况。</li>
</ul>`,readingTime:{minutes:.29,words:88},title:"",type:"article",s:"系统行为分析与稳定性"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E7%BA%BF%E6%80%A7%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B%E7%9A%84%E6%B1%82%E8%A7%A3.html",{loader:()=>B(()=>import("./线性差分方程的求解.html-DDPDMi64.js"),[]),meta:{author:["GPT-4o"],tag:["数学"],excerpt:`<h2>递推关系的本质</h2>
<p>递推关系描述一个数列中某一项与其前几项之间的关系。以常见的线性递推关系为例：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>k</mi></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_n = c_1 a_{n-1} + c_2 a_{n-2} + \\cdots + c_k a_{n-k}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.83,words:849},title:"",type:"article",s:"线性差分方程的求解"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E7%BA%BF%E6%80%A7%E9%BD%90%E6%AC%A1%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./线性齐次差分方程.html-CWxz4awm.js"),[]),meta:{tag:["数学"],excerpt:`<h3><strong>2. 线性齐次差分方程</strong></h3>
<ul>
<li><strong>定义</strong>：一般形式为：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>k</mi></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_n + c_1 a_{n-1} + c_2 a_{n-2} + \\dots + c_k a_{n-k} = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
</li>
<li><strong>解法思路</strong>：假设解的形式为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msup><mi>r</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_n = r^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，代入得到特征方程：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>r</mi><mi>k</mi></msup><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msup><mi>r</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r^k + c_1 r^{k-1} + \\dots + c_k = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9824em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0491em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
解出 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>r</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">r_1, r_2, \\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span>，通解为：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><msubsup><mi>r</mi><mn>1</mn><mi>n</mi></msubsup><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><msubsup><mi>r</mi><mn>2</mn><mi>n</mi></msubsup><mo>+</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">a_n = C_1 r_1^n + C_2 r_2^n + \\dots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span></span></p>
</li>
<li><strong>实根与复根</strong>：
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">r_i \\in \\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> 表示指数增长或衰减。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">r_i \\in \\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> 表示振荡行为。</li>
</ul>
</li>
<li><strong>与矩阵的关系</strong>：可以用矩阵形式表示递推关系，矩阵的特征值即为递推关系的根。</li>
</ul>`,readingTime:{minutes:.49,words:147},title:"",type:"article",s:"线性齐次差分方程"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E9%80%92%E6%8E%A8%E5%BC%8F%E7%9F%A9%E9%98%B5%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./递推式矩阵的特征值分析.html-alMQ6KNu.js"),[]),meta:{tag:["数学"],excerpt:`<h3>1. 特征值与递推关系的行为分类</h3>
<p>矩阵的特征值决定了递推关系的长期行为，特征值的模（绝对值）和实部（若为复数）是关键因素。以下是不同特征值类型对应的递推关系行为分析。</p>
<hr>
<h3>2. 特征值分类与行为分析</h3>
<h4><strong>(1) 实数特征值</strong></h4>
<ul>
<li>
<p><strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>λ</mi><mi mathvariant="normal">∣</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\\lambda| &gt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">λ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></strong> —— <strong>指数增长</strong></p>
<ul>
<li>模大于 1 的特征值表示递推关系随时间呈指数增长。</li>
<li>示例：斐波那契数列的主特征值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac><mo>≈</mo><mn>1.618</mn></mrow><annotation encoding="application/x-tex">\\lambda_1 = \\frac{1 + \\sqrt{5}}{2} \\approx 1.618</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.383em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.038em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.399em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">5</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.618</span></span></span></span> 表示数列快速增长。</li>
</ul>
</li>
<li>
<p><strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi mathvariant="normal">∣</mi><mi>λ</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 &lt; |\\lambda| &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">λ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></strong> —— <strong>指数衰减</strong></p>
<ul>
<li>模小于 1 的特征值表示递推关系的项逐渐减小，最终趋于 0。</li>
<li>示例：<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msup><mn>0.5</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_n = 0.5^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord">0.</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 表示逐步衰减。</li>
</ul>
</li>
<li>
<p><strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>λ</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\\lambda| = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">λ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></strong> —— <strong>线性增长或稳定</strong></p>
<ul>
<li>模等于 1 表示数列的增长速度保持稳定。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\\lambda = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 表示递推关系最终趋于常数，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\\lambda = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 表示数列在正负之间振荡但不发散。</li>
<li>示例：<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_n = (-1)^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 表示交替振荡但不发散。</li>
</ul>
</li>
<li>
<p><strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lambda = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></strong> —— <strong>完全衰减</strong></p>
<ul>
<li>表示递推关系直接收敛到 0，所有后续项均为 0。</li>
</ul>
</li>
</ul>`,readingTime:{minutes:3.28,words:985},title:"",type:"article",s:"递推式矩阵的特征值分析"}}],["/Other/%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E9%9D%9E%E9%BD%90%E6%AC%A1%E5%B7%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./非齐次差分方程.html-r7kzYze0.js"),[]),meta:{tag:["数学"],excerpt:`<h3><strong>3. 非齐次差分方程</strong></h3>
<ul>
<li><strong>基本形式</strong>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>k</mi></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_n + c_1 a_{n-1} + \\dots + c_k a_{n-k} = f(n)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></span></p>
</li>
<li><strong>解法</strong>：
<ul>
<li><strong>齐次解</strong>：与齐次差分方程类似。</li>
<li><strong>特解</strong>：尝试代入一个形式解（如多项式或指数函数）。</li>
</ul>
</li>
<li><strong>常数变易法</strong>：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msubsup><mi>r</mi><mn>1</mn><mi>n</mi></msubsup><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msubsup><mi>r</mi><mn>2</mn><mi>n</mi></msubsup><mo>+</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">a_n = C_1(n) r_1^n + C_2(n) r_2^n + \\dots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span></span></p>
其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_i(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 是关于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的函数。</li>
</ul>`,readingTime:{minutes:.31,words:92},title:"",type:"article",s:"非齐次差分方程"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/--%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0--.html",{loader:()=>B(()=>import("./--抽象代数--.html-2DnagPNA.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<div class="language-dataview line-numbers-mode" data-highlighter="shiki" data-ext="dataview" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-dataview"><span class="line"><span>LIST</span></span>
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<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:.26,words:79},title:"文档-所有文档",type:"article"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/Associativity.html",{loader:()=>B(()=>import("./Associativity.html-DJ5iA_H9.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>for all <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g, h, k \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>h</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>k</mi><mo>=</mo><mi>g</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>h</mi><mo>⋅</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g\\cdot h)\\cdot k=g\\cdot(h\\cdot k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">)</span></span></span></span></p>',readingTime:{minutes:.09,words:28},title:"",type:"article",s:"Associativity"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/Closure.html",{loader:()=>B(()=>import("./Closure.html-BJlDa9vZ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>for all <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g,h \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span> , <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>⋅</mo><mi>h</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g\\cdot h \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span></p>',readingTime:{minutes:.09,words:26},title:"",type:"article",s:"Closure"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/Identity.html",{loader:()=>B(()=>import("./Identity.html-SQwU8_xE.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>there is <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">e \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>, for all <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo>⋅</mo><mi>g</mi><mo>=</mo><mi>g</mi><mo>⋅</mo><mi>e</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">e\\cdot g=g\\cdot e=g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span></span></span></span></p>',readingTime:{minutes:.1,words:31},title:"",type:"article",s:"Identity"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/Inverse.html",{loader:()=>B(()=>import("./Inverse.html-WAvvo6i9.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>for all <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>, there is <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">h \\in G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>⋅</mo><mi>h</mi><mo>=</mo><mi>h</mi><mo>⋅</mo><mi>g</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">g\\cdot h=h\\cdot g=e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></p>',readingTime:{minutes:.09,words:28},title:"",type:"article",s:"Inverse"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E4%BB%A3%E6%95%B0%E7%BB%93%E6%9E%84.html",{loader:()=>B(()=>import("./代数结构.html-CKpyBcO8.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E7%BE%A4.html" target="_blank">群</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E7%8E%AF.html" target="_blank">环</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E5%9F%9F.html" target="_blank">域</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">向量空间</a></li>
</ul>
`,readingTime:{minutes:.07,words:21},title:"",type:"article",s:"代数结构"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E5%8A%A0%E6%B3%95%E5%B0%81%E9%97%AD%E6%80%A7.html",{loader:()=>B(()=>import("./加法封闭性.html-dz3QQ_WA.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>加法封闭性是<a class="route-link nolebase-route-link-invalid" href="#" target="_target">抽象代数</a>中的基础概念，描述集合在加法运算下的内在稳定性，是研究<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E7%BE%A4.html" target="_blank">群</a>、<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E7%8E%AF.html" target="_blank">环</a>、<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F%E5%9F%9F.html" target="_blank">域</a>等代数结构的核心条件。</p>`,readingTime:{minutes:.76,words:227},title:"",type:"article",s:"加法封闭性"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E5%8A%A0%E6%B3%95%E8%BF%90%E7%AE%97.html",{loader:()=>B(()=>import("./加法运算.html-DTPL8gtj.js"),[]),meta:{tag:["数学"],excerpt:'<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo separator="true">,</mo><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F, +)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mclose">)</span></span></span></span> 是一个交换群</p>',readingTime:{minutes:.35,words:105},title:"",type:"article",s:"加法运算"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E5%90%8C%E6%80%81.html",{loader:()=>B(()=>import("./同态.html-GxRmGwI7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>两个有单一<a class="route-link nolebase-route-link-invalid" href="#" target="_target">二元运算</a>的集合<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>和<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span>(称为原群的代数结构），同态就是映射<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\phi :X\\rightarrow Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span>使得</p>',readingTime:{minutes:.24,words:73},title:"",type:"article",s:"同态"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E5%9F%9F.html",{loader:()=>B(()=>import("./域.html-DSa8nbMM.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>域是一个集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span>，其上定义了两个运算：加法和乘法，使得这个集合在这些运算下满足以下性质：</p>`,readingTime:{minutes:1.23,words:370},title:"",type:"article",s:"域"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E7%8E%AF.html",{loader:()=>B(()=>import("./环.html-CGtLZTZZ.js"),[]),meta:{tag:["数学"],excerpt:'<p><strong>定义</strong>：环是一个集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>，其上定义了两个运算：加法和乘法，使得这个集合在这些运算下满足以下性质：</p>',readingTime:{minutes:.97,words:292},title:"",type:"article",s:"环"}}],["/Other/%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0/%E7%BE%A4.html",{loader:()=>B(()=>import("./群.html-o2M5sgHc.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>群是由一种配备二元运算的集合，其二元运算有结合律、单位元和逆元素。</p>
<p>很多自然界的变换（如平移、镜射）的汇总都符合群的定义，而某群变换下保持不变的某种性质被称为对称性；如在空间对称群的哪些变换下，面积或角度会保持不变，就是在研究立体几何的对称性。</p>
<h1>property</h1>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2FClosure.html" target="_blank">Closure</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2FAssociativity.html" target="_blank">Associativity</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2FIdentity.html" target="_blank">Identity</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2FInverse.html" target="_blank">Inverse</a></li>
</ul>`,readingTime:{minutes:.43,words:128},title:"property",type:"article"}}],["/Other/%E6%95%B0%E5%AD%A6%E5%88%86%E7%B1%BB/UI%E5%AF%BC%E8%88%AA.html",{loader:()=>B(()=>import("./UI导航.html-Cwf88pn2.js"),[]),meta:{excerpt:`
<div class="ab-note drop-shadow"><div class="ab-items ab-card ab-lay-vfall ab-lay-grid"><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>00 总论</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>01 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">数学史</a>与文献</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>03 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">数理逻辑</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">数学基础</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>05 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">组合数学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>06 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">序理论</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">格论</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">序代数结构</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>08 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">一般代数系统</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>11 <a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%95%B0%E8%AE%BA.html">数论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>12 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">域论</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">多项式</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>13 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">交换代数</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>14 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">代数几何</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>15 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">线性代数</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">多线性代数</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">矩阵论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>16 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">结合环</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">结合代数</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>17 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">结合环论</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">非结合环论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>18 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">范畴学</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">同调代数</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>19 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">K理论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>20 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">群论</a>及其推广</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>22 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">拓扑群</a>/<a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E7%BE%A4.html">李群</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>26 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">实变函数</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">实函数</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>28 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">测度</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">积分</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>30 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">复变函数</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>31 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">位势论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>32 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">多复变函数</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">解析空间</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>33 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">特殊函数</a>论</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>34 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">常微分方程</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>35 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">偏微分方程</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>37 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">动力系统</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">遍历论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>39 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">差分方程</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">函数方程</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>40 <a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%2F%E5%BA%8F%E5%88%97.html">序列</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">级数</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">发散级数</a> (求和法)</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>41 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">逼近论</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">级数展开</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>42 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">欧氏几何</a>上的<a class="route-link nolebase-route-link-invalid" href="#" target="_target">调和分析</a> (<a class="route-link nolebase-route-link-invalid" href="#" target="_target">傅立叶分析</a>)</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>43 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">抽象调和分析</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>44 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">积分变换</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">算子演算</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>45 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">积分方程</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>46 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">泛函分析</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>47 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">算子理论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>49 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">变分法</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">最优控制/最优化</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>51 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">几何学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>52 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">凸几何</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">离散几何</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>52 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">凸几何</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">离散几何</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>53 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">微分几何</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>54 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">一般拓扑学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>55 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">代数拓扑</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>57 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">流形</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">胞腔复形</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>58 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">大范围分析</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">流形</a>上的分析</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>60 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">概率论</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">随机过程</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>62 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">统计学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>65 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">数值分析</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>68 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">计算机科学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>70 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">质点力学</a>和<a class="route-link nolebase-route-link-invalid" href="#" target="_target">系统力学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>74 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">固体力学</a> (可变形)</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>76 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">流体力学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>78 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">光学</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">电磁学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>80 经典<a class="route-link nolebase-route-link-invalid" href="#" target="_target">热力学</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">热传导</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>81 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">量子理论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>82 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">统计力学</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">物质结构</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>83 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">相对论</a>与<a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E5%BC%95%E5%8A%9B.html">引力</a>理论</p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>85 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">天文学</a>和<a class="route-link nolebase-route-link-invalid" href="#" target="_target">天体物理学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>86 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">地球地理</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>90 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">运筹学</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">数学规划</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>91 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">博弈论</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">数理经济学</a>/<a class="route-link nolebase-route-link-invalid" href="#" target="_target">数理社会学</a>/数理<a class="route-link nolebase-route-link-invalid" href="#" target="_target">行为科学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>92 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">生物学</a>与其他<a class="route-link nolebase-route-link-invalid" href="#" target="_target">自然科学</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>93 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">系统论</a>与<a class="route-link nolebase-route-link-invalid" href="#" target="_target">控制论</a></p>
</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
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</div></div></div><div class="ab-items-item"><div class="ab-items-title markdown-rendered"><div><p></p>
</div></div><div class="ab-items-content markdown-rendered"><div><p>97 <a class="route-link nolebase-route-link-invalid" href="#" target="_target">数学教育</a></p>
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<div class="ab-note drop-shadow"><div class="ab-nodes min"><div class="ab-nodes-children"><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F-%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6-.html">-初等数学-</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="true"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6-.html">-高等数学-</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div><div class="ab-nodes-node" has_children="true"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F--%E5%BE%AE%E7%A7%AF%E5%88%86--.html">--微积分--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F---%E6%9E%81%E9%99%90%E4%B8%8E%E8%BF%9E%E7%BB%AD---.html">---极限与连续---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F---%E5%AF%BC%E6%95%B0---.html">---导数---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F---%E5%BE%AE%E5%88%86---.html">---微分---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F---%E7%A7%AF%E5%88%86---.html">---积分---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F---%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B---.html">---微分方程---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F---%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95---.html">---解析几何---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F---%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86---.html">---多元函数微分---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F---%E9%87%8D%E7%A7%AF%E5%88%86---.html">---重积分---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F---%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86---.html">---曲线曲面积分---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F---%E7%BA%A7%E6%95%B0---.html">---级数---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div></div></div><div class="ab-nodes-node" has_children="true"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F--%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0--.html">--线性代数--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F---%E8%A1%8C%E5%88%97%E5%BC%8F---.html">---行列式---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F---%E7%9F%A9%E9%98%B5---.html">---矩阵---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F---%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84---.html">---线性方程组---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F---%E5%90%91%E9%87%8F%E7%BB%84---.html">---向量组---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F---%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2---.html">---相似变换---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F---%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4---.html">---线性空间---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%A5%E5%85%85%2F---%E8%A1%A5%E5%85%85---.html">---补充---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div></div></div><div class="ab-nodes-node" has_children="true"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F--%E6%A6%82%E7%8E%87%E8%AE%BA--.html">--概率论--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">---概率论基本概念---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F---%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F---.html">---随机变量---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">---多维随机变量---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F---%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0---.html">---随机变量函数---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">---无穷样本的统计行为---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F---%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83---.html">---抽样分布---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F---%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1---.html">---参数估计---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F---%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C---.html">---假设检验---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">方差分析及回归</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2Fbootstrap%E6%96%B9%E6%B3%95.html">bootstrap方法</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F---%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B---.html">---随机过程---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">---马尔可夫链---</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link nolebase-route-link-invalid" href="#" target="_target">平稳随机过程</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div></div></div></div></div><div class="ab-nodes-node" has_children="true"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6%E8%BF%9B%E9%98%B6-.html">-高等数学进阶-</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F--%E5%A4%8D%E5%88%86%E6%9E%90--.html">--复分析--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0%2F--%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0--.html">--抽象代数--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%2F--%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6--.html">--离散数学--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div><div class="ab-nodes-node" has_children="false"><div class="ab-nodes-content markdown-rendered"><div><p><a class="route-link" href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F--%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0--.html">--李群李代数--</a></p>
</div></div><div class="ab-nodes-children"><div class="ab-nodes-bracket"></div><div class="ab-nodes-bracket2"></div></div></div></div></div></div></div></div>`,readingTime:{minutes:.82,words:247},title:"按学科分类",type:"article"}}],["/Other/%E6%95%B0%E5%AD%A6%E5%88%86%E7%B1%BB/%E6%A0%87%E5%87%86%E5%88%86%E7%B1%BB.html",{loader:()=>B(()=>import("./标准分类.html-CDAImCjW.js"),[]),meta:{excerpt:`
<p>分类参考:</p>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Fwww.sohu.com%2Fa%2F308230472_410558" target="_blank" rel="noopener noreferrer">一张图，带你进入数学世界</a>，这里有图</li>
<li><a href="proxy.php?url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E6%2595%25B0%25E5%25AD%25A6%25E5%25AD%25A6%25E7%25A7%2591%25E5%2588%2586%25E7%25B1%25BB%25E6%25A0%2587%25E5%2587%2586" target="_blank" rel="noopener noreferrer">wikipedia - 数学学科分类标准</a>，较严谨</li>
<li><a href="proxy.php?url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2FCategory%3A%25E6%2595%25B0%25E5%25AD%25A6%25E5%2588%2586%25E6%2594%25AF" target="_blank" rel="noopener noreferrer">wikipedia - 分类:数学分支</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Fzhuanlan.zhihu.com%2Fp%2F541201667" target="_blank" rel="noopener noreferrer">【科普】数学是如何分类的？</a>，这里会提一下每个分类里大概会有什么样的内容</li>
</ul>`,readingTime:{minutes:2.57,words:771},title:"数学标准分类",type:"article"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/--%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0--.html",{loader:()=>B(()=>import("./--李群李代数--.html-k-qi_qwd.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">Info</div>
</div>
<div class="callout-content"><p>李群与李代数是一对描述<strong>连续对称性</strong>的<strong>全局与局部</strong>结构.</p>
<ul>
<li><strong><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E7%BE%A4.html" target="_blank">李群</a> (Group, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>)</strong>: 一个连续的光滑流形, 描述几何层面上的<strong>全局 有限的对称变换</strong> (例如: 旋转 30 度).</li>
<li><strong><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E4%BB%A3%E6%95%B0.html" target="_blank">李代数</a> (Algebra, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">g</mi></mrow><annotation encoding="application/x-tex">\\mathfrak{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;vertical-align:-0.1891em;"></span><span class="mord mathfrak">g</span></span></span></span>)</strong>: 李群在单位元处的<strong>切空间</strong>, 描述<strong>局部, 线性的无穷小变换</strong> (例如: 旋转的角速度).</li>
</ul>
</div></div>`,readingTime:{minutes:4.35,words:1304},title:"文档-所有文档",type:"article"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E5%8E%84%E7%B1%B3%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./厄米矩阵.html-CqdwdYU5.js"),[]),meta:{author:["Cyletix","GPT-5"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>厄米矩阵 (Hermitian matrix) , 也称<strong>自伴随矩阵</strong>, 是共轭对称的方阵. 满足:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>A</mi><mo>†</mo></msup></mrow><annotation encoding="application/x-tex">A = A^\\dagger
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">†</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.52,words:755},title:"",type:"article",s:"厄米矩阵"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E5%8F%8D%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./反对称矩阵.html-CZmvYywu.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>转置矩阵和自身的加法逆元相等的方阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (a_{ij})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。满足：</p>`,readingTime:{minutes:4.94,words:1482},title:"叉乘表示",type:"article"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%8C%87%E6%95%B0%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./指数导数.html-BPPFfzNj.js"),[]),meta:{author:"Cyletix",tag:["数学","代数"],excerpt:`<p><strong>一维</strong>:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mi>d</mi><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">exp(td\\frac{d}{dx})f(x)=f(x+t)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mord mathnormal">d</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.17,words:51},title:"",type:"article",s:"指数导数"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%8C%87%E6%95%B0%E6%98%A0%E5%B0%84.html",{loader:()=>B(()=>import("./指数映射.html-Bf99CA9L.js"),[]),meta:{author:["Cyletix"],tag:["数学","李群","李代数"],excerpt:`<h2>定义</h2>
<p>李群 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span> 的李代数为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">g</mi></mrow><annotation encoding="application/x-tex">\\mathfrak{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;vertical-align:-0.1891em;"></span><span class="mord mathfrak">g</span></span></span></span>.<br>
指数映射定义为</p>`,readingTime:{minutes:.81,words:243},title:"",type:"article",s:"指数映射"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%9D%8E%E4%BB%A3%E6%95%B0.html",{loader:()=>B(()=>import("./李代数.html-D1lJeCV7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>李代数是一个<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>配备了一个额外的满足雅可比恒等式的二元运算 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, Y]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mclose">]</span></span></span></span>, 通常称为李括号或对易子.<br>
李代数是与<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E7%BE%A4.html" target="_blank">李群</a>相联系的代数结构, 每一种李代数理论上都对应着一个李群, 反映了该群的局部无穷小结构和性质.</p>`,readingTime:{minutes:1.97,words:591},title:"",type:"article",s:"李代数"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%9D%8E%E7%BE%A4.html",{loader:()=>B(()=>import("./李群.html-DzS4U_ej.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>李群是一个群, 同时也是一个光滑流形, 是兼具<strong>群结构</strong>与<strong>流形结构</strong>的数学对象, 它描述了<strong>连续可微的对称变换</strong>, 是连接抽象代数与微分流形的一个重要例子</p>
<h2>定义</h2>
<p>李群 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span> 是一个光滑流形, 并且群运算 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(x,y) \\mapsto xy^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↦</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 是光滑映射.<br>
它结合了群的代数结构与流形的几何结构.</p>`,readingTime:{minutes:.78,words:234},title:"",type:"article",s:"李群"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%AC%A7%E6%B0%8F%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./欧氏变换.html-Pzme_VXM.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%97%8B%E8%BD%AC%E5%8F%98%E6%8D%A2.html" target="_blank">旋转变换</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">平移变换</a></li>
</ul>
`,readingTime:{minutes:.06,words:19},title:"",type:"article",s:"欧氏变换"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./正交矩阵.html-Dw382k-U.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>正交矩阵是一个方阵，其列向量和行向量都是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E6%80%A7.html" target="_blank">正交</a>的，并且每个向量的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E8%8C%83%E6%95%B0.html" target="_blank">范数</a>为1。对于一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>，如果满足以下条件：</p>',readingTime:{minutes:1.33,words:398},title:"",type:"article",s:"正交矩阵"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%89%B9%E6%AE%8A%E6%AC%A7%E5%BC%8F%E7%BE%A4.html",{loader:()=>B(()=>import("./特殊欧式群.html-D7Q8l5qS.js"),[]),meta:{author:["Cyletix"],tag:["数学"],readingTime:{minutes:.02,words:6},title:"",type:"article",s:"特殊欧式群"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%89%B9%E6%AE%8A%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./特殊正交矩阵.html-DZsyc47X.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学","线性代数","群论"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">核心思想</div>
</div>
<div class="callout-content"><p>特殊正交矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 是定义在实数域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> 上的&quot;纯旋转&quot; 矩阵集合.<br>
它保持<strong>长度与方向</strong>不变, 即在欧几里得空间中代表旋转对称性.</p>
</div></div>`,readingTime:{minutes:2.71,words:813},title:"",type:"article",s:"特殊正交矩阵"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%89%B9%E6%AE%8A%E6%AD%A3%E4%BA%A4%E7%BE%A4.html",{loader:()=>B(()=>import("./特殊正交群.html-DGAgFF6R.js"),[]),meta:{author:["Cyletix"],tag:["数学"],readingTime:{minutes:.07,words:20},title:"",type:"article",s:"特殊正交群"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%89%B9%E6%AE%8A%E9%85%89%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./特殊酉矩阵.html-vnflpvMN.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学","线性代数","群论"],excerpt:`
<div class="callout" data-callout="quote">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-quote"><path d="M3 21c3 0 7-1 7-8V5c0-1.25-.756-2.017-2-2H4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2 1 0 1 0 1 1v1c0 1-1 2-2 2s-1 .008-1 1.031V20c0 1 0 1 1 1z"/><path d="M15 21c3 0 7-1 7-8V5c0-1.25-.757-2.017-2-2h-4c-1.25 0-2 .75-2 1.972V11c0 1.25.75 2 2 2h.75c0 2.25.25 4-2.75 4v3c0 1 0 1 1 1z"/></svg>
</div>
<div class="callout-title-inner">简介</div>
</div>
<div class="callout-content"><p>特殊酉矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 是一类特殊的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E9%85%89%E7%9F%A9%E9%98%B5.html" target="_blank">酉矩阵</a>，在复空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\mathbb{C}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 中表示保持体积和方向不变的<strong>纯旋转</strong>。满足两个条件：</p>
<ol>
<li><strong>酉矩阵 (Unitary)</strong>: 保持向量的长度（范数）不变。</li>
<li><strong>特殊 (Special)</strong>: <a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E8%A1%8C%E5%88%97%E5%BC%8F.html" target="_blank">行列式</a>为 1，保持空间的“朝向”不变。</li>
</ol>
</div></div>`,readingTime:{minutes:3.07,words:922},title:"",type:"article",s:"特殊酉矩阵"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0.html",{loader:()=>B(()=>import("./矩阵指数.html-DIfnw5bp.js"),[]),meta:{author:["Cyletix","GPT-4","GPT-5"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>矩阵指数将标量指数运算扩展到矩阵. 在矩阵李群的语境下, 指数映射由幂级数定义, 与矩阵指数一致, <strong>矩阵指数就是李群的指数映射 (Exponential Map)</strong>, 是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E7%BE%A4.html" target="_blank">李群</a>与<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%9D%8E%E4%BB%A3%E6%95%B0.html" target="_blank">李代数</a>之间的桥梁.</p>
<hr>
<h2>定义</h2>
<p>给定一个矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, 其矩阵指数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>A</mi></msup></mrow><annotation encoding="application/x-tex">e^A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span></span></span></span></span></span></span></span> 基于幂级数展开的定义:</p>`,readingTime:{minutes:7.68,words:2304},title:"",type:"article",s:"矩阵指数"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5%E7%9A%84%E8%BF%B9.html",{loader:()=>B(()=>import("./矩阵的迹.html-DWt5V7zd.js"),[]),meta:{author:["Cyletix","GPT-5"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>迹 (trace) 是定义在方阵上的一个标量函数, 表示矩阵对角线上元素的和.<br>
它在线性代数, 李群李代数, 量子力学与张量分析中都有重要作用.<br>
在李代数中, 迹常用作<strong>不变量</strong>, 例如在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">u</mi></mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\mathfrak{su}(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathfrak">su</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 中, 所有矩阵都满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">tr</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\operatorname{tr}(A)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">tr</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.</p>`,readingTime:{minutes:2.14,words:641},title:"",type:"article",s:"矩阵的迹"}}],["/Other/%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0/%E9%85%89%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./酉矩阵.html-DWWnHllJ.js"),[]),meta:{tag:["数学"],excerpt:'<p>酉矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>是指满足下列条件的<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>阶复方阵：</p>',readingTime:{minutes:1.11,words:332},title:"",type:"article",s:"酉矩阵"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/--%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6--.html",{loader:()=>B(()=>import("./--离散数学--.html-DPhYvHWD.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<div class="language-dataview line-numbers-mode" data-highlighter="shiki" data-ext="dataview" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-dataview"><span class="line"><span>LIST</span></span>
<span class="line"><span>WHERE file.folder=this.file.folder</span></span>
<span class="line"><span>AND contains(dlink,link(this.file.name))</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:.1,words:30},title:"文档-所有文档",type:"article"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB.html",{loader:()=>B(()=>import("./二元关系.html-h4kkfWIN.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"二元关系"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%8F%8C%E9%93%BE%E6%A0%91.html",{loader:()=>B(()=>import("./双链树.html-BHjBxYfF.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"双链树"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%9B%BE%E7%9A%84%E5%90%8C%E6%9E%84.html",{loader:()=>B(()=>import("./图的同构.html-CbI_3kK7.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"图的同构"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%9B%BE%E7%9A%84%E7%9F%A9%E9%98%B5%E8%A1%A8%E7%A4%BA.html",{loader:()=>B(()=>import("./图的矩阵表示.html-aaIf2hpC.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"图的矩阵表示"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%AD%90%E5%9B%BE%E4%B8%8E%E8%A1%A5%E5%9B%BE.html",{loader:()=>B(()=>import("./子图与补图.html-afBIZzI-.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"子图与补图"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%AE%B9%E6%96%A5%E5%8E%9F%E7%90%86.html",{loader:()=>B(()=>import("./容斥原理.html-BKW-V51y.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"容斥原理"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%AF%B9%E5%81%B6.html",{loader:()=>B(()=>import("./对偶.html-JDNS89U3.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"对偶"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%B8%83%E5%B0%94%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./布尔矩阵.html-D82JBIts.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"布尔矩阵"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E5%BA%8F%E5%88%97.html",{loader:()=>B(()=>import("./序列.html-B0KeCMBv.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"序列"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E6%8E%92%E5%88%97.html",{loader:()=>B(()=>import("./排列.html-3tiU1fnP.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"排列"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E6%A0%BC.html",{loader:()=>B(()=>import("./格.html-BykLfCql.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"格"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E6%B1%89%E8%AF%BA%E5%A1%94%E5%9B%BE.html",{loader:()=>B(()=>import("./汉诺塔图.html-Bamcxc9r.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"汉诺塔图"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E7%BB%84%E5%90%88.html",{loader:()=>B(()=>import("./组合.html-u_2Yq3pG.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"组合"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E7%BB%B4%E6%81%A9%E5%9B%BE.html",{loader:()=>B(()=>import("./维恩图.html-CO6XWNMx.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"维恩图"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E9%9B%86%E5%90%88.html",{loader:()=>B(()=>import("./集合.html-CSPdtgYT.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],excerpt:`<p>集合是数学中最基本的概念, 较难给出严格精角定义.</p>
<p>通常将若干个可确定, 可分辨的对象构成的无序整体称为集合 (set) , 常用大写英文字母 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">,</mo><mi>C</mi><mo separator="true">,</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo separator="true">,</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">A, B, C, X, Y, Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span></span></span></span> 等表示.</p>`,readingTime:{minutes:.82,words:245},title:"",type:"article",s:"集合"}}],["/Other/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6/%E9%9B%86%E5%90%88%E7%9A%84%E8%BF%90%E7%AE%97.html",{loader:()=>B(()=>import("./集合的运算.html-DsbAxP62.js"),[]),meta:{author:["Zacharia2"],tag:["数学"],readingTime:{minutes:.03,words:10},title:"",type:"article",s:"集合的运算"}}],["/Other/%E8%AF%B4%E6%98%8E/MOC%E5%8E%9F%E5%88%99.html",{loader:()=>B(()=>import("./MOC原则.html-Bh0Bp6v-.js"),[]),meta:{tag:["规则"],excerpt:`<ul>
<li>What is this topic?</li>
<li>What's the path?</li>
<li>Why should I care?</li>
</ul>
`,readingTime:{minutes:.05,words:15},title:"",type:"article",s:"MOC原则"}}],["/Other/%E8%AF%B4%E6%98%8E/Markdown%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./Markdown数学公式.html-8E9SGy-Y.js"),[]),meta:{author:["DanielGavin","Cyletix","SituChengxiang","ArcherReilly"],tag:["markdown","数学","公式","规则"],excerpt:`
<h2>行内与独行</h2>
<ol>
<li>行内公式: 将公式插入到本行内, 符号: <code>$公式内容$</code>, 如: <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">xyz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn>0</mn><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">0_{m \\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8528em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\\emptyset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">0</mi></mrow><annotation encoding="application/x-tex">\\boldsymbol{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mn>0</mn><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\\vec{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9274em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9274em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">0</span></span><span style="top:-3.2134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span></li>
<li>独行公式: 将公式插入到新的一行内, 并且居中, 符号: <code>$$公式内容$$</code>, 如: $$xyz$$</li>
</ol>`,readingTime:{minutes:9.27,words:2780},title:"一、基本符号",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/README(old).html",{loader:()=>B(()=>import("./README(old).html-DDXnZpBD.js"),[]),meta:{tag:["数学"],excerpt:`
<p>基于Obsidian建立公开共享的数学知识库！<br>
PKMer知识管理交流群(QQ): 825255377<br>
PKMer-Math开发者交流群(QQ): 782017903</p>
<h2>前言</h2>
<p>本库亦在用通俗的语言描述 梳理高等数学的主要内容, 整理为Obsidian风格的markdown文件, 内容上会使用大量Obsidian特色风格语法, 包括但不限于:</p>
<ul>
<li>使用Obsidian双链语法, 组织目录和相关内容, 构成一个Obsidian的数学知识网络.</li>
<li>使用Callout语法, 为重点内容制作简易美观的词条卡片.</li>
<li>使用块引用语法, 引用原子化的内容组织更高层文章</li>
</ul>`,readingTime:{minutes:9.69,words:2908},title:"Pkmer-Math",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/gitignore%E6%A8%A1%E6%9D%BF.html",{loader:()=>B(()=>import("./gitignore模板.html-B7yYdXRZ.js"),[]),meta:{excerpt:`<p>这是一个通用的 .gitignore 文件模板，适用于大多数情况。将此内容保存为 <code>.gitignore</code> 放在根目录下, 根据需求自定义规则, 可以根据需要自定义此文件，<strong>禁止同步到仓库</strong></p>
<div class="language-gitignore line-numbers-mode" data-highlighter="shiki" data-ext="gitignore" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-gitignore"><span class="line"><span># 排除自己, 必须添加此项!  </span></span>
<span class="line"><span>.gitignore</span></span>
<span class="line"><span></span></span>
<span class="line"><span># 同步.obsidian文件夹下指定的子文件夹,名称前加"!", 这些插件都暂时没有用到, 将来找机会剔除</span></span>
<span class="line"><span>.obsidian/*   # 不同步obsidian所有配置项</span></span>
<span class="line"><span>.obsidian/app.json</span></span>
<span class="line"><span>!.obsidian/appearance.json   # 在不同步所有配置的前提下,需要同步的反例加上!</span></span>
<span class="line"><span>.obsidian/bookmarks.json</span></span>
<span class="line"><span>!.obsidian/community-plugins.json</span></span>
<span class="line"><span>!.obsidian/core-plugins.json</span></span>
<span class="line"><span>!.obsidian/graph.json</span></span>
<span class="line"><span>!.obsidian/hotkeys.json</span></span>
<span class="line"><span>.obsidian/graph.json</span></span>
<span class="line"><span>.obsidian/types.json</span></span>
<span class="line"><span>.obsidian/workspace.json</span></span>
<span class="line"><span></span></span>
<span class="line"><span></span></span>
<span class="line"><span>!.obsidian/snippets</span></span>
<span class="line"><span>!.obsidian/plugins/*</span></span>
<span class="line"><span>!.obsidian/plugins/dataview</span></span>
<span class="line"><span>!.obsidian/plugins/templater-obsidian</span></span>
<span class="line"><span># !.obsidian/plugins/obsidian-git/</span></span>
<span class="line"><span># !.obsidian/plugins/obsidian-linter/</span></span>
<span class="line"><span>!.obsidian/plugins/quickadd/</span></span>
<span class="line"><span># !.obsidian/plugins/obsidian-outliner/</span></span>
<span class="line"><span># !.obsidian/plugins/table-editor-obsidian/</span></span>
<span class="line"><span># !.obsidian/plugins/obsidian-latex-suite/</span></span>
<span class="line"><span></span></span>
<span class="line"><span></span></span>
<span class="line"><span># assert文件夹</span></span>
<span class="line"><span># _assets_/*</span></span>
<span class="line"><span>_assets_/草稿</span></span>
<span class="line"><span></span></span>
<span class="line"><span></span></span>
<span class="line"><span># 其他ob相关文件夹</span></span>
<span class="line"><span>.history</span></span>
<span class="line"><span>.trash</span></span>
<span class="line"><span></span></span>
<span class="line"><span># 针对Mac系统的忽略</span></span>
<span class="line"><span>.DS_Store</span></span>
<span class="line"><span></span></span>
<span class="line"><span># 子模块相关, 暂时没有用到</span></span>
<span class="line"><span>.gitmodules</span></span>
<span class="line"><span></span></span>
<span class="line"><span># vscode配置文件夹</span></span>
<span class="line"><span>.vscode</span></span>
<span class="line"><span></span></span>
<span class="line"><span># node_modules文件夹</span></span>
<span class="line"><span>node_modules</span></span>
<span class="line"><span></span></span>
<span class="line"><span># package.json和包锁文件</span></span>
<span class="line"><span>package.json</span></span>
<span class="line"><span>package-lock.json</span></span>
<span class="line"><span></span></span>
<span class="line"><span># mcp相关</span></span>
<span class="line"><span>*mcp*</span></span>
<span class="line"><span></span></span>
<span class="line"><span># 日志文件</span></span>
<span class="line"><span>*.log</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:.8,words:240},title:"",type:"article",s:"gitignore模板"}}],["/Other/%E8%AF%B4%E6%98%8E/tikz%E7%BB%98%E5%9B%BE%E6%A0%87%E5%87%86.html",{loader:()=>B(()=>import("./tikz绘图标准.html-CrjyIIDl.js"),[]),meta:{tag:["数学","规则"],excerpt:`
<h2>环境配置</h2>
<p>代码类型: tikz</p>
<div class="language- line-numbers-mode" data-highlighter="shiki" data-ext style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-"><span class="line"><span>\\usepackage{tikz}</span></span>
<span class="line"><span>\\begin{document}</span></span>
<span class="line"><span>\\begin{tikzpicture}</span></span>
<span class="line"><span>...</span></span>
<span class="line"><span>\\end{tikzpicture}</span></span>
<span class="line"><span>\\end{document}</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:21.72,words:6517},title:"简介",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E4%BB%93%E5%BA%93%E6%96%87%E4%BB%B6%E7%BB%93%E6%9E%84.html",{loader:()=>B(()=>import("./仓库文件结构.html-I5EXokoE.js"),[]),meta:{tag:["规则"],excerpt:`<p>本文描述仓库文件树的物理结构</p>
<hr>
<h2>配置文件目录树</h2>
<div class="language- line-numbers-mode" data-highlighter="shiki" data-ext style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-"><span class="line"><span>├─ .github                # GitHub Actions 自动化配置目录</span></span>
<span class="line"><span>│  └─ workflows           # CI/CD 工作流（自动构建与发布网站）</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ .obsidian              # Obsidian 的主配置文件夹</span></span>
<span class="line"><span>│  ├─ plugins             # 所用插件目录（dataview / templater / quickadd / tikzjax 等）</span></span>
<span class="line"><span>│  ├─ snippets            # CSS 代码片段，控制排版风格</span></span>
<span class="line"><span>│  └─ themes              # 主题目录（当前使用 GitHub Theme / Vanilla AMOLED）</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ .vscode                # VS Code 环境配置，用于本地开发与 Markdown 预览</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ _assets_               # 内部资源与模板配置文件夹</span></span>
<span class="line"><span>│  ├─ Callouts            # Callout 风格模板（信息块格式定义）</span></span>
<span class="line"><span>│  ├─ dataview查询        # Dataview 查询模板及常用语句</span></span>
<span class="line"><span>│  ├─ Images              # 图片与图表资源</span></span>
<span class="line"><span>│  ├─ QuickAdd            # 快速添加模板（笔记/例题/定义）</span></span>
<span class="line"><span>│  ├─ Template            # Markdown 模板（章节/概念/词条格式）</span></span>
<span class="line"><span>│  ├─ scripts             # 自动化脚本（批量生成、文件重命名等）</span></span>
<span class="line"><span>│  └─ todo                # 临时待办与任务记录</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ 微积分                 # 高等数学主体之一：微积分系统</span></span>
<span class="line"><span>│  ├─ 极限 / 导数 / 微分 / 积分 / 多元函数微分 等核心章节</span></span>
<span class="line"><span>│  ├─ 重积分 / 曲线曲面积分 / 解析几何 / 级数 等进阶部分</span></span>
<span class="line"><span>│  └─ 以函数分析与几何直观为主线展开</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ 线性代数               # 高等数学主体之二：线性代数系统</span></span>
<span class="line"><span>│  ├─ 行列式 / 矩阵 / 向量组 / 线性空间 等核心内容</span></span>
<span class="line"><span>│  ├─ 相似变换 / 特征值 / 线性方程组 等进阶章节</span></span>
<span class="line"><span>│  └─ 含“补充”小节，用于记录扩展性例题与证明</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ 概率论                 # 高等数学主体之三：概率与统计系统</span></span>
<span class="line"><span>│  ├─ 概率论基础 / 随机变量 / 随机变量函数 等核心内容</span></span>
<span class="line"><span>│  ├─ 极限定理 / 参数估计 / 假设检验 / 方差分析及回归</span></span>
<span class="line"><span>│  ├─ 抽样分布 / 随机过程 等专题</span></span>
<span class="line"><span>│  └─ 与后续统计学习方法衔接</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>├─ Other                  # 扩展与前瞻内容（非高数核心，但与之关联）</span></span>
<span class="line"><span>│  ├─ 说明                 # 本仓库规范、协作方式、插件配置等说明文档</span></span>
<span class="line"><span>│  ├─ 例题与练习           # 整理常见题型与典型推导</span></span>
<span class="line"><span>│  ├─ pdf参考资料          # 教材与文献 PDF（仅供学习使用）</span></span>
<span class="line"><span>│  ├─ 初等数学              # 作为高数前置知识（概念补充）</span></span>
<span class="line"><span>│  ├─ 实分析 / 复分析 / 离散数学 / 抽象代数 等后续课程内容</span></span>
<span class="line"><span>│  ├─ 李群李代数 / 微分流形 / 泛函分析 / 偏微分方程 等高阶专题</span></span>
<span class="line"><span>│  └─ 供未来内容扩展与学科交叉使用</span></span>
<span class="line"><span>│</span></span>
<span class="line"><span>└─ 其他配置               # 杂项、规则与导出配置（如 .clinerules / 结构检查文件等）</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:2.6,words:781},title:"",type:"article",s:"仓库文件结构"}}],["/Other/%E8%AF%B4%E6%98%8E/%E4%BD%BF%E7%94%A8%E6%AD%A5%E9%AA%A4.html",{loader:()=>B(()=>import("./使用步骤.html-BwmL-PBg.js"),[]),meta:{tag:["规则"],excerpt:`<p>本仓库的使用分两种:</p>
<h2>作为本地Obsidian仓库打开</h2>
<ol>
<li>克隆此仓库到本地<div class="language-shell line-numbers-mode" data-highlighter="shiki" data-ext="shell" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-shell"><span class="line"><span style="--shiki-light:#4078F2;--shiki-dark:#61AFEF">git</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> clone</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> https://github.com/PKM-er/Pkmer-Math.git</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div></div></div>若希望自定义本地仓库的名称(如想要文件夹名称为Math, 而不是PKM-Math):<div class="language-shell line-numbers-mode" data-highlighter="shiki" data-ext="shell" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-shell"><span class="line"><span style="--shiki-light:#4078F2;--shiki-dark:#61AFEF">git</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> clone</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> https://github.com/PKM-er/Pkmer-Math.git</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> Math</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div></div></div></li>
<li>使用Obsidian打开下载的本地仓库文件夹</li>
<li>根据个人习惯, 选择任意一个主页面板使用:
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F-%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6-.html" target="_blank">-高等数学-</a>: 仅学习高等数学, 不进行开发, 不系统学习进阶知识</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FHome.html" target="_blank">Home</a>: 兼顾高等数学学习和轻度开发</li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">新版目录.base</a>: 查看和管理所有文件及YAML元数据</li>
<li>除此之外, 还可以使用和查看自定义dataview</li>
</ul>
</li>
</ol>`,readingTime:{minutes:.73,words:220},title:"",type:"article",s:"使用步骤"}}],["/Other/%E8%AF%B4%E6%98%8E/%E5%85%B3%E4%BA%8E%E5%90%88%E4%BD%9C%E5%BC%80%E5%8F%91.html",{loader:()=>B(()=>import("./关于合作开发.html-ChBLFKq_.js"),[]),meta:{tag:["规则"],excerpt:`
<p>本项目在 GitHub 开源，为了更好地进行合作开发，计划制定相关教程和规范。以下是当前开发原则、内容规范及开发流程。规范会根据需求不断调整，欢迎提出改进建议！</p>
<hr>
<h2>开发原则</h2>
<ol>
<li><strong>简洁易懂</strong>：确保代码和文档简洁明了，方便快速理解重点。</li>
<li><strong>保证正确性</strong>：对自己的修改负责，可以多次修改，直到合适为止。</li>
<li><strong>奥卡姆剃刀原则</strong>：如无必要，勿增实体。尽量基于 Obsidian 原生功能进行开发，避免过度依赖复杂插件。</li>
<li><strong>优先记录</strong>：若时间紧迫，可将内容先归类到对应一级文件夹，后续再整理完善。</li>
</ol>`,readingTime:{minutes:3.83,words:1148},title:"合作开发指南",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E5%85%B3%E4%BA%8E%E7%9B%AE%E5%BD%95%E7%BB%93%E6%9E%84.html",{loader:()=>B(()=>import("./关于目录结构.html-C_W-s6Po.js"),[]),meta:{excerpt:`<p>我说一下我是怎么用Obsidian的。</p>
<p>首先，我会写一堆原子化的笔记（当然不是创作上的时间顺序）。然后，我会用其他的笔记来引用这些原子化的笔记，写成一个完整的文档(等价于泡泡)。接着，写一个目录性质的md文件(性质上也属于泡泡)，里面写具体目录结构的链接，这些目录文档原子化笔记形成了一整个学科，这就是整个结构的样子。</p>
<p>但是，这些文档不是必须要用文档内的链接(所谓的硬链接)去组织结构的。文档内的链接只是那种沿着目录结构走的，最相关的一些近邻文档(可能相当于所谓的有序？)</p>
<p>就是链接和链接有区分，这里其实有两种链接：一种是用于组织、描述一个学科、一个学科完整介绍的链接，这种链接是服务于目录结构的，学科这个组织顺序的；另一种链接是它和一些其他无关的、和这个学科目录无关的其他引申的 推广的相关知识。</p>`,readingTime:{minutes:2.55,words:766},title:"",type:"article",s:"关于目录结构"}}],["/Other/%E8%AF%B4%E6%98%8E/%E5%86%85%E5%AE%B9%E8%A7%84%E8%8C%83.html",{loader:()=>B(()=>import("./内容规范.html-BJ7KEmtA.js"),[]),meta:{tag:["规则"],excerpt:`<p>本文描述内容层级的构建原则和划分方法</p>
<hr>
<h1>构建原则</h1>
<p>充分利用 Obsidian 的特色语法, 包括但不限于:</p>
<ul>
<li>使用双链语法, 组织目录与相关内容, 构建一个数学知识的网络化结构.</li>
<li>利用 Callout 语法, 为重点内容创建简洁美观的卡片式词条.</li>
<li>通过块引用语法, 将原子化的内容引用到更高层次的文章中, 以实现高效的知识整合与关联.<br>
为了便于知识传播, 使用TikzJax绘制图形, 参照<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E8%AF%B4%E6%98%8E%2Ftikz%E7%BB%98%E5%9B%BE%E6%A0%87%E5%87%86.html" target="_blank">tikz绘图标准</a>.<br>
禁止大量使用不必要的图片,此处不是图片垃圾场, 不要倒垃圾!!!!</li>
</ul>`,readingTime:{minutes:6.02,words:1805},title:"构建原则",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E5%8E%86%E5%8F%B2%E9%87%8D%E5%86%99%E5%90%8E%E7%9A%84%E5%90%8C%E6%AD%A5%E8%AF%B4%E6%98%8E.html",{loader:()=>B(()=>import("./历史重写后的同步说明.html-DGvX0k5Z.js"),[]),meta:{tag:["规则","Git"],excerpt:`
<p>2026-03-22 仓库进行过一次历史重写，用于清理 <code>_assets_</code> 与 <code>.obsidian</code> 的旧历史记录，并保留当前最新状态。</p>
<p>这意味着：</p>
<ul>
<li>远端 <code>main</code> 的提交历史已经变化。</li>
<li>之前基于旧历史 <code>clone</code> 的本地仓库，需要手动同步一次。</li>
<li><code>fork</code> 仓库不会自动更新，但如果要继续跟随上游，也需要手动同步。</li>
</ul>
<hr>
<h2>谁需要处理</h2>
<h3>1. 新 clone 的用户</h3>`,readingTime:{minutes:3.28,words:983},title:"历史重写后的同步说明",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E5%90%8C%E6%AD%A5%E9%85%8D%E7%BD%AE%E6%B3%A8%E6%84%8F%E4%BA%8B%E9%A1%B9.html",{loader:()=>B(()=>import("./同步配置注意事项.html-C38JWmKc.js"),[]),meta:{tag:["数学"],excerpt:`<p>由于仓库使用<code>.gitignore</code>管理忽略文件, 并且<code>.gitignore</code>文件本身参与了同步, 会出现一种比较危险的情况:</p>
<ul>
<li>你配置了<code>.gitignore</code>, 使得配置文件夹如<code>.obsidian</code>不参与git追踪</li>
<li>当你想查看历史提交记录时, 你checkout切换了分支</li>
<li><strong>这时, 你会惊奇地发现, 你的配置文件没有了</strong></li>
<li>哪怕切换回main分支, 没有参与git追踪的所有文件都不会回来了.</li>
</ul>`,readingTime:{minutes:3.03,words:908},title:"",type:"article",s:"同步配置注意事项"}}],["/Other/%E8%AF%B4%E6%98%8E/%E6%8B%9B%E5%8B%9F%E5%85%AC%E5%91%8A.html",{loader:()=>B(()=>import("./招募公告.html-D0AOKYoI.js"),[]),meta:{tag:["规则"],excerpt:`
<h2>项目现状</h2>
<ul>
<li>建立了较为完善的文档规范和协作流程</li>
<li>已基本完成微积分、线性代数、概率论等基础数学内容的框架搭建</li>
<li>正在持续完善和扩展各领域内容(复分析, 离散数学等)</li>
</ul>
<h2>我们需要</h2>
<ul>
<li>数学内容贡献者：负责数学知识的整理和编写</li>
<li>文档维护者：协助维护文档规范和格式</li>
<li>AI工具使用者：擅长使用AI工具辅助内容创作和优化</li>
<li>其他支持：欢迎任何对数学知识整理感兴趣的朋友</li>
</ul>
<h2>需要掌握的技术</h2>
<ol>
<li>Markdown语法和Latex公式</li>
<li>Git版本控制</li>
<li>AI工具, 辅助内容创作和优化</li>
<li>QuickAddjs, 但不是必须</li>
</ol>`,readingTime:{minutes:.95,words:284},title:"PKM-Math 项目招募",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E6%8F%92%E4%BB%B6%E9%85%8D%E7%BD%AE%E8%AF%B4%E6%98%8E.html",{loader:()=>B(()=>import("./插件配置说明.html-BLOWCxdY.js"),[]),meta:{tag:["规则"],excerpt:`
<p>设计理念允许不同的Obsidian配置, 以适应不同的人的使用体验. 其中有必要的相同插件, 有适应不同人群使用习惯的插件.</p>
<h2>个性化配置原则</h2>
<ul>
<li>推荐插件
<ul>
<li><a href="#dataview">dataview</a> 只能在章节目录中使用, 同时保留手写的目录</li>
<li><a href="#Templater">Templater</a> 方便新建笔记</li>
<li><a href="#ObsidianGit">ObsidianGit</a> 方便Github合作开发, 当然你也可以用vscode</li>
<li><a href="#TikZJax">TikZJax</a>: 渲染三维图像</li>
<li><a href="#Extended%20MathJax">Extended MathJax</a>: 闭合曲线积分符号的支持</li>
</ul>
</li>
<li>允许使用轻量级插件(如快捷表格操作, 数学公式等)
<ul>
<li><a href="#Force%20note%20view%20mode">Force note view mode</a></li>
<li><s><a href="#No%20more%20flickering%20inline%20math">No more flickering inline math</a></s></li>
</ul>
</li>
<li>不推荐使用任何破坏性插件(quickadd,Linter等需要谨慎操作, 每次用完后需要关闭插件), 但如果熟悉的话,允许使用
<ul>
<li><a href="#Linter">Linter</a></li>
<li><a href="#QuickAdd">QuickAdd</a></li>
</ul>
</li>
<li>知识词条页面严禁使用html, css, javascript, 非知识词条的组织页面可以酌情使用</li>
</ul>`,readingTime:{minutes:2.19,words:657},title:"关于插件和配置",type:"article"}}],["/Other/%E8%AF%B4%E6%98%8E/%E6%95%B0%E5%AD%A6%E5%B8%B8%E7%94%A8%E6%97%A5%E8%AF%AD.html",{loader:()=>B(()=>import("./数学常用日语.html-DsIXCFSe.js"),[]),meta:{tag:["数学","日语"],excerpt:`<ol>
<li>仮定する (かていする) - 假设</li>
<li>証明する (しょうめいする) - 证明</li>
<li>求める (もとめる) - 求解</li>
<li>計算する (けいさんする) - 计算</li>
<li>変数 (へんすう) - 变量</li>
<li>方程式 (ほうていしき) - 方程式</li>
<li>解く (とく) - 解答</li>
<li>置き換える (おきかえる) - 替换</li>
<li>代入する (だいにゅうする) - 代入</li>
<li>微分する (びぶんする) - 微分</li>
<li>積分する (せきぶんする) - 积分</li>
<li>導出する (どうしゅつする) - 导出</li>
<li>等式 (とうしき) - 等式</li>
<li>不等式 (ふとうしき) - 不等式</li>
<li>同値 (どうち) - 同值</li>
<li>無限 (むげん) - 无限</li>
<li>極限 (きょくげん) - 极限</li>
<li>漸近 (ぜんきん) - 渐近</li>
<li>収束する (しゅうそくする) - 收敛</li>
<li>発散する (はっさんする) - 发散</li>
<li>行列 (ぎょうれつ) - 矩阵</li>
<li>ベクトル - 向量</li>
<li>固有値 (こゆうち) - 特征值</li>
<li>固有ベクトル (こゆうベクトル) - 特征向量</li>
<li>線形代数 (せんけいだいすう) - 线性代数</li>
<li>解析する (かいせきする) - 分析</li>
<li>函数 (かんすう) - 函数</li>
<li>微分方程式 (びぶんほうていしき) - 微分方程式</li>
<li>確率 (かくりつ) - 概率</li>
<li>統計 (とうけい) - 统计</li>
<li>期待値 (きたいち) - 期望值</li>
<li>分散 (ぶんさん) - 方差</li>
<li>標準偏差 (ひょうじゅんへんさ) - 标准差</li>
<li>相関 (そうかん) - 相关</li>
<li>回帰 (かいき) - 回归</li>
<li>最適化 (さいてきか) - 优化</li>
<li>近似する (きんじする) - 近似</li>
<li>解析解 (かいせきかい) - 解析解</li>
<li>数値解 (すうちかい) - 数值解</li>
<li>境界条件 (きょうかいじょうけん) - 边界条件</li>
<li>初期条件 (しょきじょうけん) - 初始条件</li>
<li>特異点 (とくいてん) - 奇点</li>
<li>演算 (えんざん) - 运算</li>
<li>組み合わせ (くみあわせ) - 组合</li>
<li>順列 (じゅんれつ) - 排列</li>
<li>グラフ - 图表</li>
<li>座標 (ざひょう) - 坐标</li>
<li>直線 (ちょくせん) - 直线</li>
<li>曲線 (きょくせん) - 曲线</li>
<li>解析学 (かいせきがく) - 解析学</li>
</ol>`,readingTime:{minutes:.96,words:288},title:"",type:"article",s:"数学常用日语"}}],["/Other/%E8%AF%B4%E6%98%8E/%E6%9C%AC%E5%9C%B0%E6%8E%92%E9%99%A4%E6%96%87%E4%BB%B6.html",{loader:()=>B(()=>import("./本地排除文件.html-CsJbkMIg.js"),[]),meta:{tag:["规则"],excerpt:`<p>本指南用于在本地工作区中排除远程仓库的特定目录，而不影响远程仓库本身。</p>
<h2>1.  <strong>进入仓库根目录</strong></h2>
<pre><code>打开终端或命令行，\`cd\` 到你的本地 Git 仓库根目录。
</code></pre>
<h2>2. <strong>启用 Sparse Checkout</strong></h2>
<div class="language-bash line-numbers-mode" data-highlighter="shiki" data-ext="bash" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-bash"><span class="line"><span style="--shiki-light:#4078F2;--shiki-dark:#61AFEF">git</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> sparse-checkout</span><span style="--shiki-light:#50A14F;--shiki-dark:#98C379"> init</span><span style="--shiki-light:#986801;--shiki-dark:#D19A66"> --cone</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div></div></div>`,readingTime:{minutes:1.83,words:550},title:"",type:"article",s:"本地排除文件"}}],["/Other/%E8%AF%B4%E6%98%8E/%E7%AC%A6%E5%8F%B7%E4%B9%A6%E5%86%99%E8%A7%84%E8%8C%83.html",{loader:()=>B(()=>import("./符号书写规范.html-lYRXeRRO.js"),[]),meta:{tag:["数学"],excerpt:`<table>
<thead>
<tr>
<th>符号</th>
<th>LaTeX 示例</th>
<th>说明</th>
</tr>
</thead>
<tbody>
<tr>
<td>常量</td>
<td><code>$a, x, f(x)$</code></td>
<td><strong>LaTeX 默认处理为斜体</strong>。无需额外修饰操作。</td>
</tr>
<tr>
<td>向量</td>
<td><code>$\\mathbf{v}$</code> 或 <code>$\\vec{v}$</code></td>
<td><strong>工程/应用数学</strong>偏好加粗 <code>\\mathbf</code>；<strong>理论数学/物理</strong>偏好箭头 <code>\\vec</code>。都可以, 全文须统一</td>
</tr>
<tr>
<td>矩阵</td>
<td><code>$\\mathbf{A}, \\mathbf{B}$</code></td>
<td>建议使用大写加粗罗马体, 但不强制, 不用也没有关系</td>
</tr>
<tr>
<td>张量</td>
<td><code>$\\boldsymbol{T}$</code> 或 <code>$\\mathbb{T}$</code></td>
<td>建议使用加粗斜体 <code>\\boldsymbol</code> 或黑板加粗 <code>\\mathbb</code>, 不用也没有关系</td>
</tr>
<tr>
<td>集合</td>
<td><code>$\\mathbb{R}, \\mathbb{Z}, \\mathbb{N}$</code></td>
<td>必须使用黑板加粗 <code>\\mathbb</code>。元素用普通斜体。</td>
</tr>
<tr>
<td>特殊函数</td>
<td><code>$\\sin, \\log, \\exp$</code></td>
<td>必须使用反斜杠转义，以保持<strong>直立罗马体</strong>。</td>
</tr>
<tr>
<td>微分符号</td>
<td><code>$\\mathrm{d}x$</code></td>
<td>推荐使用直立体 <code>\\mathrm{d}</code>，区分于变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span>, 不用也没有关系</td>
</tr>
<tr>
<td>常量</td>
<td><code>$i, e, \\pi$</code></td>
<td>复数单位、自然底数、圆周率，通常直接使用默认斜体。</td>
</tr>
</tbody>
</table>`,readingTime:{minutes:.77,words:231},title:"",type:"article",s:"符号书写规范"}}],["/Other/%E8%AF%B4%E6%98%8E/%E9%97%AD%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%E7%AC%A6%E5%8F%B7%E7%9A%84%E8%A1%A8%E7%A4%BA%E6%96%B9%E6%B3%95.html",{loader:()=>B(()=>import("./闭曲面积分符号的表示方法.html-CRCtmFPl.js"),[]),meta:{tag:["数学","规则"],excerpt:`<p>由于Obsidian不支持二重曲面积分符号<code>\\oiint</code>的渲染, 因此下面提供一种折中的方式, 可以在Obsidian中渲染出二重积分的符号</p>
<div class="language- line-numbers-mode" data-highlighter="shiki" data-ext style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-"><span class="line"><span>\\int\\kern{-17mu}{\\unicode{x25CB}}\\kern{-20mu}\\int_{C}</span></span></code></pre>
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<div class="callout-title-icon">
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<div class="callout-title-inner">Bug</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:0,words:1},title:"",type:"article",s:"Callouts-Bug"}}],["/assets_/Callouts/Callouts-ChatGPT.html",{loader:()=>B(()=>import("./Callouts-ChatGPT.html-BCMMn_YI.js"),[]),meta:{excerpt:`
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</div></details>`,readingTime:{minutes:.01,words:2},title:"",type:"article",s:"Callouts-ChatGPT"}}],["/assets_/Callouts/Callouts-Danger.html",{loader:()=>B(()=>import("./Callouts-Danger.html-BPlxzXZy.js"),[]),meta:{excerpt:`
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<div class="callout-title-inner">Danger</div>
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<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-x"><path d="M18 6 6 18"/><path d="m6 6 12 12"/></svg>
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<div class="callout-title-inner">Failure</div>
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<div class="callout-content"><p></p>
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<div class="callout-title">
<div class="callout-title-icon">
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<div class="callout-title-inner">Info</div>
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<div class="callout-content"><p></p>
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<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
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<div class="callout-title-inner">Note</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:0,words:1},title:"",type:"article",s:"Callouts-Note"}}],["/assets_/Callouts/Callouts-Question.html",{loader:()=>B(()=>import("./Callouts-Question.html-COqkpLNX.js"),[]),meta:{excerpt:`
<div class="callout" data-callout="question">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-help-circle"><circle cx="12" cy="12" r="10"/><path d="M9.09 9a3 3 0 0 1 5.83 1c0 2-3 3-3 3"/><path d="M12 17h.01"/></svg>
</div>
<div class="callout-title-inner">Question</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:0,words:1},title:"",type:"article",s:"Callouts-Question"}}],["/assets_/Callouts/Callouts-Quote.html",{loader:()=>B(()=>import("./Callouts-Quote.html-CV082t_H.js"),[]),meta:{excerpt:`
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</div>
<div class="callout-title-inner">Quote</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p></p>
</div></details>`,readingTime:{minutes:0,words:1},title:"",type:"article",s:"Callouts-Quote"}}],["/assets_/Callouts/Callouts-Success.html",{loader:()=>B(()=>import("./Callouts-Success.html-CbZ4D30s.js"),[]),meta:{excerpt:`
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<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-check"><path d="M20 6 9 17l-5-5"/></svg>
</div>
<div class="callout-title-inner">Success</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:0,words:1},title:"",type:"article",s:"Callouts-Success"}}],["/assets_/Callouts/Callouts-Tips.html",{loader:()=>B(()=>import("./Callouts-Tips.html-D2CP0F0j.js"),[]),meta:{excerpt:`
<details class="callout" data-callout="tip" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-flame"><path d="M8.5 14.5A2.5 2.5 0 0 0 11 12c0-1.38-.5-2-1-3-1.072-2.143-.224-4.054 2-6 .5 2.5 2 4.9 4 6.5 2 1.6 3 3.5 3 5.5a7 7 0 1 1-14 0c0-1.153.433-2.294 1-3a2.5 2.5 0 0 0 2.5 2.5z"/></svg>
</div>
<div class="callout-title-inner">Tip</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p></p>
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<div class="callout" data-callout="todo">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-check-circle-2"><circle cx="12" cy="12" r="10"/><path d="m9 12 2 2 4-4"/></svg>
</div>
<div class="callout-title-inner">Todo</div>
</div>
<div class="callout-content"><p></p>
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<div class="callout" data-callout="warning">
<div class="callout-title">
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</div>
<div class="callout-title-inner">Warning</div>
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<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">Note</div>
</div>
<div class="callout-content"><p></p>
<div class="language-md line-numbers-mode" data-highlighter="shiki" data-ext="md" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-md"><span class="line"><span style="--shiki-light:#A0A1A7;--shiki-light-font-style:italic;--shiki-dark:#5C6370;--shiki-dark-font-style:inherit">> </span><span style="--shiki-light:#986801;--shiki-light-font-style:italic;--shiki-dark:#5C6370;--shiki-dark-font-style:inherit">[</span><span style="--shiki-light:#4078F2;--shiki-light-font-style:italic;--shiki-dark:#61AFEF;--shiki-dark-font-style:inherit">!note</span><span style="--shiki-light:#986801;--shiki-light-font-style:italic;--shiki-dark:#5C6370;--shiki-dark-font-style:inherit">]</span></span>
<span class="line"><span style="--shiki-light:#A0A1A7;--shiki-light-font-style:italic;--shiki-dark:#5C6370;--shiki-dark-font-style:inherit">></span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div></div></div></div></div>`,readingTime:{minutes:.33,words:100},title:"更多",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/---%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86---.html",{loader:()=>B(()=>import("./---多元函数微分---.html-CvzREU4D.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ol>
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<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95.html" target="_blank">最小二乘法</a></li>
</ol>`,readingTime:{minutes:.4,words:121},title:"文档-所有文档",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/Schwarz%20%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./Schwarz 定理.html-C5J8qroo.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>对于大多数常见函数，混合偏导数是相等的。这由 <strong>Schwarz 定理</strong>（Schwarz's theorem）或称 <strong>Clairaut 定理</strong>（Clairaut's theorem）所保证。该定理表明，如果一个函数的混合偏导数在一个点及其附近连续，那么这两个混合偏导数是相等的。</p>
<h1>定理陈述</h1>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 是定义在开集 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">U \\subset \\mathbb{R}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 上的二次连续可导函数，即函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 具有连续的二阶偏导数。则对于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 中的任意点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，有：</p>`,readingTime:{minutes:1.33,words:399},title:"定理陈述",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/n%E7%BB%B4%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./n维空间.html-Cn8Ae0aK.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 维空间是多元函数的定义域所在集合, 可视为实数域上的线性空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 的几何模型.<br>
分析学在此基础上引入<strong>距离, 邻域, 极限, 连续</strong>等概念, 以研究函数的变化规律.</p>`,readingTime:{minutes:1.42,words:426},title:"",type:"article",s:"n维空间"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E4%BA%8C%E5%85%83%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./二元泰勒公式.html-IPs3VYIX.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:'<p>二元泰勒公式提供了一个二元函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 附近的二阶近似。是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html" target="_blank">泰勒公式</a>的推广</p>',readingTime:{minutes:1.83,words:548},title:"",type:"article",s:"二元泰勒公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%81%8F%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./偏导数.html-CJEzXCuN.js"),[]),meta:{author:["Cyletix","夜幕下的启明星"],tag:["数学"],excerpt:`
<p>设<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z=f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_{0},y_{0})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>的某个邻域内有定义，当<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>固定在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">y_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，而<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>处有增量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span>，相应的函数增量 为:<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_{0}+\\Delta x,y_{0})-f(x_{0},y_{0})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，若</p>`,readingTime:{minutes:2.28,words:685},title:"偏导数定义",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%85%A8%E5%BE%AE%E5%88%86.html",{loader:()=>B(()=>import("./全微分.html-BtDN66iJ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<h2>全增量</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\Delta z=f(x+\\Delta x,y+\\Delta y)-f(x,y)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.98,words:293},title:"预备知识",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%90%91%E9%87%8F%E5%80%BC%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./向量值函数.html-BAlA-54f.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>向量值函数是指那些其输出为向量的函数，通常形式为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">r</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\\mathbf{r}(t) = \\langle f(t), g(t), h(t) \\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">r</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)⟩</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 是实数函数。这种类型的函数在物理学和工程学中非常重要，常用来描述在三维空间中的动态系统，如粒子的运动路径。</p>',readingTime:{minutes:.7,words:211},title:"",type:"article",s:"向量值函数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5.html",{loader:()=>B(()=>import("./多元函数基本概念.html-BBJ6bzvm.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">概要</div>
</div>
<div class="callout-content"><p>从单变量函数到高维分析的过渡, 主要回答两个根本问题:</p>
<ol>
<li>高维空间中如何描述点与集合？</li>
<li>高维空间函数又是什么形式？</li>
</ol>
<p>本节从几何视角引入<strong>点集</strong>与<strong>空间结构</strong>, 这些定义共同支撑后续多元函数的&quot;极限&quot; &quot;连续&quot; &quot;偏导&quot; &quot;梯度&quot; 等所有分析概念.</p>
</div></div>`,readingTime:{minutes:1.31,words:394},title:"",type:"article",s:"多元函数基本概念"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%E5%BA%94%E7%94%A8.html",{loader:()=>B(()=>import("./多元函数微分应用.html-BpLu-Qar.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E5%90%91%E9%87%8F%E5%80%BC%E5%87%BD%E6%95%B0.html" target="_blank">向量值函数</a></p>
<h1>空间曲线的切线和法平面</h1>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E7%A9%BA%E9%97%B4%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%88%87%E7%BA%BF%E5%92%8C%E6%B3%95%E5%B9%B3%E9%9D%A2.html" target="_blank">空间曲线的切线和法平面</a></p>
`,readingTime:{minutes:.15,words:44},title:"向量值函数",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E6%9E%81%E5%80%BC.html",{loader:()=>B(()=>import("./多元函数极值.html-7p14Za8X.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>判断极值</h2>
<p>使用<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E6%B5%B7%E5%A1%9E%E7%9F%A9%E9%98%B5.html" target="_blank">海塞矩阵</a>的行列式判断函数在极值点的性质</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>H</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><mo>=</mo><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><msub><mi>f</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>−</mo><mo stretchy="false">(</mo><msub><mi>f</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">D= |H| = 
\\begin{vmatrix}
\\frac{\\partial^2 f}{\\partial x^2} &amp; \\frac{\\partial^2 f}{\\partial x \\partial y} \\\\
\\frac{\\partial^2 f}{\\partial y \\partial x} &amp; \\frac{\\partial^2 f}{\\partial y^2}
\\end{vmatrix} 
= f_{xx} f_{yy} - (f_{xy})^2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1023em;vertical-align:-1.3011em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:5em;"></span><span style="width:0.333em;height:3em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3em" viewBox="0 0 333 3000"><path d="M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:5em;"></span><span style="width:0.333em;height:3em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3em" viewBox="0 0 333 3000"><path d="M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1502em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.22,words:667},title:"",type:"article",s:"多元函数极值"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E5%AE%9A%E4%B9%89.html",{loader:()=>B(()=>import("./多元函数的定义.html-DFTKaFms.js"),[]),meta:{author:["Cyletix"],tag:["数学","多元函数"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">简介</div>
</div>
<div class="callout-content"><p>多元函数是单变量函数的自然推广. 在理解<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E5%B9%B3%E9%9D%A2%E7%82%B9%E9%9B%86.html" target="_blank">平面点集</a>与 <a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2Fn%E7%BB%B4%E7%A9%BA%E9%97%B4.html" target="_blank">n维空间</a>后, 我们可以把&quot;输入是一个数&quot; 的函数, 扩展为&quot;输入是多个数(一个n维点或向量)&quot; 的情形.<br>
它建立了高维空间中数量关系的桥梁, 是后续极限, 连续, 偏导数与多重积分的基础.</p>
</div></div>`,readingTime:{minutes:1.27,words:380},title:"",type:"article",s:"多元函数的定义"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./多元函数的极限.html-CNTpjn87.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">学习导语</div>
</div>
<div class="callout-content"><p>多元函数的极限在<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E5%AE%9A%E4%B9%89.html" target="_blank">多元函数的定义</a>的基础上, 引入极限的思想, 说明函数值在高维空间中是否会趋向某一固定数值.</p>
</div></div>`,readingTime:{minutes:2.97,words:891},title:"",type:"article",s:"多元函数的极限"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E8%BF%9E%E7%BB%AD%E6%80%A7.html",{loader:()=>B(()=>import("./多元函数的连续性.html-BSUFIk-t.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">学习导语</div>
</div>
<div class="callout-content"><p>多元函数的连续性在<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E9%99%90.html" target="_blank">多元函数的极限</a>的基础上, 研究函数在某点是否&quot;无突变&quot; .<br>
它连接了&quot;极限的存在&quot; 与&quot;函数自身的取值&quot; , 是分析函数可微性的必要前提.</p>
</div></div>`,readingTime:{minutes:3.23,words:969},title:"",type:"article",s:"多元函数的连续性"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC.html",{loader:()=>B(()=>import("./多元复合函数求导.html-CDuU00wL.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h3>定义和基本概念</h3>
<p>设有一个多变量复合函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 又是其他变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 的函数，即 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x = g(u, v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = h(u, v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span>。我们想找到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 对 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 的偏导数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>u</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial z}{\\partial u}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>v</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial z}{\\partial v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>。</p>`,readingTime:{minutes:1.81,words:544},title:"",type:"article",s:"多元复合函数求导"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%A4%9A%E5%85%83%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC.html",{loader:()=>B(()=>import("./多元隐函数求导.html-5ZUvxfZM.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
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<div class="callout-content"><p>本节讨论<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E9%9A%90%E5%87%BD%E6%95%B0%E5%AE%9A%E7%90%86.html" target="_blank">隐函数定理</a>在多元求导中的应用.<br>
当多个变量通过方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo separator="true">,</mo><mo>…</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F(x, y, z, \\ldots)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 相关联时,<br>
即使函数形式无法显式写出, 仍可以通过偏导数关系确定一个变量对其他变量的变化率.<br>
这是多元微分法的重要工具, 其逻辑基础源于连续性与可微性.</p>
</div></div>`,readingTime:{minutes:4.64,words:1391},title:"",type:"article",s:"多元隐函数求导"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E5%B9%B3%E9%9D%A2%E7%82%B9%E9%9B%86.html",{loader:()=>B(()=>import("./平面点集.html-DMD_TRrE.js"),[]),meta:{author:["Cyletix"],tag:["数学","空间概念"],excerpt:`<p>平面上任意一点可由有序数对 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 表示。<br>
所有点的集合称为<strong>平面点集</strong>。它是多元函数研究的几何基础。</p>`,readingTime:{minutes:.68,words:205},title:"",type:"article",s:"平面点集"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./拉普拉斯方程.html-DC6U4XBZ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>ϕ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\nabla^{2}\\phi=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:.48,words:143},title:"定义",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%96%B9%E5%90%91%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./方向导数.html-juqp6CNc.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`
<p>方向导描述了多变量函数在某一点沿特定方向的变化率。直观地说，方向导数告诉我们，当我们沿着某个方向微小地移动时，函数值的变化量。<br>
假设有一个定义在欧几里得空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 中的可微函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1, x_2, ..., x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，我们想计算这个函数在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 处沿着单位向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 方向的方向导数。</p>`,readingTime:{minutes:.82,words:246},title:"简介",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%9B%B2%E9%9D%A2%E7%9A%84%E5%88%87%E5%B9%B3%E9%9D%A2.html",{loader:()=>B(()=>import("./曲面的切平面.html-DyGTN5-Z.js"),[]),meta:{author:["Cyletix","GPT-4","Gemini"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>设二元函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 处可微，则过点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0, f(x_0, y_0))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span> 且与曲面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 相切的平面称为函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 处的切平面。</p>`,readingTime:{minutes:3,words:900},title:"",type:"article",s:"曲面的切平面"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95.html",{loader:()=>B(()=>import("./最小二乘法.html-BSqtvC5T.js"),[]),meta:{tag:["数学"],excerpt:`<p>这是微积分中使用的最小二乘法, 与概率论中的最小二乘法稍有不同.</p>
<p>对于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>个样本<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_i,y_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, 我们希望找到一个直线可以最好地接近样本结果</p>`,readingTime:{minutes:.43,words:130},title:"",type:"article",s:"最小二乘法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%A2%AF%E5%BA%A6.html",{loader:()=>B(()=>import("./梯度.html-BJjtlfj-.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>梯度是一个标量函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 的偏导数组成的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%90%91%E9%87%8F%E5%9C%BA.html" target="_blank">向量场</a>, 梯度给出了空间各点函数增长最快的方向. 梯度是一种关于多元导数的概括, 可以看作一元函数导数的推广.</p>`,readingTime:{minutes:1.11,words:333},title:"",type:"article",s:"梯度"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E6%B5%B7%E5%A1%9E%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./海塞矩阵.html-JKx_VcNI.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于二元函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，海塞矩阵定义如下：</p>',readingTime:{minutes:.3,words:90},title:"",type:"article",s:"海塞矩阵"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E7%A9%BA%E9%97%B4%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%88%87%E7%BA%BF%E5%92%8C%E6%B3%95%E5%B9%B3%E9%9D%A2.html",{loader:()=>B(()=>import("./空间曲线的切线和法平面.html-CjaNTqt9.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>空间曲线是通过向量值函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">r</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\mathbf{r}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">r</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 描述的一条曲线。例如，考虑空间中的曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">r</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>t</mi><mo separator="true">,</mo><msup><mi>t</mi><mn>2</mn></msup><mo separator="true">,</mo><msup><mi>t</mi><mn>3</mn></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\\mathbf{r}(t) = \\langle t, t^2, t^3 \\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">r</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span>，我们可以通过这个函数探索曲线的不同特性。</p>',readingTime:{minutes:1.2,words:361},title:"",type:"article",s:"空间曲线的切线和法平面"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E9%9A%90%E5%87%BD%E6%95%B0%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./隐函数定理.html-DvZEn0v0.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">隐函数定理</div>
</div>
<div class="callout-content"><p>对于一个由关系</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x, y) = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>表示的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E9%9A%90%E5%87%BD%E6%95%B0.html" target="_blank">隐函数</a>，如果它在某一点的偏导数满足一定条件，则在该点的邻域内，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 可以表示为关于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的<a class="route-link nolebase-route-link-invalid" href="#" target="_target">显函数</a>：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = h(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>
<p>这样就将隐函数关系转化为常见的函数关系形式。</p>
</div></div>`,readingTime:{minutes:.39,words:118},title:"",type:"article",s:"隐函数定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E9%9B%85%E5%8F%AF%E6%AF%94%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./雅可比矩阵.html-DWaWz10d.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>%%来自 Wikipedia%%<br>
在向量分析中，雅可比矩阵是函数的一阶<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E5%81%8F%E5%AF%BC%E6%95%B0.html" target="_blank">偏导数</a>以一定方式排列成的矩阵。<br>
当其为方形矩阵时，其行列式称为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E9%9B%85%E5%8F%AF%E6%AF%94%E8%A1%8C%E5%88%97%E5%BC%8F.html" target="_blank">雅可比行列式</a>。</p>
<p>其重要性在于，如果函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">f</mi><mo>:</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">\\mathbf{f} : \\mathbb{R}^n \\to\\mathbb{R}^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf" style="margin-right:0.109em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 可微，在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的雅可比矩阵即为该函数在该点的最佳线性逼近。因此，雅可比矩阵是单变量实数<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E5%87%BD%E6%95%B0%E7%9A%84%E5%BE%AE%E5%88%86.html" target="_blank">函数的微分</a>在向量值多变量函数上的推广。</p>`,readingTime:{minutes:1.63,words:489},title:"",type:"article",s:"雅可比矩阵"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86/%E9%9B%85%E5%8F%AF%E6%AF%94%E8%A1%8C%E5%88%97%E5%BC%8F.html",{loader:()=>B(()=>import("./雅可比行列式.html-BOmR68jL.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>雅可比行列式是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E9%9B%85%E5%8F%AF%E6%AF%94%E7%9F%A9%E9%98%B5.html" target="_blank">雅可比矩阵</a>为方阵(<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>)时的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E8%A1%8C%E5%88%97%E5%BC%8F.html" target="_blank">行列式</a>。</p>`,readingTime:{minutes:1.18,words:355},title:"",type:"article",s:"雅可比行列式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/---%E5%AF%BC%E6%95%B0---.html",{loader:()=>B(()=>import("./---导数---.html-5UzxFM6G.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%AF%BC%E6%95%B0.html" target="_blank">导数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%8F%AF%E5%AF%BC%E4%B8%8E%E8%BF%9E%E7%BB%AD%E7%9A%84%E5%85%B3%E7%B3%BB.html" target="_blank">可导与连续的关系</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html" target="_blank">求导法则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%8F%8D%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html" target="_blank">反函数求导法则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html" target="_blank">复合函数求导法则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%B8%B8%E7%94%A8%E6%B1%82%E5%AF%BC%E5%85%AC%E5%BC%8F.html" target="_blank">常用求导公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E9%AB%98%E9%98%B6%E5%AF%BC%E6%95%B0.html" target="_blank">高阶导数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC.html" target="_blank">微积分/导数/隐函数求导</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B%E6%B1%82%E5%AF%BC.html" target="_blank">参数方程求导</a></li>
</ol>`,readingTime:{minutes:.39,words:118},title:"目录",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E4%B9%98%E6%B3%95%E5%AE%9A%E5%88%99.html",{loader:()=>B(()=>import("./乘法定则.html-B_JGuEgw.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{d}{dx}\\left[f(x)g(x)\\right] = f&#x27;(x)g(x) + f(x)g&#x27;(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.13,words:39},title:"",type:"article",s:"乘法定则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%8D%95%E4%BE%A7%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./单侧导数.html-Ui_B8Vgr.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">单侧导数</div>
</div>
<div class="callout-content"><p></p>
<ul>
<li><strong>左导数</strong>: <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>f</mi><mo lspace="0em" rspace="0em">−</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msubsup><mi>x</mi><mn>0</mn><mo>−</mo></msubsup></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f&#x27;_{-}(x_0) = \\lim_{x \\to x_0^-} \\frac{f(x) - f(x_0)}{x - x_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1072em;vertical-align:-0.3053em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4857em;vertical-align:-1.0587em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.2593em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.821em;"><span style="top:-2.1885em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span><span style="top:-2.9043em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3115em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0587em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li>
<li><strong>右导数</strong>: <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>f</mi><mo lspace="0em" rspace="0em">+</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msubsup><mi>x</mi><mn>0</mn><mo>+</mo></msubsup></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f&#x27;_{+}(x_0) = \\lim_{x \\to x_0^+} \\frac{f(x) - f(x_0)}{x - x_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1072em;vertical-align:-0.3053em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4857em;vertical-align:-1.0587em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.2593em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.821em;"><span style="top:-2.1885em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span><span style="top:-2.9043em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3115em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0587em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li>
</ul>
</div></div>`,readingTime:{minutes:.29,words:88},title:"",type:"article",s:"单侧导数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B%E6%B1%82%E5%AF%BC.html",{loader:()=>B(()=>import("./参数方程求导.html-BCbPGh3Q.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<p>与<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC.html" target="_blank">微积分/导数/隐函数求导</a>类似, 实质是对中间变量t的隐函数求导数, 如:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>y</mi><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\\begin{cases} 
{x=\\phi(t)}\\\\
{y=\\psi(t)}
\\end{cases}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">ψ</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.29,words:87},title:"参数方程确定的函数",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%8F%8D%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./反函数求导法则.html-BXkY9TTR.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:'<p>如果函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在区间<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span>内单调可导, 且<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;(y)\\neq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 则反函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f^{-1}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span>也可导, 且</p>',readingTime:{minutes:.43,words:130},title:"",type:"article",s:"反函数求导法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%8F%AF%E5%AF%BC%E4%B8%8E%E8%BF%9E%E7%BB%AD%E7%9A%84%E5%85%B3%E7%B3%BB.html",{loader:()=>B(()=>import("./可导与连续的关系.html-CZrDX4KX.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p><em>函数在某点连续</em> 是 <em>函数在该点可导</em> 的必要非充分条件:<br>
可导必定连续, 连续不一定可导</p>
`,readingTime:{minutes:.15,words:44},title:"",type:"article",s:"可导与连续的关系"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./复合函数求导法则.html-DUeRUMI-.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 可导, 而 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 可导,那么复合函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">[</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex">y=f\\big[g(x)\\big]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">]</span></span></span></span></span> 在点x可导, 且其导数为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx}=f&#x27;(u)\\cdot g&#x27;(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx}=\\frac{dy}{du}\\cdot \\frac{du}{dx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>',readingTime:{minutes:.32,words:96},title:"",type:"article",s:"复合函数求导法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%AF%B9%E6%95%B0%E6%B1%82%E5%AF%BC%E6%B3%95.html",{loader:()=>B(()=>import("./对数求导法.html-CF2P4SNY.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="tip">
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<div class="callout-title-inner">对数求导法</div>
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<div class="callout-content"><p>当函数形式为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">y=f(x)^{g(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">g</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> (幂指函数) 或涉及多项的复杂乘积、商、根式时, 可使用对数求导法简化计算:</p>
<ol>
<li>在方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 两边同时取自然对数, 得到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mi>y</mi><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\\ln y = \\ln[f(x)]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)]</span></span></span></span>.</li>
<li>利用对数性质 (如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mi>b</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mi>ln</mi><mo>⁡</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\\ln(a^b) = b \\ln a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>⁡</mo><mi>a</mi><mo>−</mo><mi>ln</mi><mo>⁡</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\\ln(\\frac{a}{b}) = \\ln a - \\ln b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span>) 简化右侧表达式.</li>
<li>将 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\ln y = g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> (简化后的表达式) 视为隐函数, 两边同时对 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 求导, 得到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{1}{y} \\cdot \\frac{dy}{dx} = g&#x27;(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3262em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>.</li>
<li>解出 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>y</mi><mo>⋅</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} = y \\cdot g&#x27;(x) = f(x) \\cdot g&#x27;(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>.</li>
</ol>
</div></div>`,readingTime:{minutes:.59,words:177},title:"",type:"article",s:"对数求导法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%AF%BC%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./导函数.html-Bwo-ScWr.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">导函数</div>
</div>
<div class="callout-content"><p>若 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span> 内每一点都可导，则得到<strong>导函数</strong>：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">f&#x27;(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1791em;vertical-align:-0.7521em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mrel mtight">→</span><span class="mord mtight">0</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">h</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">h</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>导函数反映了原函数在每一点的变化率。</p>
</div></div>`,readingTime:{minutes:.24,words:71},title:"",type:"article",s:"导函数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./导数.html-DbzTP2Ff.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>导数是微积分学的核心概念之一，描述了函数在某一点的<strong>瞬时变化率</strong>。通过导数，可以研究函数的动态变化趋势，如函数的单调性、极值等。</p>
<h2>定义</h2>
<p>导数是微积分学的核心概念之一，描述了函数在某一点的<strong>瞬时变化率</strong>。</p>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">&quot;导数&quot;的两种含义</div>
</div>
<div class="callout-content"><p>&quot;导数&quot;一词常常混用，既可指<strong>在一点的导数</strong>（数值），也可指<strong>导函数</strong>（函数）。</p>
</div></div>`,readingTime:{minutes:2.13,words:640},title:"",type:"article",s:"导数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%B8%B8%E7%94%A8%E6%B1%82%E5%AF%BC%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./常用求导公式.html-lus_W0Ei.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>幂指函数</h2>
<p>其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 都是常数：</p>`,readingTime:{minutes:.69,words:207},title:"",type:"article",s:"常用求导公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E5%B8%B8%E8%A7%81%E5%87%BD%E6%95%B0%E5%AF%BC%E6%95%B0%E9%80%9A%E9%A1%B9.html",{loader:()=>B(()=>import("./常见函数导数通项.html-CKENHwpO.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<h3>多项式函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">f(x) = x^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span></h3>',readingTime:{minutes:.57,words:172},title:"",type:"article",s:"常见函数导数通项"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./指数函数.html-DoAYAFC7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>通过微积分构造的函数, 无法用代数方法建立. 形如:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">y=a^x
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.84,words:553},title:"",type:"article",s:"指数函数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E6%B1%82%E5%AF%BC%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./求导法则.html-D-qSMsgF.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>满足<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E7%BA%BF%E6%80%A7.html" target="_blank">线性</a></p>
<ol>
<li></li>
</ol>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">[</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>±</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msup><mi>u</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>±</mo><msup><mi>v</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[u(x)\\pm v(x)]&#x27; = u&#x27;(x)\\pm v&#x27;(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.19,words:56},title:"",type:"article",s:"求导法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%93%BE%E5%BC%8F%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./链式法则.html-BHGA4Ahr.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true">(</mo><mi>f</mi><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo fence="true">)</mo></mrow><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{d}{dx}\\left(f[g(x)]\\right) = f&#x27;[g(x)]g&#x27;(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)]</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)]</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.07,words:21},title:"",type:"article",s:"链式法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%99%A4%E6%B3%95%E5%AE%9A%E5%88%99.html",{loader:()=>B(()=>import("./除法定则.html-CwH3XnD0.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true">(</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{d}{dx}\\left(\\frac{f(x)}{g(x)}\\right) = \\frac{f&#x27;(x)g(x) - f(x)g&#x27;(x)}{[g(x)]^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3649em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4289em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.12,words:36},title:"",type:"article",s:"除法定则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%9A%90%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./隐函数.html-BQYd6mls.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">隐函数</div>
</div>
<div class="callout-content"><p>隐函数是由<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E9%9A%90%E5%BC%8F%E6%96%B9%E7%A8%8B.html" target="_blank">隐式方程</a>间接定义的函数。<br>
例如, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">y=\\sqrt{1-x^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1266em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9134em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.8734em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1266em;"><span></span></span></span></span></span></span></span></span> 是由方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^2+y^2-1=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 所确定的函数。<br>
而直接用自变量表达的函数称为<strong>显函数</strong>, 如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y=\\cos x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span>。</p>
</div></div>`,readingTime:{minutes:.29,words:86},title:"",type:"article",s:"隐函数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC.html",{loader:()=>B(()=>import("./隐函数求导.html-BGBAYn8y.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>在微积分中, 我们不仅需要处理 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 形式的显函数, 还会遇到由方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F(x,y)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 定义的隐函数. 本文档主要探讨如何在不将隐函数“显化”的情况下求出其导数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>`,readingTime:{minutes:2.55,words:766},title:"",type:"article",s:"隐函数求导"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%9A%90%E5%BC%8F%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./隐式方程.html-BRkacMB7.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>隐式方程是形如</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x_1, x_2, \\cdots, x_n) = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>的<a class="route-link nolebase-route-link-invalid" href="#" target="_target">关系</a>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 是<a class="route-link nolebase-route-link-invalid" href="#" target="_target">多元函数</a>。</p>
</div></div>`,readingTime:{minutes:.11,words:34},title:"",type:"article",s:"隐式方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%AF%BC%E6%95%B0/%E9%AB%98%E9%98%B6%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./高阶导数.html-D1lRsE-l.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<p>y的n阶导数记作</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mfrac><mrow><msup><mi>d</mi><mi>n</mi></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">y^{(n)}=\\frac{d^ny}{dx^n}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1324em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.66,words:197},title:"定义",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/---%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B---.html",{loader:()=>B(()=>import("./---微分方程---.html-GnYmnliG.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5.html" target="_blank">微分方程基本概念</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%8F%AF%E5%88%86%E7%A6%BB%E5%8F%98%E9%87%8F%E7%9A%84%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">可分离变量的微分方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E9%BD%90%E6%AC%A1%E6%96%B9%E7%A8%8B.html" target="_blank">齐次方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E4%B8%80%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">一阶线性微分方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E9%AB%98%E9%98%B6%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E9%99%8D%E9%98%B6.html" target="_blank">高阶微分方程降阶</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E9%AB%98%E9%98%B6%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">高阶齐次线性微分方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B8%B8%E7%B3%BB%E6%95%B0%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">常系数线性微分方程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E6%AC%A7%E6%8B%89%E6%96%B9%E7%A8%8B.html" target="_blank">欧拉方程</a></li>
</ol>`,readingTime:{minutes:4.52,words:1355},title:"目录",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E4%B8%80%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./一阶线性微分方程.html-DkD-zNAo.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>定义</h2>
<p>一阶线性微分方程可定义为以下标准形式：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>=</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \\frac{dy}{dx} + P(x)y = Q(x) 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:1.51,words:452},title:"伯努利方程",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E4%B8%80%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E9%80%9A%E8%A7%A3.html",{loader:()=>B(()=>import("./一阶线性微分方程通解.html-BalL6POM.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>C</mi><msup><mi>e</mi><mrow><mo>−</mo><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow></msup><mo>∫</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow></msup><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">y=Ce^{ -\\int P(x) \\, dx  }+e^{ -\\int P(x) \\, dx  }\\int Q(x)e^{ \\int P(x) \\, dx  } \\, dx 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0598em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9765em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop op-symbol small-op mtight" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9765em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop op-symbol small-op mtight" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9765em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop op-symbol small-op mtight" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight" style="margin-right:0.1389em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>`,readingTime:{minutes:.1,words:30},title:"",type:"article",s:"一阶线性微分方程通解"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E4%BA%8C%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./二阶线性微分方程.html-BmM-iCkV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">二阶线性微分方程</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:1.94,words:583},title:"定义",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E4%BA%8C%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84.html",{loader:()=>B(()=>import("./二阶线性微分方程解的结构.html-B22IGiQ2.js"),[]),meta:{tag:["数学"],excerpt:`
<p>要求解二阶非齐次线性微分方程, 则需要先讨论齐次的情况</p>
<p>二阶齐次线性微分方程</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;+P(x)y&#x27;+Q(x)y=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:3.08,words:923},title:"解的结构",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E4%BC%AF%E5%8A%AA%E5%88%A9%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./伯努利方程.html-BoCVYGF9.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>伯努利方程是一阶非线性微分方程, 其标准形式为:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>y</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">y&#x27;+P(x)=Q(x)y^n
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.07,words:620},title:"",type:"article",s:"伯努利方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%8F%8C%E6%9B%B2%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./双曲函数.html-CL1C4NJn.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>双曲函数的定义域是实数，其自变量的值叫做双曲角。双曲函数出现于某些重要的线性微分方程的解中，譬如说定义<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E6%82%AC%E9%93%BE%E7%BA%BF.html" target="_blank">悬链线</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E6%96%B9%E7%A8%8B.html" target="_blank">拉普拉斯方程</a>。</p>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%BC%A6.html" target="_blank">双曲正弦</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E5%8F%8C%E6%9B%B2%E4%BD%99%E5%BC%A6.html" target="_blank">双曲余弦</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%88%87.html" target="_blank">双曲正切</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E5%8F%8D%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%BC%A6.html" target="_blank">反双曲正弦</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E5%8F%8D%E5%8F%8C%E6%9B%B2%E4%BD%99%E5%BC%A6.html" target="_blank">反双曲余弦</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E5%8F%8D%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%88%87.html" target="_blank">反双曲正切</a></li>
</ul>`,readingTime:{minutes:1.58,words:475},title:"双曲函数恒等式",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%8F%AF%E5%88%86%E7%A6%BB%E5%8F%98%E9%87%8F%E7%9A%84%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./可分离变量的微分方程.html-BIAbBclL.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>微分方程的一般形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} = P(x,y)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:1.63,words:490},title:"",type:"article",s:"可分离变量的微分方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%8F%AF%E5%8C%96%E4%B8%BA%E9%BD%90%E6%AC%A1%E7%9A%84%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./可化为齐次的方程.html-BL76TFD9.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">Info</div>
</div>
<div class="callout-content"><p></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mi>y</mi><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub></mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx}=\\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:1.05,words:316},title:"",type:"article",s:"可化为齐次的方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%A4%8D%E7%B3%BB%E6%95%B0%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./复系数齐次线性微分方程.html-B6YOSVi5.js"),[]),meta:{tag:["数学"],excerpt:`<h2>复系数微分方程的特征</h2>
<p>当特征方程有复系数时，可能会得到两个不共轭的复数根，这在常规的实系数微分方程中是不会出现的。在处理这种情况时，需要考虑的因素和解的形式会有所不同。<br>
复系数的线性微分方程的形式：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>b</mi><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mi>c</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> a \\frac{d^2 y}{dx^2} + b \\frac{dy}{dx} + c y = 0 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">cy</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:6.46,words:1939},title:"",type:"article",s:"复系数齐次线性微分方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B8%B8%E7%B3%BB%E6%95%B0%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./常系数线性微分方程.html-CC7Q2cuq.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B8%B8%E7%B3%BB%E6%95%B0%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">常系数齐次线性微分方程</a></p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B8%B8%E7%B3%BB%E6%95%B0%E9%9D%9E%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">常系数非齐次线性微分方程</a></p>
`,readingTime:{minutes:.11,words:33},title:"",type:"article",s:"常系数线性微分方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B8%B8%E7%B3%BB%E6%95%B0%E9%9D%9E%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./常系数非齐次线性微分方程.html-DD81y9JB.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>二阶常系数非齐次线性微分方程的一般形式</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mi>p</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>q</mi><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;+py&#x27;+qy=f(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:6.54,words:1961},title:"待定系数法",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%B8%B8%E7%B3%BB%E6%95%B0%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./常系数齐次线性微分方程.html-BBRatd1V.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;+P(x)y&#x27;+Q(x)y=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9963em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:1.5,words:451},title:"相关链接",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5.html",{loader:()=>B(()=>import("./微分方程基本概念.html-B0HB9BCg.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>微分方程描述了一个未知函数与其<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%AF%BC%E6%95%B0%2F%E5%AF%BC%E6%95%B0.html" target="_blank">导数</a>之间的关系</p>
<ol>
<li><strong>定义</strong>：微分方程是包含未知函数及其导数的方程。如果方程中包含的是偏导数，那么这个方程被称为偏微分方程；如果只包含普通导数，则是常微分方程。</li>
<li><strong>阶数</strong>：微分方程的阶数是方程中出现的最高导数的阶数。例如，含有二阶导数的方程是二阶微分方程。</li>
<li><strong>线性与非线性</strong>：如果方程中的未知函数及其导数项的乘积仅以线性方式出现，则称为线性微分方程；否则，它是非线性的。</li>
<li><strong>解</strong>：微分方程的解是满足方程的函数。对于常微分方程，解通常是一个表达式；对于偏微分方程，解可能更为复杂。</li>
<li><strong>初始条件和边界条件</strong>：解决微分方程时，通常需要初始条件（针对常微分方程）或边界条件（针对偏微分方程）来确定特定的解。</li>
<li><strong>应用</strong>：微分方程在物理学（如牛顿定律）、工程学（如热力学和流体动力学）、生物学（如种群动态）等领域有广泛应用。</li>
<li><strong>数值方法</strong>：对于无法找到解析解的微分方程，可以使用数值方法（如欧拉法、龙格-库塔法）来近似解决。</li>
</ol>`,readingTime:{minutes:1.24,words:372},title:"",type:"article",s:"微分方程基本概念"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%82%AC%E9%93%BE%E7%BA%BF.html",{loader:()=>B(()=>import("./悬链线.html-ClRRPNJy.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>悬链线（catenary）是描述一条由自身重力影响而形成的绳索或链条悬挂在两个支点之间的形状的曲线。悬链线的方程在直角坐标系中的一般形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mi>cosh</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mfrac><mi>x</mi><mi>a</mi></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mrow><mo fence="true">(</mo><msup><mi>e</mi><mfrac><mi>x</mi><mi>a</mi></mfrac></msup><mo>+</mo><msup><mi>e</mi><mfrac><mi>x</mi><mi>a</mi></mfrac></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y = a \\cosh\\left(\\frac{x}{a}\\right) =\\frac{a}{2}\\left(e^{\\frac{x}{a}} + e^{\\frac{x}{a}}\\right)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.836em;vertical-align:-0.686em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cosh</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8971em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8971em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></p>`,readingTime:{minutes:.69,words:207},title:"",type:"article",s:"悬链线"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%9C%97%E6%96%AF%E5%9F%BA%E8%A1%8C%E5%88%97%E5%BC%8F.html",{loader:()=>B(()=>import("./朗斯基行列式.html-Cp5DDNlG.js"),[]),meta:{tag:["数学"],excerpt:`<p>朗斯基行列式（Wronskian determinant）用来判断一组函数是否线性相关，在微分方程中非常有用。</p>
<h2>定义</h2>
<p>给定<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>个可微函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y_1(x), y_2(x), \\dots, y_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，其朗斯基行列式<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(y_1, y_2, \\dots, y_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>定义为：</p>`,readingTime:{minutes:.79,words:237},title:"",type:"article",s:"朗斯基行列式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E6%AC%A7%E6%8B%89%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./欧拉方程.html-Cwi4Ry2x.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>变系数的线性微分方程，一般说来都是不容易求解的。但是有些特殊的变系数线性微分方程，则可以通过变量代换化为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2F%E5%B8%B8%E7%B3%BB%E6%95%B0%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html" target="_blank">常系数线性微分方程</a>，因而容易求解，欧拉方程就是其中的一种</p>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">欧拉方程</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:2.66,words:797},title:"求解",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E9%AB%98%E9%98%B6%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E9%99%8D%E9%98%B6.html",{loader:()=>B(()=>import("./高阶微分方程降阶.html-D6cR4edf.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>从这节开始将讨论二阶以上的微分方程.<br>
如果高阶可以降阶到一阶,就可以用一阶微分方程的求解方法<br>
以下是三个类型的容易降阶的高阶微分方程组</p>
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{(n)}=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0824em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;=f(x,y&#x27;)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;=f(y,y&#x27;)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li>
</ol>`,readingTime:{minutes:1.57,words:471},title:"",type:"article",s:"高阶微分方程降阶"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E9%AB%98%E9%98%B6%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./高阶齐次线性微分方程.html-B59cjS79.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>高阶线性微分方程</h2>
<p>有以下一般形式</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>d</mi><mi>n</mi></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>d</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a_n(x)\\frac{d^n y}{dx^n} + a_{n-1}(x)\\frac{d^{n-1} y}{dx^{n-1}} + \\cdots + a_1(x)\\frac{dy}{dx} + a_0(x)y = f(x) 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:2.22,words:665},title:"",type:"article",s:"高阶齐次线性微分方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/%E9%BD%90%E6%AC%A1%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./齐次方程.html-DdG2yyuf.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>如果一阶微分方程可化为下列形式, 那么就称这个方程为一阶齐次方程.</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mfrac><mi>y</mi><mi>x</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx}=\\phi(\\frac{y}{x})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:1.09,words:327},title:"",type:"article",s:"齐次方程"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/---%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86---.html",{loader:()=>B(()=>import("./---曲线曲面积分---.html-CBHLhaCj.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%BC%A7%E5%BE%AE%E5%88%86.html" target="_blank">弧微分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E4%BF%9D%E5%AE%88%E5%9C%BA.html" target="_blank">保守场</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%A0%87%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86.html" target="_blank">标量场的曲线积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%90%91%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86.html" target="_blank">向量场的曲线积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%A0%BC%E6%9E%97%E5%85%AC%E5%BC%8F.html" target="_blank">格林公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%A0%87%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86.html" target="_blank">标量场的曲面积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%90%91%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86.html" target="_blank">向量场的曲面积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E9%AB%98%E6%96%AF%E5%85%AC%E5%BC%8F.html" target="_blank">高斯公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E9%80%9A%E9%87%8F.html" target="_blank">通量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%95%A3%E5%BA%A6.html" target="_blank">散度</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%96%AF%E6%89%98%E5%85%8B%E6%96%AF%E5%85%AC%E5%BC%8F.html" target="_blank">斯托克斯公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E7%8E%AF%E6%B5%81%E9%87%8F.html" target="_blank">环流量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%97%8B%E5%BA%A6.html" target="_blank">旋度</a></li>
</ul>`,readingTime:{minutes:.37,words:111},title:"文档-所有文档",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E4%B8%A4%E7%B1%BB%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%E7%9A%84%E5%85%B3%E7%B3%BB.html",{loader:()=>B(()=>import("./两类曲面积分的关系.html-DsKQCYw5.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∬</mo><mi>S</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>+</mo><mi>Q</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>+</mo><mi>R</mi><mi>cos</mi><mo>⁡</mo><mi>γ</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>S</mi><mo>=</mo><mo>∬</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mi>d</mi><mi>z</mi><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>z</mi><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex"> \\iint_S (P\\cos\\alpha+Q\\cos\\beta+R\\cos\\gamma) \\, dS =\\iint P(x,y,z)dydz+Q(x,y,z)dxdz+R(x,y,z)dxdy
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2717em;vertical-align:-0.9117em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.001em;">∬</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4334em;"><span style="top:-1.7883em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9117em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.222em;vertical-align:-0.862em;"></span><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.001em;">∬</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span></span></p>`,readingTime:{minutes:.21,words:64},title:"",type:"article",s:"两类曲面积分的关系"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E4%BF%9D%E5%AE%88%E5%9C%BA.html",{loader:()=>B(()=>import("./保守场.html-narlx3uD.js"),[]),meta:{tag:["数学"],excerpt:'<p>保守场是一个特殊的向量场. 如果向量场 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> 是某个标量函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span> 的梯度，即：</p>',readingTime:{minutes:.94,words:283},title:"",type:"article",s:"保守场"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%90%91%E9%87%8F%E5%9C%BA.html",{loader:()=>B(()=>import("./向量场.html-BCh2Sf0c.js"),[]),meta:{tag:["数学"],excerpt:`<p>在空间中的每一点都分配一个向量的场。表示方向和大小，例如风速场。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>F</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>F</mi><mi>y</mi></msub><mo separator="true">,</mo><msub><mi>F</mi><mi>z</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \\mathbf{F}(x, y, z) = (F_x, F_y, F_z) 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.25,words:74},title:"",type:"article",s:"向量场"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%90%91%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./向量场的曲线积分.html-ZhrAD9NQ.js"),[]),meta:{tag:["数学"],excerpt:`<p>二维或三维空间中，沿给定曲线对向量场进行积分，计算向量场与曲线切线方向的点乘在曲线上的积分。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><mi>C</mi></msub><mi mathvariant="bold">F</mi><mo>⋅</mo><mi>d</mi><mi mathvariant="bold">r</mi></mrow><annotation encoding="application/x-tex"> \\int_C \\mathbf{F} \\cdot d\\mathbf{r} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">r</span></span></span></span></span></p>`,readingTime:{minutes:1.24,words:372},title:"",type:"article",s:"向量场的曲线积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%90%91%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./向量场的曲面积分.html-7TMvXyM7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>三维空间中，沿给定曲面对向量场进行积分，评估向量场的法向分量穿过曲面的流量。解决流量分布不均匀的流体在有向曲面上流过的体积. 这是最一般的表达形式，适用于任何形状的曲面和任意向量场</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∮</mo><mi>S</mi></msub><mi mathvariant="bold">F</mi><mo>⋅</mo><mi>d</mi><mi mathvariant="bold">S</mi></mrow><annotation encoding="application/x-tex">\\oint_S \\mathbf{F} \\cdot d\\mathbf{S} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∮</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">S</span></span></span></span></span></p>`,readingTime:{minutes:13.28,words:3983},title:"一般形式",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%90%91%E9%87%8F%E5%BE%AE%E5%88%86%E7%AE%97%E5%AD%90.html",{loader:()=>B(()=>import("./向量微分算子.html-Bq53kdWC.js"),[]),meta:{tag:["数学"],excerpt:`<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> (nabla) 是向量微分算子, 算子作用于n维空间的向量, 同样得到一个n维向量.<br>
对有正交坐标系的空间中每个维度求一阶偏导数所组成. 在三维直角坐标系中定义为:</p>`,readingTime:{minutes:.84,words:251},title:"",type:"article",s:"向量微分算子"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%90%91%E9%87%8F%E5%BE%AE%E5%88%86%E7%AE%97%E5%AD%90%E6%81%92%E7%AD%89%E5%BC%8F.html",{loader:()=>B(()=>import("./向量微分算子恒等式.html-FNHP1-10.js"),[]),meta:{author:["GPT-5"],tag:["数学","向量分析","恒等式"],excerpt:`<h2>定义</h2>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">向量微分算子的基本恒等式</div>
</div>
<div class="callout-content"><p>在连续可微的场中，梯度、旋度、散度之间满足以下恒等关系：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∇</mi><mo>×</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∇</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="2em"/><mi mathvariant="normal">∇</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></mrow><annotation encoding="application/x-tex">\\nabla \\times (\\nabla \\phi) = 0, \\qquad
\\nabla \\cdot (\\nabla \\times \\mathbf{F}) = 0.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">∇</span><span class="mord mathnormal">ϕ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">F</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.</span></span></span></span></span></p>
<p>它们表明：</p>
<ul>
<li>任何梯度场的旋度恒为零；</li>
<li>任何旋度场的散度恒为零。</li>
</ul>
</div></div>`,readingTime:{minutes:2.11,words:634},title:"",type:"article",s:"向量微分算子恒等式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E5%BC%A7%E5%BE%AE%E5%88%86.html",{loader:()=>B(()=>import("./弧微分.html-BRO6z3DP.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><mi>s</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mn>2</mn></msup></mrow></msqrt><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">ds=\\sqrt {1+{y&#x27;}^{2}}dx
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.4624em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3776em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6779em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8819em;"><span style="top:-3.1308em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.3376em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
c53.7,-170.3,84.5,-266.8,92.5,-289.5z
M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4624em;"><span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>`,readingTime:{minutes:.43,words:130},title:"弧微分公式",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E7%AE%97%E5%AD%90.html",{loader:()=>B(()=>import("./拉普拉斯算子.html-Dl605b_q.js"),[]),meta:{tag:["数学"],excerpt:'<p>拉普拉斯算子<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\nabla^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>使用<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%90%91%E9%87%8F%E5%BE%AE%E5%88%86%E7%AE%97%E5%AD%90.html" target="_blank">向量微分算子</a> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> 定义, 是其特定组合(两次应用 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> 并取<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%95%A3%E5%BA%A6.html" target="_blank">散度</a>)，表示为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\nabla^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span>。</p>',readingTime:{minutes:.47,words:140},title:"",type:"article",s:"拉普拉斯算子"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%95%A3%E5%BA%A6.html",{loader:()=>B(()=>import("./散度.html-Uj4AIpNe.js"),[]),meta:{author:["GPT-5"],tag:["数学","向量分析","公式"],excerpt:`<h2>定义</h2>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">散度（divergence）</div>
</div>
<div class="callout-content"><p>向量场的散度是该点附近体积元内向量场流出量与体积元体积之比的极限，描述场的局部“源”或“汇”的强度。<br>
散度可由向量微分算子表示为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">div</mi><mo>⁡</mo><mi mathvariant="bold">F</mi><mo>=</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi mathvariant="bold">F</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\\operatorname{div}\\mathbf{F} = \\nabla \\cdot \\mathbf{F}.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.0139em;">div</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span><span class="mord">.</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:1.26,words:377},title:"",type:"article",s:"散度"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%96%AF%E6%89%98%E5%85%8B%E6%96%AF%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./斯托克斯公式.html-zb6niMlh.js"),[]),meta:{tag:["数学","公式"],excerpt:`<h2>定义</h2>
<p>斯托克斯公式最常见的形式定义在三维空间, 将曲面上的曲线积分与向量场作用在该曲面上的旋度的面积积分联系起来。三维空间的斯托克斯公式的标准形式</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∮</mo><mrow><mi mathvariant="normal">∂</mi><mi>S</mi></mrow></msub><mi mathvariant="bold">F</mi><mo>⋅</mo><mi>d</mi><mi mathvariant="bold">r</mi><mo>=</mo><msub><mo>∬</mo><mi>S</mi></msub><mo stretchy="false">(</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mi mathvariant="bold">F</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi mathvariant="bold">S</mi></mrow><annotation encoding="application/x-tex">\\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∮</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4258em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.0576em;">S</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2717em;vertical-align:-0.9117em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.001em;">∬</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4334em;"><span style="top:-1.7883em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9117em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">F</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">S</span></span></span></span></span></p>`,readingTime:{minutes:4.47,words:1341},title:"",type:"article",s:"斯托克斯公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%97%8B%E5%BA%A6.html",{loader:()=>B(()=>import("./旋度.html-CT6Ju8hA.js"),[]),meta:{author:["Cyletix","GPT-5"],tag:["数学"],excerpt:`<h2>定义</h2>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">旋度（curl）</div>
</div>
<div class="callout-content"><p>对于向量场 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">v</mi></mrow><annotation encoding="application/x-tex">\\mathbf{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.016em;">v</span></span></span></span>，其旋转程度由旋度描述，定义为</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">curl</mi><mo>⁡</mo><mi mathvariant="bold">v</mi><mo>=</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mi mathvariant="bold">v</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\\operatorname{curl}\\mathbf{v} = \\nabla \\times \\mathbf{v}.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm">curl</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="mord">.</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:1.21,words:362},title:"",type:"article",s:"旋度"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%8E%87.html",{loader:()=>B(()=>import("./曲率.html-B7B_BboG.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>曲率K表示连续曲线在某点的弯曲程度</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>K</mi><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>s</mi><mo>→</mo><mn>0</mn></mrow></munder><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">∣</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>α</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>s</mi></mrow></mfrac><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">∣</mo><mo>=</mo><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">∣</mo><mfrac><mrow><mi>d</mi><mi>α</mi></mrow><mrow><mi>d</mi><mi>s</mi></mrow></mfrac><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">∣</mo></mrow><annotation encoding="application/x-tex">K=\\lim_{\\Delta s\\rightarrow0}\\Big|\\frac{\\Delta\\alpha}{\\Delta s}\\Big|=\\Big|\\frac{d\\alpha}{ds}\\Big|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1047em;vertical-align:-0.7443em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3557em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">s</span><span class="mrel mtight">→</span><span class="mord mtight">0</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7443em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.15em;"><span style="top:-3.15em;"><span class="pstrut" style="height:3.8em;"></span><span style="width:0.333em;height:1.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.8em" viewBox="0 0 333 1800"><path d="M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.65em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.15em;"><span style="top:-3.15em;"><span class="pstrut" style="height:3.8em;"></span><span style="width:0.333em;height:1.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.8em" viewBox="0 0 333 1800"><path d="M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.65em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.15em;"><span style="top:-3.15em;"><span class="pstrut" style="height:3.8em;"></span><span style="width:0.333em;height:1.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.8em" viewBox="0 0 333 1800"><path d="M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.65em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">s</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.15em;"><span style="top:-3.15em;"><span class="pstrut" style="height:3.8em;"></span><span style="width:0.333em;height:1.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.8em" viewBox="0 0 333 1800"><path d="M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.65em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.44,words:132},title:"",type:"article",s:"曲率"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E5%A4%96%E7%A7%AF%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./曲线外积积分.html-C7UV5Cw5.js"),[]),meta:{tag:["数学"],excerpt:`<h2>示例</h2>
<p>给定三维向量场：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mrow><mo fence="true">(</mo><mo>−</mo><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mn>4</mn></mfrac><mo fence="true">)</mo></mrow><mi mathvariant="bold">i</mi><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="bold">j</mi><mo>+</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F} = \\left(-\\frac{xy}{4}\\right)\\mathbf{i} + (z-x)\\mathbf{j} + (x+y)\\mathbf{k}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.836em;vertical-align:-0.686em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathbf">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mord mathbf">k</span></span></span></span></span></p>`,readingTime:{minutes:1.14,words:341},title:"",type:"article",s:"曲线外积积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%A0%87%E9%87%8F%E5%9C%BA.html",{loader:()=>B(()=>import("./标量场.html-DELOFOTu.js"),[]),meta:{tag:["数学"],excerpt:`<p>在空间中的每一点都分配一个标量的场。表示大小，例如温度分布。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x, y, z) 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.15,words:44},title:"",type:"article",s:"标量场"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%A0%87%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./标量场的曲线积分.html-GOjPQ4k3.js"),[]),meta:{tag:["数学"],excerpt:`
<p>用于解决不均质光滑曲线的质量问题</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><mi>L</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\\int_L f(x,y)ds
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">s</span></span></span></span></span></p>`,readingTime:{minutes:1.15,words:346},title:"同济版",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%A0%87%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./标量场的曲面积分.html-q22paRl7.js"),[]),meta:{tag:["数学"],excerpt:`<p>计算<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%A0%87%E9%87%8F%E5%9C%BA.html" target="_blank">标量场</a>在曲面上的积分, 评估曲面上函数值乘以曲面元素面积的总和, 适用于求解物理量在曲面上的分布.</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∬</mo><mi>S</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\\iint_{S} f(x, y, z) \\, dS 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2717em;vertical-align:-0.9117em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.001em;">∬</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4334em;"><span style="top:-1.7883em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">S</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9117em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span></span></p>`,readingTime:{minutes:2.27,words:681},title:"Wikipedia",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E6%A0%BC%E6%9E%97%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./格林公式.html-DYjH4kpR.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>将闭曲线上的线积分转化为由该闭曲线围成的区域上的双重积分，广泛应用于场论与流体力学。</p>
<p>若连通域D内的函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>具有一阶连续的偏导数</p>`,readingTime:{minutes:1.61,words:483},title:"",type:"article",s:"格林公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E7%8E%AF%E6%B5%81%E9%87%8F.html",{loader:()=>B(()=>import("./环流量.html-w2au8wNQ.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>环流量是指沿闭合曲线的向量场的线积分：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo>=</mo><msub><mo>∮</mo><mi>C</mi></msub><mi>v</mi><mo>⋅</mo><mi>d</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\\Gamma = \\oint_C v \\cdot dr
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2719em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∮</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span></span></p>`,readingTime:{minutes:.45,words:135},title:"",type:"article",s:"环流量"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E9%80%9A%E9%87%8F.html",{loader:()=>B(()=>import("./通量.html-B306_-Yb.js"),[]),meta:{tag:["数学"],excerpt:`<p>向量场通过曲面的流量/流动强度，表达为场的向量与面元法向量的点积的面积积分。计算通量时，需考虑向量场的方向和表面的法向方向, 一般定义由内向外为正方向。通量属于<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E5%90%91%E9%87%8F%E5%9C%BA%E7%9A%84%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86.html" target="_blank">第二类曲面积分</a></p>
<h2>公式</h2>
<p>对于向量场 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> 通过曲面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 的通量，定义为：</p>`,readingTime:{minutes:2.24,words:671},title:"",type:"article",s:"通量"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86%E5%92%8C%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%E4%B8%AD%E7%9A%84%E9%9D%A2%E7%A7%AF%E5%85%83%E7%B4%A0%E7%AD%94%E7%96%91.html",{loader:()=>B(()=>import("./重积分和曲面积分中的面积元素答疑.html-B1M1DIov.js"),[]),meta:{tag:["数学"],excerpt:`<h3>曲面参数化</h3>
<p>在三维空间中，曲面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 通常可以通过两个参数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 来描述。这意味着曲面上的每一点都可以由这两个参数唯一确定。具体地，曲面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 的参数化表示为：</p>`,readingTime:{minutes:14.19,words:4256},title:"",type:"article",s:"重积分和曲面积分中的面积元素答疑"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86/%E9%AB%98%E6%96%AF%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./高斯公式.html-B8srasLs.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>常见的形式定义在三维欧氏空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\\mathbb{R}^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 中, 描述三维向量场通过闭合曲面的通量等于场在封闭体积内散度的体积积分。<br>
它同斯托克斯定理一样是更一般的广义斯托克斯定理在三维的特例。在<a class="route-link nolebase-route-link-invalid" href="#" target="_target">-电磁学-</a>和流体动力学均有应用。<br>
严格来说，高斯散度定理也只是在三维欧氏空间中呈现出“通量–体积”这种物理意义，它在任意维度的流形上有不同几何解释。</p>`,readingTime:{minutes:1.49,words:447},title:"",type:"article",s:"高斯公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/---%E6%9E%81%E9%99%90%E4%B8%8E%E8%BF%9E%E7%BB%AD---.html",{loader:()=>B(()=>import("./---极限与连续---.html-CgHJyacY.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%98%A0%E5%B0%84%E4%B8%8E%E5%87%BD%E6%95%B0.html" target="_blank">映射与函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B%E6%80%A7.html" target="_blank">一致收敛性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%9E%81%E9%99%90.html" target="_blank">极限</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%97%A0%E7%A9%B7.html" target="_blank">无穷</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%9E%81%E9%99%90%E8%BF%90%E7%AE%97%E6%B3%95%E5%88%99.html" target="_blank">极限运算法则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%9E%81%E9%99%90%E5%AD%98%E5%9C%A8%E5%87%86%E5%88%99.html" target="_blank">极限存在准则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E8%BF%9E%E7%BB%AD%E6%80%A7.html" target="_blank">连续性</a></li>
</ol>`,readingTime:{minutes:.23,words:70},title:"文档-未连回",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B%E6%80%A7.html",{loader:()=>B(()=>import("./一致收敛性.html-UXOc5Y-R.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>一致收敛是分析数学中描述函数序列收敛行为的一个概念。与点态收敛相比，一致收敛提供了更强的收敛保证，使得收敛行为在整个定义域上是统一的。</p>
<h2>点态收敛</h2>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E7%82%B9%E6%80%81%E6%94%B6%E6%95%9B.html" target="_blank">点态收敛</a></p>
<h2>一致收敛</h2>
<p>函数序列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{f_n(x)\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)}</span></span></span></span>在集合<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>上一致收敛到函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，如果对于任意正数ε，存在自然数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow/><mi>N</mi></mrow><annotation encoding="application/x-tex">{} N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span>（依赖于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">ε</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span>但不依赖于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>），使得当<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">n≥N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span>时，对<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>中所有<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>满足</p>`,readingTime:{minutes:1.63,words:489},title:"DeepSeek-R1",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E4%B8%80%E8%87%B4%E8%BF%9E%E7%BB%AD%E6%80%A7.html",{loader:()=>B(()=>import("./一致连续性.html-DdpSzCpc.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>一致连续性</h2>
<p>一致连续性是比一般连续性更严格的条件。它不仅要求函数在某点附近连续，还要求函数在整个定义域内的连续性具有一致的标准。</p>
<h2>定义</h2>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 是一个定义在集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">D \\subseteq {R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span></span> 上的函数。如果对于任意的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\epsilon &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，存在一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\delta &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 使得对任意的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x, y \\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>，只要满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">|x - y| &lt; \\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span></span></span></span>，就有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">|f(x) - f(y)| &lt; \\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>，则称 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上一致连续。<br>
用数学符号表示为：</p>`,readingTime:{minutes:1.55,words:465},title:"",type:"article",s:"一致连续性"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E4%BB%8B%E5%80%BC%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./介值定理.html-bY-43nL3.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">介值定理</div>
</div>
<div class="callout-content"><p>零点定理的一般形式。如果函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在闭区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上连续，且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> 是介于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 之间的任意一个数，那么在开区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 内至少存在一点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.046em;">ξ</span></span></span></span>，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f(\\xi)=C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.046em;">ξ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span>。</p>
</div></div>`,readingTime:{minutes:.27,words:81},title:"",type:"article",s:"介值定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./函数极限.html-uO5OE95-.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">ε-δ定义</div>
</div>
<div class="callout-content"><p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的去心邻域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>U</mi><mo>∘</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U^\\circ(x_0,\\delta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mclose">)</span></span></span></span> 有定义：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi mathvariant="normal">∀</mi><mi>ε</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mtext> </mtext><mi mathvariant="normal">∃</mi><mi>δ</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mtext> </mtext><mn>0</mn><mo>&lt;</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>δ</mi><mo>⇒</mo><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>ε</mi></mrow><annotation encoding="application/x-tex">\\lim\\limits_{x\\to x_0}f(x)=A \\iff \\forall \\varepsilon&gt;0,\\ \\exists \\delta&gt;0,\\ 0&lt;|x-x_0|&lt;\\delta \\Rightarrow |f(x)-A|&lt;\\varepsilon
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5501em;vertical-align:-0.8001em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.37,words:110},title:"",type:"article",s:"函数极限"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%8D%95%E8%B0%83%E6%9C%89%E7%95%8C%E5%87%86%E5%88%99.html",{loader:()=>B(()=>import("./单调有界准则.html-BLz-gOyl.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p><strong>单调有界函数/数列必有极限</strong></p>
</div></div>`,readingTime:{minutes:.41,words:122},title:"",type:"article",s:"单调有界准则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%8F%8C%E6%9B%B2%E4%BD%99%E5%BC%A6.html",{loader:()=>B(()=>import("./双曲余弦.html-DLpC--OA.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">ch(x)=\\dfrac{e^{ x }+e^{ -x }}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1343em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4483em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.08,words:25},title:"",type:"article",s:"双曲余弦"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%88%87.html",{loader:()=>B(()=>import("./双曲正切.html-S9yyFg8W.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">th(x)=\\dfrac{sh(x)}{ch(x)}=\\dfrac{e^{ x }-e^{ -x }}{e^{ x }+e^{ -x }}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2177em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4483em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6973em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.1,words:31},title:"",type:"article",s:"双曲正切"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%BC%A6.html",{loader:()=>B(()=>import("./双曲正弦.html-BZoViRsb.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">sh(x)=\\dfrac{e^{ x }-e^{ -x }}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1343em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4483em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.08,words:23},title:"",type:"article",s:"双曲正弦"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E5%A4%B9%E9%80%BC%E5%87%86%E5%88%99.html",{loader:()=>B(()=>import("./夹逼准则.html-BwCaHjTu.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>若数列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">{x_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">{y_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">{z_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> 从某项起满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>x</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>z</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n \\leq x_n \\leq z_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">{y_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">{z_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> 极限为a<br>
则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{x_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 的极限存在, 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\lim _{n \\to \\infty} x_n=a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></p>
</div></div>`,readingTime:{minutes:.44,words:133},title:"",type:"article",s:"夹逼准则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./数列极限.html-BdZ8owbA.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">ε-N定义</div>
</div>
<div class="callout-content"><p>设数列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{x_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>，若存在常数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 满足：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>ε</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mtext> </mtext><mi mathvariant="normal">∃</mi><mi>N</mi><mo>∈</mo><msup><mi mathvariant="double-struck">N</mi><mo>+</mo></msup><mo separator="true">,</mo><mtext> </mtext><mi>n</mi><mo>&gt;</mo><mi>N</mi><mo>⇒</mo><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>ε</mi></mrow><annotation encoding="application/x-tex">\\forall \\varepsilon&gt;0,\\ \\exists N\\in\\mathbb{N}^+,\\ n&gt;N \\Rightarrow |x_n-a|&lt;\\varepsilon
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0158em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span></span></p>
<p>则称数列<strong>收敛于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></strong>，记作 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\\lim\\limits_{n\\to\\infty}x_n=a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>；否则称数列<strong>发散</strong>。</p>
</div></div>`,readingTime:{minutes:.32,words:97},title:"",type:"article",s:"数列极限"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%97%A0%E7%A9%B7.html",{loader:()=>B(()=>import("./无穷.html-B6GaRfqH.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>本文分为三部分</p>
<ul>
<li><a href="#%E6%97%A0%E7%A9%B7%E5%B0%8F">无穷小</a></li>
<li><a href="#%E6%97%A0%E7%A9%B7%E5%A4%A7">无穷大</a></li>
<li><a href="#%E4%B8%BB%E9%83%A8">主部</a></li>
</ul>
<hr>
<h1>无穷小</h1>
<h2>简介</h2>
<p>无穷小是极限理论的核心概念，在微积分、渐近分析中用于描述趋近于零的变化量。</p>
<h2>定义</h2>

<details class="callout" data-callout="info" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">形式化定义</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p>在自变量的某个变化过程（如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \\to x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \\to \\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span>）中：</p>
<ul>
<li>若 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim \\alpha(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则称 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\alpha(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为该过程中的<strong>无穷小量</strong></li>
<li>特别地，零函数是永恒的无穷小</li>
</ul>
</div></details>`,readingTime:{minutes:2.2,words:659},title:"无穷小",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%97%A0%E7%A9%B7%E5%B0%8F%E7%9A%84%E6%AF%94%E8%BE%83.html",{loader:()=>B(()=>import("./无穷小的比较.html-BDNz0Ted.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>定义</h2>

<details class="callout" data-callout="info" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">无穷小比较定义</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo separator="true">,</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\\alpha,\\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span> 为同一过程中的无穷小：</p>
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mi>β</mi><mi>α</mi></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim \\frac{\\beta}{\\alpha}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ⇒ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi><mo>=</mo><mi>o</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\beta=o(\\alpha)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">o</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span></span>（高阶无穷小）</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mi>β</mi><mi>α</mi></mfrac><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\\lim \\frac{\\beta}{\\alpha}=\\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> ⇒ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\beta=O(\\alpha)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span></span>（低阶无穷小）</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mi>β</mi><mi>α</mi></mfrac><mo>=</mo><mi>c</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim \\frac{\\beta}{\\alpha}=c\\neq0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ⇒ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 同阶</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mi>β</mi><msup><mi>α</mi><mi>k</mi></msup></mfrac><mo>=</mo><mi>c</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>c</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo separator="true">,</mo><mi>k</mi><mo>&gt;</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\lim \\frac{\\beta}{\\alpha^k}=c, (c\\neq0,k&gt;0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2896em;vertical-align:-0.3574em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> ⇒ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 阶无穷小</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mi>β</mi><mi>α</mi></mfrac><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\\lim \\frac{\\beta}{\\alpha}=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mop">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> ⇒ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi><mo>∼</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">\\beta \\sim \\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>（等价无穷小）</li>
</ul>
</div></details>`,readingTime:{minutes:1.22,words:365},title:"DeepSeek-R1",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%98%A0%E5%B0%84%E4%B8%8E%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./映射与函数.html-CiBhrMhn.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E6%98%A0%E5%B0%84.html%23%25E5%25AE%259A%25E4%25B9%2589" target="_blank">映射</a></p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E5%87%BD%E6%95%B0.html%23%25E5%25AE%259A%25E4%25B9%2589" target="_blank">函数</a></p>
`,readingTime:{minutes:.06,words:19},title:"",type:"article",s:"映射与函数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9C%80%E5%80%BC%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./最值定理.html-B__VgDnx.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">最值定理</div>
</div>
<div class="callout-content"><p>如果函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在闭区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上连续，则它在该区间上必定<strong>有界</strong>，并且能够取得<strong>最大值</strong>和<strong>最小值</strong>。</p>
</div></div>`,readingTime:{minutes:.2,words:60},title:"",type:"article",s:"最值定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9C%89%E7%95%8C%E6%80%A7.html",{loader:()=>B(()=>import("./有界性.html-DYnwUWD_.js"),[]),meta:{author:["SituChengxiang"],tag:["数学"],excerpt:`<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的定义域为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> ，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上有定义。<br>
如果存在数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">K1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mord">1</span></span></span></span> ，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f(x) \\leq K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对任意 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x \\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 都成立，则称函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上有上界。<br>
反之，如果存在数字 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">K_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mi>K</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f(x) \\geq K_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对任意 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x \\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 都成立，则称函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上有下界，而 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">K_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 称为函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上的一个下界。<br>
如果存在正数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> ，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">|f(x)| \\leq M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> 对任意 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x\\in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 都成立，则称函数在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上有界。如果这样的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> 不存在，就称函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 上无界；等价于，无论对于任何正数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> ，总存在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x_1 \\in X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> ，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>&gt;</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">|f(x_1)|&gt;M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> ，那么函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 上无界。<br>
此外，函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 上有界的充（分必）要条件是它在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 上既有上界也有下界。</p>`,readingTime:{minutes:.77,words:232},title:"",type:"article",s:"有界性"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./极限.html-BeKzIyEP.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`<p>本章分为三部分：</p>
<ul>
<li><a href="#%E6%95%B0%E5%88%97%E7%9A%84%E6%9E%81%E9%99%90">数列的极限</a></li>
<li><a href="#%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E9%99%90">函数的极限</a></li>
<li><a href="#%E6%9E%81%E9%99%90%E8%BF%90%E7%AE%97%E7%9A%84%E6%9C%AC%E8%B4%A8">极限运算的本质</a></li>
</ul>
<hr>
<h1>数列的极限</h1>
<p>数列极限是分析学的基础概念，通过ε-N语言严格定义收敛性，为后续研究<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E7%BA%A7%E6%95%B0%E6%94%B6%E6%95%9B.html" target="_blank">级数收敛</a>、<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E8%BF%9E%E7%BB%AD%E6%80%A7.html" target="_blank">连续性</a>奠定基础。</p>`,readingTime:{minutes:1.98,words:594},title:"数列的极限",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9E%81%E9%99%90%E5%AD%98%E5%9C%A8%E5%87%86%E5%88%99.html",{loader:()=>B(()=>import("./极限存在准则.html-BiUddADK.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>以下是判定极限存在的两个准则,以及作为应用准则的例子</p>
<hr>
<h2>准则1:<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E5%A4%B9%E9%80%BC%E5%87%86%E5%88%99.html" target="_blank">夹逼准则</a></h2>
<p>若 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mover accent="true"><mi>U</mi><mo>˚</mo></mover><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\\in \\mathring{U}(x_{0},r)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.3472em;"><span class="mord">˚</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mclose">)</span></span></span></span>或<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>&gt;</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">|x|&gt;M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span>时, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)\\leq f(x)\\leq h(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><br>
且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub><mi mathvariant="normal">/</mi><mi mathvariant="normal">∞</mi></mrow></munder><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub><mi mathvariant="normal">/</mi><mi mathvariant="normal">∞</mi></mrow></munder><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\lim_{ x \\to x_{0}/\\infty }g(x)=\\lim_{ x \\to x_{0}/\\infty }h(x)=A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.716em;vertical-align:-0.966em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.309em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">/∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.716em;vertical-align:-0.966em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.309em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">/∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span><br>
则<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub><mi mathvariant="normal">/</mi><mi mathvariant="normal">∞</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim_{ x \\to x_{0}/\\infty }f(x)=A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.716em;vertical-align:-0.966em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.309em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">/∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.966em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> , 极限存在</p>`,readingTime:{minutes:.78,words:234},title:"极限存在准则",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9E%81%E9%99%90%E8%BF%90%E7%AE%97%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./极限运算法则.html-P8ALPkL6.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>极限运算属于线性泛函，所以极限的运算满足<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%8F%AF%E5%8A%A0%E6%80%A7.html" target="_blank">可加性</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E9%BD%90%E6%AC%A1.html" target="_blank">齐次</a>性，因此可以推导出无穷小之间的运算关系。</p>
<h2>线性泛函</h2>
<p>在这之前提到过：<br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9E%81%E9%99%90%2F%E6%9E%81%E9%99%90.html%23%25E6%259E%2581%25E9%2599%2590%25E8%25BF%2590%25E7%25AE%2597%25E7%259A%2584%25E6%259C%25AC%25E8%25B4%25A8%23" target="_blank">极限</a><br>
因此，极限运算的性质应该从<strong>泛函</strong>的角度来分析。</p>`,readingTime:{minutes:2.23,words:670},title:"",type:"article",s:"极限运算法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E6%9F%AF%E8%A5%BF%E6%9E%81%E9%99%90%E5%AD%98%E5%9C%A8%E5%87%86%E5%88%99.html",{loader:()=>B(()=>import("./柯西极限存在准则.html-_FUCPotl.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>数列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{x_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 收敛的<strong>充分必要条件</strong>是:<br>
对于任意给定的正数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> ,存在正整数N, 使得当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>&gt;</mo><mi>N</mi><mo separator="true">,</mo><mi>n</mi><mo>&gt;</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">m&gt;N, n&gt;N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> 时, 有</p>`,readingTime:{minutes:.2,words:59},title:"",type:"article",s:"柯西极限存在准则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E7%82%B9%E6%80%81%E6%94%B6%E6%95%9B.html",{loader:()=>B(()=>import("./点态收敛.html-lyvpmZtZ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>函数序列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{f_n(x)\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)}</span></span></span></span>在集合D上点态收敛到函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，如果对于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>中每个<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，序列中的函数值<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>随<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>增大趋向于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</p>',readingTime:{minutes:.19,words:58},title:"",type:"article",s:"点态收敛"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E7%AC%AC%E4%B8%80%E4%B8%AA%E9%87%8D%E8%A6%81%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./第一个重要极限.html-C_O3tL71.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.11,words:33},title:"",type:"article",s:"第一个重要极限"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E7%AC%AC%E4%BA%8C%E4%B8%AA%E9%87%8D%E8%A6%81%E6%9E%81%E9%99%90.html",{loader:()=>B(()=>import("./第二个重要极限.html-C-m6MkXj.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<h2>定义</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><msup><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">)</mo><mi>x</mi></msup><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\\lim _{x \\to \\infty} \\Big( 1 + \\frac{1}{x}\\Big)^x=e
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.85em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="delimsizing size2">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2043em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></span></p>`,readingTime:{minutes:1.28,words:385},title:"推论1",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E7%BA%A7%E6%95%B0%E6%94%B6%E6%95%9B.html",{loader:()=>B(()=>import("./级数收敛.html-BBzEE-dE.js"),[]),meta:{tag:["数学"],readingTime:{minutes:.03,words:8},title:"",type:"article",s:"级数收敛"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E8%B4%B9%E9%A9%AC%E5%BC%95%E7%90%86.html",{loader:()=>B(()=>import("./费马引理.html-DAkpKDxK.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">必要条件</div>
</div>
<div class="callout-content"><p>设函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>点可导 ，且在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>处取得极值 。则<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><msup><mrow/><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f^{&#x27;}(x_{0})=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1925em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9425em;"><span style="top:-2.9425em;margin-right:0.05em;"><span class="pstrut" style="height:2.5795em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8278em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p>
<ul>
<li>反之不一定,如<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">y=x^{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span>,在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>导数为0,但不是极值。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">y= \\lvert x \\rvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∣</span><span class="mord mathnormal">x</span><span class="mclose">∣</span></span></span></span> 在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>处不可导,但<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>处取极小值。</li>
</ul>
</div></div>`,readingTime:{minutes:.26,words:79},title:"",type:"article",s:"费马引理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E8%BF%9E%E7%BB%AD%E6%80%A7.html",{loader:()=>B(()=>import("./连续性.html-DgteEl92.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`
<h2>简介</h2>
<p>连续性是函数的一种基本性质，直观地描述了函数图像没有“中断”或“跳跃”。如果一个函数的图像可以用一笔画出，没有任何间断，那么它就是连续的。这一概念是微分学乃至整个微积分学的基石。</p>
<h2>定义</h2>
<h3>点的连续性 (Continuity at a Point)</h3>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的某一邻域内有定义。函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 连续，有以下三种等价的定义形式：</p>`,readingTime:{minutes:3.15,words:944},title:"连续性",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E9%82%BB%E5%9F%9F.html",{loader:()=>B(()=>import("./邻域.html-Czg84gAc.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ul>
<li><strong>δ邻域</strong>：<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><mi>δ</mi><mo separator="true">,</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(x_0,\\delta)=(x_0-\\delta,x_0+\\delta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mclose">)</span></span></span></span></li>
<li><strong>去心邻域</strong>：<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>U</mi><mo>∘</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>∖</mo><mo stretchy="false">{</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U^\\circ(x_0,\\delta)=U(x_0,\\delta)\\setminus\\{x_0\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span></li>
</ul>`,readingTime:{minutes:.14,words:41},title:"",type:"article",s:"邻域"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E6%9E%81%E9%99%90/%E9%9B%B6%E7%82%B9%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./零点定理.html-dgYIMfFC.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">零点定理</div>
</div>
<div class="callout-content"><p>如果函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在闭区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上连续，且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 异号（即 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(a) \\cdot f(b) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），那么在开区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 内至少存在一点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.046em;">ξ</span></span></span></span>，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(\\xi)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.046em;">ξ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。</p>
</div></div>`,readingTime:{minutes:.22,words:67},title:"",type:"article",s:"零点定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/---%E7%A7%AF%E5%88%86---.html",{loader:()=>B(()=>import("./---积分---.html-lcUMq1iA.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86.html" target="_blank">不定积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E5%AE%9A%E7%A7%AF%E5%88%86.html" target="_blank">定积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%E8%A1%A8.html" target="_blank">积分表</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%E4%B8%8A%E9%99%90%E7%9A%84%E5%87%BD%E6%95%B0%E5%8F%8A%E5%85%B6%E5%AF%BC%E6%95%B0.html" target="_blank">积分上限的函数及其导数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E7%89%9B%E9%A1%BF-%E8%8E%B1%E5%B8%83%E5%B0%BC%E8%8C%A8%E5%85%AC%E5%BC%8F.html" target="_blank">牛顿-莱布尼茨公式</a></li>
</ul>`,readingTime:{minutes:.39,words:116},title:"不定积分",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/ILATE%20%E8%A7%84%E5%88%99.html",{loader:()=>B(()=>import("./ILATE 规则.html-CSjSeVUv.js"),[]),meta:{tag:["数学"],excerpt:'<p>分部积分中, 在选择哪个函数作 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 时, 有一个被广泛使用的经验法则, 称为 <strong>ILATE 规则</strong>. 该规则为选择 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 的函数类型设定了优先级, 排在前面的函数求导后形式会变得更简单 (变为代数式) , 而排在后面的函数 (如指数, 三角函数) 积分后形式依然简单且可控.</p>',readingTime:{minutes:.73,words:220},title:"",type:"article",s:"ILATE 规则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./不定积分.html-JX2qDqoK.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<p>不定积分是微积分中的核心概念之一，主要涉及求函数的原函数。在微积分中，不定积分与导数是互逆的过程。若函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的原函数，则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的导数等于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。数学上表示为：</p>`,readingTime:{minutes:.68,words:203},title:"不定积分概述",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E4%BC%BD%E9%A9%AC%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./伽马函数.html-SBYKbbyk.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner"><strong>定义:</strong></div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.53,words:160},title:"伽马函数",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95.html",{loader:()=>B(()=>import("./分部积分法.html-E_-GVnEK.js"),[]),meta:{author:["Cyletix","Wikipedia","Gemini"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>分部积分法, 是求解积分问题的一种重要方法. 它源于函数乘积的求导法则, 常用于对两个不同类型函数乘积的积分. 通过该方法, 可以将一个复杂的积分问题转化为一个更简单, 可直接求解的积分问题.</p>
<p>其思想核心是将待积函数看作一个函数的导数与另一个函数的乘积, 从而进行转化. 在离散的级数分析中, 其对应的方法称为<strong>分部求和</strong> (Summation by parts) .</p>
<h2>定义</h2>
<p>分部积分法包括不定积分和定积分两种形式<br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E5%85%AC%E5%BC%8F.html" target="_blank">分部积分法公式</a></p>`,readingTime:{minutes:2.31,words:693},title:"",type:"article",s:"分部积分法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./分部积分法公式.html-Dydi8hDs.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>若函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u=u(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v=v(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 连续可导, 则：<br>
<strong>不定积分形式</strong></p>`,readingTime:{minutes:.19,words:58},title:"",type:"article",s:"分部积分法公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E7%9A%84%E6%8E%A8%E8%AE%BA.html",{loader:()=>B(()=>import("./分部积分的推论.html-D2qoiZ9M.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><msup><mi>e</mi><mrow><mi>m</mi><mi>t</mi></mrow></msup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><msup><mi>e</mi><mrow><mi>m</mi><mi>t</mi></mrow></msup><mfrac><mrow><mi>m</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\int e^{mt} \\cos(nt) dt = e^{mt}\\frac{m\\cos(nt)+n\\sin(nt)}{m^2 + {n^2}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1963em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:1.44,words:432},title:"余弦-指数",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%8C%BA%E9%97%B4%E5%86%8D%E7%8E%B0.html",{loader:()=>B(()=>import("./区间再现.html-6mZI6ty1.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>定积分问题常用技巧</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\int_a^bf(x)dx=\\int_a^bf(a+b-x)dx
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>`,readingTime:{minutes:.15,words:46},title:"",type:"article",s:"区间再现"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%8F%8D%E5%8F%8C%E6%9B%B2%E4%BD%99%E5%BC%A6.html",{loader:()=>B(()=>import("./反双曲余弦.html-DT9jMDm1.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mi>r</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=arch(x)=\\ln(x+\\sqrt{ x^{2}-1 })
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">c</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3123em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0623em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.0223em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">t</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.1,words:30},title:"",type:"article",s:"反双曲正切"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%8F%8D%E5%8F%8C%E6%9B%B2%E6%AD%A3%E5%BC%A6.html",{loader:()=>B(()=>import("./反双曲正弦.html-ZTQlXv6i.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mi>r</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=arsh(x)=\\ln(x+\\sqrt{ x^{2}+1 })
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3123em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0623em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.0223em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E6%97%A0%E7%A9%B7%E9%99%90%E7%9A%84%E5%8F%8D%E5%B8%B8%E7%A7%AF%E5%88%86.html" target="_blank">无穷限的反常积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E6%97%A0%E7%95%8C%E5%87%BD%E6%95%B0%E7%9A%84%E5%8F%8D%E5%B8%B8%E7%A7%AF%E5%88%86.html" target="_blank">无界函数的反常积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%A7%AF%E5%88%86%2F%E4%BC%BD%E9%A9%AC%E5%87%BD%E6%95%B0.html" target="_blank">伽马函数</a></li>
</ol>
`,readingTime:{minutes:.12,words:36},title:"",type:"article",s:"反常积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%AE%9A%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./定积分.html-C-bifOBf.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<p>主要用于计算函数在特定区间上的“总效应”，比如面积、体积或其他累积量。定积分是对不定积分概念的扩展，涉及到具体的积分区间。数学上，定积分表示为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\int_a^b f(x)\\,dx
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>`,readingTime:{minutes:.45,words:134},title:"定积分概述",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%AE%9A%E7%A7%AF%E5%88%86%E6%8D%A2%E5%85%83.html",{loader:()=>B(()=>import("./定积分换元.html-CMg1f9G4.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mi>α</mi><mi>β</mi></msubsup><mi>f</mi><mo stretchy="false">[</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\\int _{a}^{b}f(x) \\, dx=\\int _{\\alpha}^{\\beta}f[\\phi(t)]\\phi&#x27;(t) \\, dt  
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0528em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">[</span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)]</span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></p>`,readingTime:{minutes:1.78,words:534},title:"",type:"article",s:"定积分换元"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E5%B8%B8%E8%A7%81%E7%A7%AF%E5%88%86%E5%BD%A2%E5%BC%8F%E5%A4%84%E7%90%86.html",{loader:()=>B(()=>import("./常见积分形式处理.html-Da81p1-k.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>定积分计算中常见的积分形式以及处理方式总结：</p>
<h3>1. <strong>三角函数的平方</strong></h3>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mspace width="1em"/><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\\cos^2\\theta = \\frac{1 + \\cos(2\\theta)}{2}, \\quad \\sin^2\\theta = \\frac{1 - \\cos(2\\theta)}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:2.02,words:607},title:"",type:"article",s:"常见积分形式处理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E6%97%A0%E7%95%8C%E5%87%BD%E6%95%B0%E7%9A%84%E5%8F%8D%E5%B8%B8%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./无界函数的反常积分.html-DgmKRPa2.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>无界函数的反常积分涉及在积分区间或被积函数中存在无界性的情况。</p>
<h3>定义</h3>
<p>对于函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，如果在某个区间<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span>内的某一点<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a &lt; c &lt; b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>）上函数无界，即<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mi>c</mi></mrow></msub><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\\lim_{x \\to c} |f(x)| = \\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mathnormal mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span>，那么在这个区间的积分被称为无界函数的反常积分。</p>`,readingTime:{minutes:1.13,words:339},title:"",type:"article",s:"无界函数的反常积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E6%97%A0%E7%A9%B7%E9%99%90%E7%9A%84%E5%8F%8D%E5%B8%B8%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./无穷限的反常积分.html-CEvGD2ij.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msubsup><mo>∫</mo><mi>a</mi><mi>t</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\lim_{ t \\to \\infty }\\int_{a}^{t}f(x)\\, dx  
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4554em;vertical-align:-0.9119em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5435em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>`,readingTime:{minutes:.26,words:78},title:"",type:"article",s:"无穷限的反常积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E6%B2%83%E5%88%A9%E6%96%AF%E4%B9%98%E7%A7%AF.html",{loader:()=>B(()=>import("./沃利斯乘积.html-nxDPid79.js"),[]),meta:{tag:["数学","公式"],excerpt:`<h2>定义</h2>

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">沃利斯积 (Wallis Product)</div>
</div>
<div class="callout-content"><p>该公式给出了圆周率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span> 的一种无穷乘积表示形式：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>=</mo><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>⋅</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>⋅</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>⋅</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>⋅</mo><mfrac><mn>6</mn><mn>5</mn></mfrac><mo>⋅</mo><mfrac><mn>6</mn><mn>7</mn></mfrac><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\\frac{\\pi}{2} = \\prod_{n=1}^{\\infty} \\frac{2n}{2n-1} \\cdot \\frac{2n}{2n+1} = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></span></p>
<p>也可以写作：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>=</mo><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\pi}{2} = \\prod_{n=1}^{\\infty} \\frac{4n^2}{4n^2 - 1} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:1.4,words:420},title:"",type:"article",s:"沃利斯乘积"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E6%B2%83%E5%88%A9%E6%96%AF%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./沃利斯积分.html-DCpTxsQ2.js"),[]),meta:{author:["Cyletix","Gemini","Wikipedia"],tag:["公式","数学"],excerpt:`<h2>简介</h2>
<p>沃利斯公式（Wallis Formula），在中文教材中常因其递推计算的形式被称为<strong>点火公式</strong>。该名称通常指代两个密切相关的数学成果：</p>
<ol>
<li><strong>沃利斯积分</strong>：一个计算正弦函数（或余弦函数）高次幂在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, \\frac{\\pi}{2}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">]</span></span></span></span> 区间上定积分的递推公式。</li>
<li><strong>沃利斯积</strong>：一个通过无穷乘积来表示圆周率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span> 的著名公式。</li>
</ol>`,readingTime:{minutes:1.91,words:574},title:"",type:"article",s:"沃利斯积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E6%B5%B7%E7%BB%B4%E8%B5%9B%E9%81%AE%E7%9B%96%E6%B3%95.html",{loader:()=>B(()=>import("./海维赛遮盖法.html-fv5kmMbT.js"),[]),meta:{author:["Cyletix"],tag:["数学","代数"],excerpt:`
<div class="callout" data-callout="tip">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-flame"><path d="M8.5 14.5A2.5 2.5 0 0 0 11 12c0-1.38-.5-2-1-3-1.072-2.143-.224-4.054 2-6 .5 2.5 2 4.9 4 6.5 2 1.6 3 3.5 3 5.5a7 7 0 1 1-14 0c0-1.153.433-2.294 1-3a2.5 2.5 0 0 0 2.5 2.5z"/></svg>
</div>
<div class="callout-title-inner">tip</div>
</div>
<div class="callout-content"><p></p>
<p>这不是留数定理, 留数定理属于复分析求积分相关内容</p>
</div></div>`,readingTime:{minutes:1.15,words:345},title:"",type:"article",s:"海维赛遮盖法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E7%89%9B%E9%A1%BF-%E8%8E%B1%E5%B8%83%E5%B0%BC%E8%8C%A8%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./牛顿-莱布尼茨公式.html-2z16DLMs.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\int _{a}^{b}f(x) \\, dx=F(b)-F(a) 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.511em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.22,words:67},title:"",type:"article",s:"牛顿-莱布尼茨公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86%E4%B8%8A%E9%99%90%E7%9A%84%E5%87%BD%E6%95%B0%E5%8F%8A%E5%85%B6%E5%AF%BC%E6%95%B0.html",{loader:()=>B(()=>import("./积分上限的函数及其导数.html-zDmT90g6.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在区间<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span>上连续, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x\\in[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span>, 考虑如下定积分</p>',readingTime:{minutes:.5,words:150},title:"",type:"article",s:"积分上限的函数及其导数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86%E8%A1%A8.html",{loader:()=>B(()=>import("./积分表.html-CQqA8RB2.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>幂指函数</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mi>k</mi><mi>d</mi><mi>x</mi><mo>=</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\\int k dx = kx + C
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span></span></p>`,readingTime:{minutes:1.46,words:437},title:"",type:"article",s:"积分表"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E7%AC%AC%E4%B8%80%E7%B1%BB%E6%8D%A2%E5%85%83%E6%B3%95.html",{loader:()=>B(()=>import("./第一类换元法.html-JKyYvBm0.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`<h2>简介</h2>
<p><strong>第一类换元积分法</strong>，通常也被称为 <strong>u-换元法</strong>（u-substitution）或<strong>凑微分法</strong>，是积分计算中最基本的方法。其本质是链式求导法则的逆运算，通过引入一个中间变量，将被积函数转化为更简单、可以直接使用基本积分公式的形式。<br>
形如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\int f(g(x)) g&#x27;(x) \\, dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span> 的积分。通过识别出复合函数的“内层” <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 及其导数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g&#x27;(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，我们可以简化积分过程。</p>`,readingTime:{minutes:2.63,words:788},title:"",type:"article",s:"第一类换元法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E7%AC%AC%E4%BA%8C%E7%B1%BB%E6%8D%A2%E5%85%83%E6%B3%95.html",{loader:()=>B(()=>import("./第二类换元法.html-WNHjuzzw.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:'<p>第二类换元法的主要思路是将 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span> 换元后里面的内容提取到外面, 得到容易求得的积分后再将变量换回<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>.</p>',readingTime:{minutes:1.23,words:370},title:"经典类型",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%A7%AF%E5%88%86/%E9%AB%98%E6%96%AF%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./高斯积分.html-y2biIAPU.js"),[]),meta:{tag:["数学","公式"],excerpt:`<h2>标准形式</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\\int_{-\\infty}^{+\\infty} e^{-x^2} \\, dx = \\sqrt{\\pi}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1908em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8492em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span><span style="top:-2.8092em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1908em;"><span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.09,words:327},title:"",type:"article",s:"高斯积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/---%E7%BA%A7%E6%95%B0---.html",{loader:()=>B(()=>import("./---级数---.html-DZPN4yL-.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%B8%B8%E6%95%B0%E9%A1%B9%E7%BA%A7%E6%95%B0.html" target="_blank">常数项级数</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">常数项级数审敛法</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E8%B0%83%E5%92%8C%E7%BA%A7%E6%95%B0.html" target="_blank">调和级数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%87%BD%E6%95%B0%E9%A1%B9%E7%BA%A7%E6%95%B0.html" target="_blank">函数项级数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%B9%82%E7%BA%A7%E6%95%B0.html" target="_blank">幂级数</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">函数展开成幂级数</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">微分方程的幂级数解法</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E6%AC%A7%E6%8B%89%E5%85%AC%E5%BC%8F.html" target="_blank">欧拉公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%87%A0%E4%BD%95%E7%BA%A7%E6%95%B0.html" target="_blank">几何级数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0.html" target="_blank">傅里叶级数</a></li>
</ul>`,readingTime:{minutes:1.52,words:455},title:"目录",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/p-%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./p-级数.html-CnXChjWp.js"),[]),meta:{tag:["数学"],excerpt:'<p><strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span>-级数</strong> 是一种特殊形式的无穷级数，它的通项由正整数的幂倒数构成，形式如下：</p>',readingTime:{minutes:1.26,words:377},title:"",type:"article",s:"p-级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./傅里叶级数.html-Cs6G0QEq.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h3>傅立叶级数的三角形式</h3>
<p>周期为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span></span></span></span> 的函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 可以使用傅立叶级数表示为：</p>`,readingTime:{minutes:6.41,words:1924},title:"",type:"article",s:"傅里叶级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%85%AC%E5%BC%8F-%E6%B3%B0%E5%8B%92%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./公式-泰勒级数.html-aETy4yVq.js"),[]),meta:{tag:["数学","公式"],excerpt:'<p>如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处无穷可导且余项 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mi>R</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle\\lim_{n \\to \\infty} R_n(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.45em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则函数可展开为泰勒级数：</p>',readingTime:{minutes:.19,words:57},title:"",type:"article",s:"公式-泰勒级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%87%A0%E4%BD%95%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./几何级数.html-DH4CU5Qq.js"),[]),meta:{author:["Cyletix","DeepSeekV3","GPT-4o","Wikipedia"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>几何级数是无穷多个项的总和，这些连续项之间的公比<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span>是恒定的.</p>`,readingTime:{minutes:2.56,words:769},title:"",type:"article",s:"几何级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%87%A0%E4%BD%95%E7%BA%A7%E6%95%B0%E7%A4%BA%E4%BE%8B.html",{loader:()=>B(()=>import("./几何级数示例.html-x4u0nivu.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(x) = \\frac{1}{1-x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 是一个<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%87%A0%E4%BD%95%E7%BA%A7%E6%95%B0.html" target="_blank">几何级数</a>，可以通过其基本性质直接求出其麦克劳林展开。麦克劳林展开实际上是泰勒展开在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 处的特殊情形。对于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1}{1-x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，我们可以通过直接将其视为无穷级数来进行展开，前提是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|x| &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 以确保级数收敛。</p>',readingTime:{minutes:1.46,words:438},title:"",type:"article",s:"几何级数示例"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%87%BD%E6%95%B0%E9%A1%B9%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./函数项级数.html-BsOCRqPz.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>相比泰勒级数, 函数项级数是更广义的概念</p>
<h2>定义</h2>
<p>函数项级数是由函数组成的级数，将一列函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">f_0(x), f_1(x), \\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span> 相加，得到的表达式称为函数项级数。其一般形式为：</p>`,readingTime:{minutes:1.51,words:453},title:"",type:"article",s:"函数项级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%8D%B7%E7%A7%AF.html",{loader:()=>B(()=>import("./卷积.html-BmJnkKtW.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>卷积描述了两个函数(信号)如何通过叠加来影响彼此, 是两个函数的二元算子, 使用<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span>表示</p>',readingTime:{minutes:.74,words:221},title:"",type:"article",s:"卷积"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%8F%8C%E9%98%B6%E4%B9%98.html",{loader:()=>B(()=>import("./双阶乘.html-B9ppIstN.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学"],excerpt:`<h2>定义</h2>
<ul>
<li>对于奇数的双阶乘 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">!</mo><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">(2n-1)!!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)!!</span></span></span></span>：</li>
</ul>`,readingTime:{minutes:1.63,words:489},title:"",type:"article",s:"双阶乘"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%B8%B8%E6%95%B0%E9%A1%B9%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./常数项级数.html-CScGUYdM.js"),[]),meta:{tag:["数学"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E7%AD%89%E6%AF%94%E7%BA%A7%E6%95%B0.html" target="_blank">等比级数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2Fp-%E7%BA%A7%E6%95%B0.html" target="_blank">p-级数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E8%B0%83%E5%92%8C%E7%BA%A7%E6%95%B0.html" target="_blank">调和级数</a></li>
<li></li>
</ul>
`,readingTime:{minutes:.06,words:17},title:"",type:"article",s:"常数项级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%B9%82%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./幂级数.html-BNDCRj-C.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>幂级数是一种特殊的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%87%BD%E6%95%B0%E9%A1%B9%E7%BA%A7%E6%95%B0.html" target="_blank">函数项级数</a>, 每一项都是函数自变量取值下的函数值，其形式为多项式的无限推广</p>
<h2>定义</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\sum_{n=0}^\\infty a_n (x - x_0)^n
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.54,words:762},title:"",type:"article",s:"幂级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%B9%82%E7%BA%A7%E6%95%B0%E8%BF%90%E7%AE%97%E8%A7%84%E5%88%99.html",{loader:()=>B(()=>import("./幂级数运算规则.html-DmsNIrKo.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h3>1. 加法与减法</h3>
<p>给定两个幂级数：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\sum_{n=0}^{\\infty} a_n (x-c)^n
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.3,words:391},title:"",type:"article",s:"幂级数运算规则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E5%B9%BF%E4%B9%89%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./广义二项式定理.html-Dywfrzis.js"),[]),meta:{tag:["数学"],excerpt:`<h2>简介</h2>
<p>广义二项式定理是普通<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86.html" target="_blank">二项式定理</a>的推广，可视作<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%87%A0%E4%BD%95%E7%BA%A7%E6%95%B0.html" target="_blank">几何级数</a>在任意实指数下的形式化延伸。适用于幂指数为任意实数或复数的情况。在数学分析和无穷级数领域中，这是一个重要的工具，尤其在处理类似 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(1 - u)^{-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span> 这种高次幂分母时。</p>`,readingTime:{minutes:2.17,words:651},title:"",type:"article",s:"广义二项式定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E6%94%B6%E6%95%9B%E5%8D%8A%E5%BE%84%E8%AE%A1%E7%AE%97.html",{loader:()=>B(()=>import("./收敛半径计算.html-7h-k1JWM.js"),[]),meta:{tag:["数学"],excerpt:'<p>计算使级数收敛的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的范围通常涉及找出该级数的收敛半径和收敛区间。对于幂级数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\sum_{n=0}^{\\infty} a_n x^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，我们可以使用以下方法来确定其收敛性：</p>',readingTime:{minutes:1.99,words:596},title:"",type:"article",s:"收敛半径计算"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E6%AC%A7%E6%8B%89%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./欧拉公式.html-DjtJYyz4.js"),[]),meta:{author:["Zacharia2","Cyletix"],tag:["数学","公式"],excerpt:`<p>欧拉公式是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%A4%8D%E5%88%86%E6%9E%90%2F--%E5%A4%8D%E5%88%86%E6%9E%90--.html" target="_blank">--复分析--</a>中的一个基本公式，它建立了<a class="route-link nolebase-route-link-invalid" href="#" target="_target">复指数函数</a>与<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0.html" target="_blank">三角函数</a>之间的联系</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex"> e^{i\\theta} = \\cos\\theta + i\\sin\\theta 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">θ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span></span></p>`,readingTime:{minutes:.73,words:218},title:"",type:"article",s:"欧拉公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E6%B3%B0%E5%8B%92%E5%B1%95%E5%BC%80%E4%B8%AD%E7%9A%84%E5%8F%8C%E9%98%B6%E4%B9%98.html",{loader:()=>B(()=>import("./泰勒展开中的双阶乘.html-Bgp5M2R7.js"),[]),meta:{author:["Cyletix","Gemini2.5Pro"],tag:["数学"],excerpt:'<p>在<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%B9%BF%E4%B9%89%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86.html" target="_blank">广义二项式定理</a>中，当指数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 为分数（特别是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\\pm 1/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">±</span><span class="mord">1/2</span></span></span></span>）时，其泰勒展开式的系数会系统性地产生<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E7%BA%A7%E6%95%B0%2F%E5%8F%8C%E9%98%B6%E4%B9%98.html" target="_blank">双阶乘</a>。</p>',readingTime:{minutes:1.1,words:331},title:"",type:"article",s:"泰勒展开中的双阶乘"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E6%B3%B0%E5%8B%92%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./泰勒级数.html-CVQ2fMNK.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学","公式"],excerpt:`<h2>简介</h2>
<p>泰勒级数是一种特殊的幂级数, 泰勒级数将一个函数表示为无穷幂级数的形式, 其系数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 由函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的导数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{f^{(n)}(x_0)}{n!}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5067em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1617em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mclose mtight">!</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9667em;"><span style="top:-2.9667em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5357em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 确定. 如果一个函数在某点无限可导, 并且其泰勒公式的余项在该点附近收敛于零, 那么这个函数就可以由其泰勒级数完全表示.</p>`,readingTime:{minutes:1.17,words:351},title:"",type:"article",s:"泰勒级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E7%AD%89%E6%AF%94%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./等比级数.html-CaryGA-m.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>由<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%2F%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97.html" target="_blank">等比数列</a>定义的级数</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>a</mi><msup><mi>q</mi><mi>i</mi></msup><mo>=</mo><mi>a</mi><mo>+</mo><mi>a</mi><mi>q</mi><mo>+</mo><mi>a</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>a</mi><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\\sum_{i=0}^\\infty aq^i=a+aq+aq^2+\\cdots+aq^{n-1}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.19,words:58},title:"",type:"article",s:"等比级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E7%BA%A7%E6%95%B0%E6%94%B6%E6%95%9B%E6%9D%A1%E4%BB%B6.html",{loader:()=>B(()=>import("./级数收敛条件.html-C7KB1leg.js"),[]),meta:{tag:["数学"],excerpt:'<p>级数收敛的条件指的是判定一个级数是否收敛的不同方法和准则。一个级数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\\sum_{n=1}^{\\infty} a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的收敛性取决于其部分和序列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">S_n = \\sum_{k=1}^{n} a_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是否收敛到一个有限值。以下是一些常用的判别准则：</p>',readingTime:{minutes:2.08,words:625},title:"",type:"article",s:"级数收敛条件"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E7%BA%A7%E6%95%B0/%E8%B0%83%E5%92%8C%E7%BA%A7%E6%95%B0.html",{loader:()=>B(()=>import("./调和级数.html-ynrIgeGf.js"),[]),meta:{tag:["数学"],excerpt:`<h2>背景</h2>
<p>从直观上理解，一个级数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\\sum a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的收敛性至少要求其通项 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 趋近于0，即<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lim_{n \\to \\infty} a_n = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，但这只是<strong>必要条件</strong>而不是充分条件。也就是说，通项趋近于0并不保证级数一定收敛，调和级数就是一个非常典型的反例。</p>`,readingTime:{minutes:2.62,words:786},title:"",type:"article",s:"调和级数"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/---%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95---.html",{loader:()=>B(()=>import("./---解析几何---.html-2pmIbqE_.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F.html" target="_blank">向量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%90%91%E9%87%8F%E8%BF%90%E7%AE%97.html" target="_blank">向量运算</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%A9%BA%E9%97%B4%E5%B9%B3%E9%9D%A2.html" target="_blank">空间平面</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%A9%BA%E9%97%B4%E7%9B%B4%E7%BA%BF.html" target="_blank">空间直线</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%A9%BA%E9%97%B4%E6%9B%B2%E9%9D%A2.html" target="_blank">空间曲面</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%A9%BA%E9%97%B4%E6%9B%B2%E7%BA%BF.html" target="_blank">空间曲线</a></li>
</ul>`,readingTime:{minutes:.24,words:72},title:"文档-所有文档",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%86%85%E7%A7%AF.html",{loader:()=>B(()=>import("./内积.html-DnyoT1zl.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F.html" target="_blank">向量</a>与向量的二元运算, 映射结果到实数</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>a</mi><mo>⃗</mo></mover><mo>⋅</mo><mover accent="true"><mi>b</mi><mo>⃗</mo></mover><mo>=</mo><mo>∣</mo><mover accent="true"><mi>a</mi><mo>⃗</mo></mover><mo>∣</mo><mo>∣</mo><mover accent="true"><mi>b</mi><mo>⃗</mo></mover><mo>∣</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\vec{a}\\cdot \\vec{b}=\\mid \\vec{a}\\mid\\mid \\vec{b}\\mid \\cos\\theta
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2274em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">b</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2274em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">b</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span></span></p>`,readingTime:{minutes:.43,words:129},title:"内积",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%88%87%E5%B9%B3%E9%9D%A2%E7%9A%84%E6%B3%95%E5%90%91%E9%87%8F.html",{loader:()=>B(()=>import("./切平面的法向量.html-BMdWWM1K.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>三维空间中的隐函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F(x, y, z) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 定义了一个二维曲面。在曲面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span> 上某点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span> 的<strong>切平面</strong>是曲面在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span> 附近的最佳线性逼近。该切平面的法向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>n</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\\vec{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.714em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">n</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span> 平行于<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E6%A2%AF%E5%BA%A6.html" target="_blank">梯度</a> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\\nabla F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span>，梯度指向函数值变化最快的方向，并且正交于切平面。</p>`,readingTime:{minutes:1.27,words:382},title:"",type:"article",s:"切平面的法向量"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%8F%B3%E6%89%8B%E8%9E%BA%E6%97%8B%E5%AE%9A%E5%88%99.html",{loader:()=>B(()=>import("./右手螺旋定则.html-B4N_e-I_.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>右手四指握紧，拇指伸直，形成由<strong>旋向</strong>和<strong>轴向</strong>两个特征完全确定的圆柱形</p>
<ol>
<li>旋向：圆柱正方向为，四指旋向为起点在圆柱轴线上的向量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1\\rightarrow n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>轴向：拇指伸直的方向为轴线正方向</li>
</ol>`,readingTime:{minutes:.27,words:82},title:"",type:"article",s:"右手螺旋定则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%90%91%E9%87%8F%E8%BF%90%E7%AE%97.html",{loader:()=>B(()=>import("./向量运算.html-CuM-Zbq6.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%86%85%E7%A7%AF.html%23%25E5%2586%2585%25E7%25A7%25AF" target="_blank">内积</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%A4%96%E7%A7%AF.html%23%25E5%25A4%2596%25E7%25A7%25AF" target="_blank">外积</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E6%B7%B7%E5%90%88%E7%A7%AF.html%23%25E6%25B7%25B7%25E5%2590%2588%25E7%25A7%25AF" target="_blank">混合积</a></p>
`,readingTime:{minutes:.08,words:24},title:"",type:"article",s:"向量运算"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E5%A4%96%E7%A7%AF.html",{loader:()=>B(()=>import("./外积.html-BOcviHSH.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>n维<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>中(<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n\\geq 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>), <a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F.html" target="_blank">向量</a>与向量的二元运算, 映射结果到<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>，得到一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">n(n-1)/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">/2</span></span></span></span> 维度的新向量(<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>时得到的实际上是一个标量)<br>
外积和内积一样依赖于欧几里德空间的度量，但与内积之不同的是，外积还依赖于定向或右手定则</p>`,readingTime:{minutes:1.92,words:577},title:"外积",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E6%B7%B7%E5%90%88%E7%A7%AF.html",{loader:()=>B(()=>import("./混合积.html-CtfYmXtk.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>混合积</h2>
<p>混合积是指三个向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">{c}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span></span> 的数量积。它通常用来计算三个向量所定义的平行六面体的体积。混合积的定义如下：</p>`,readingTime:{minutes:2.04,words:612},title:"示例",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E7%A9%BA%E9%97%B4%E5%B9%B3%E9%9D%A2.html",{loader:()=>B(()=>import("./空间平面.html-DioWD-cx.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>一般形式</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mi>x</mi><mo>+</mo><mi>B</mi><mi>y</mi><mo>+</mo><mi>C</mi><mi>z</mi><mo>+</mo><mi>D</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F(x,y,z)=Ax+By+Cz+D=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:.87,words:262},title:"",type:"article",s:"空间平面"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E7%A9%BA%E9%97%B4%E6%9B%B2%E7%BA%BF.html",{loader:()=>B(()=>import("./空间曲线.html-v8EuNe23.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<h3>一般方程</h3>
<p>空间曲线通常由两个非线性方程组定义，空间中的点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 必须满足的条件：</p>`,readingTime:{minutes:2.19,words:657},title:"",type:"article",s:"空间曲线"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E7%A9%BA%E9%97%B4%E6%9B%B2%E9%9D%A2.html",{loader:()=>B(()=>import("./空间曲面.html-Bp2IFNCB.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`
<p>在空间解析几何中, 关于曲面的研究有两个基本问题:</p>
<ol>
<li>已知曲面作为点的几何轨迹时, 建立这个曲面方程</li>
<li>已知坐标间的方程, 研究方程表示的曲面形状</li>
</ol>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%A9%BA%E9%97%B4%E5%B9%B3%E9%9D%A2.html" target="_blank">空间平面</a>属于第一类, 空间曲面就是此类问题的推广.</p>
<h1>常见曲面</h1>
<h2>球面</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.3,words:391},title:"背景",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E7%A9%BA%E9%97%B4%E7%9B%B4%E7%BA%BF.html",{loader:()=>B(()=>import("./空间直线.html-DLC06QBC.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:`<h2>一般式</h2>
<p>空间直线由两平面相交来定义, 将两空间平面方程联立</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>A</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>A</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \\left\\{
\\begin{aligned}
A_1x+B_1y+C_1z+D_1=0 \\\\
A_2x+B_2y+C_2z+D_2=0
\\end{aligned}
\\right.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6em;"><span style="top:-3.76em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.26em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:1.14,words:343},title:"",type:"article",s:"空间直线"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84.html",{loader:()=>B(()=>import("./线性映射.html-yKyIK4AC.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`
<p>在线性代数中，多重线性映射是有多个向量变量而对每个变量都是线性的函数。</p>
<p>n个变量的多线性映射也叫做n重线性映射。</p>
<p>如果所有变量属于同一个空间，可以考虑对称、反对称和交替的n重线性映射。后两个是一致的，如果底层的环（或域）有不同于二的特征，否则前两个是一致的。</p>
<p>一般讨论可见多重线性代数。</p>
<h2>示例</h2>
<ul>
<li>在实数域上的内积（点积）是两个变量的对称双线性函数，</li>
<li>矩阵的行列式是方矩阵的列（或行）的斜对称多重线性函数。</li>
<li>矩阵的迹数是方矩阵的列（或行）的多重线性函数。</li>
<li>双线性映射是多重线性映射。</li>
</ul>`,readingTime:{minutes:6.29,words:1887},title:"Wikipedia",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/---%E9%87%8D%E7%A7%AF%E5%88%86---.html",{loader:()=>B(()=>import("./---重积分---.html-BxtGqnBh.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E4%BA%8C%E9%87%8D%E7%A7%AF%E5%88%86.html" target="_blank">二重积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E4%B8%89%E9%87%8D%E7%A7%AF%E5%88%86.html" target="_blank">三重积分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E9%9D%A2%E9%9D%A2%E7%A7%AF.html" target="_blank">曲面面积</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E8%B4%A8%E5%BF%83.html" target="_blank">质心</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E8%BD%AC%E5%8A%A8%E6%83%AF%E9%87%8F.html" target="_blank">转动惯量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E5%BC%95%E5%8A%9B.html" target="_blank">引力</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E9%87%8D%E7%A7%AF%E5%88%86%2F%E5%90%AB%E5%8F%82%E9%87%8D%E7%A7%AF%E5%88%86.html" target="_blank">含参重积分</a></li>
</ul>`,readingTime:{minutes:.25,words:75},title:"文档-所有文档",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E4%B8%89%E9%87%8D%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./三重积分.html-Cd3w0SrC.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>三重积分用于计算在三维空间区域上的函数的积分。假设我们有一个函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span>，并且我们希望在区域<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>上对该函数进行积分。三重积分的表示形式如下：</p>',readingTime:{minutes:2.51,words:752},title:"",type:"article",s:"三重积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E4%BA%8C%E9%87%8D%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./二重积分.html-Czf3DfUh.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>二重积分用于计算在平面区域上的函数的积分。假设我们有一个函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，并且我们希望在区域<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>上对该函数进行积分。二重积分的表示形式如下：</p>`,readingTime:{minutes:1.72,words:517},title:"",type:"article",s:"二重积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E4%BA%8C%E9%87%8D%E7%A7%AF%E5%88%86%E6%8D%A2%E5%85%83%E6%B3%95.html",{loader:()=>B(()=>import("./二重积分换元法.html-_9mjbR12.js"),[]),meta:{tag:["数学"],excerpt:`<p>在换元法中，可以通过适当的变量替换，简化复杂的积分计算</p>
<ol>
<li>选择合适的新变量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u, v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span>，使得原积分区域和函数形式变得简单。</li>
<li>计算<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E5%BE%AE%E5%88%86%2F%E9%9B%85%E5%8F%AF%E6%AF%94%E8%A1%8C%E5%88%97%E5%BC%8F.html" target="_blank">雅可比行列式</a><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span></span></span></span>，用于转换积分形式。</li>
<li>将原积分转换为新变量下的积分：</li>
</ol>`,readingTime:{minutes:.42,words:126},title:"",type:"article",s:"二重积分换元法"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E5%90%AB%E5%8F%82%E9%87%8D%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./含参重积分.html-B8tbGNQZ.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p>含参重积分表示带有参数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>的函数在某个区域<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>内的积分：</p>`,readingTime:{minutes:.56,words:168},title:"",type:"article",s:"含参重积分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E5%BC%95%E5%8A%9B.html",{loader:()=>B(()=>import("./引力.html-HAXPh-zD.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<p>引力描述两质量之间的相互作用力。</p>
<h4>1. 牛顿万有引力定律</h4>
<p>两点质量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">m_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>和<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">m_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>间的引力大小为：</p>`,readingTime:{minutes:1.19,words:356},title:"",type:"article",s:"引力"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E6%97%8B%E8%BD%AC%E4%BD%93%E7%9A%84%E4%BD%93%E7%A7%AF.html",{loader:()=>B(()=>import("./旋转体的体积.html-DPOEz3np.js"),[]),meta:{tag:["数学"],excerpt:`<h2>直角坐标</h2>
<ol>
<li>计算旋转切面的面积，直接利用圆的面积定义</li>
<li>计算切面沿旋转轴的累积</li>
</ol>
<h2>柱面坐标</h2>
<p>绕<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>轴旋转的体积计算可以用柱坐标系<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo separator="true">,</mo><mi>θ</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r, \\theta, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span>来表示，其中<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span>是半径，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span>是旋转角度，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>是高度。公式可以表示为：</p>`,readingTime:{minutes:.52,words:156},title:"",type:"article",s:"旋转体的体积"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E6%9B%B2%E9%9D%A2%E9%9D%A2%E7%A7%AF.html",{loader:()=>B(()=>import("./曲面面积.html-DFBI-aei.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<p>曲面面积积分是曲面积分的一种特例，专门用于计算三维空间中曲面的面积。它的基本思想是通过积分计算出曲面上每一个微小区域的面积，并将它们累加得到整个曲面的面积。</p>
<h3>1. 定义</h3>
<p>假设<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>是一个光滑的曲面，曲面的面积可以通过以下公式计算：</p>`,readingTime:{minutes:2.01,words:604},title:"",type:"article",s:"曲面面积"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E6%9F%B1%E5%9D%90%E6%A0%87%E7%B3%BB%E7%9A%84%E4%BE%9D%E8%B5%96%E5%85%B3%E7%B3%BB.html",{loader:()=>B(()=>import("./柱坐标系的依赖关系.html-SrrnIA_P.js"),[]),meta:{tag:["数学"],excerpt:`<p>在柱坐标系下，计算体积的三重积分时，变量之间的依赖关系指的是在积分过程中不同变量的取值范围或变化范围是如何相互关联的。理解这些依赖关系对于正确设定积分上下限至关重要。</p>
<h3>1. 什么是依赖关系？</h3>
<p>依赖关系在三重积分中指的是一个积分变量的取值范围或变化范围受到另一个变量的影响。例如，在柱坐标系 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo separator="true">,</mo><mi>θ</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r, \\theta, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 中，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> 的取值范围可能会依赖于角度 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span>，而 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 的范围可能会依赖于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> 的值。</p>`,readingTime:{minutes:2.25,words:675},title:"",type:"article",s:"柱坐标系的依赖关系"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E6%9F%B1%E9%9D%A2%E5%9D%90%E6%A0%87.html",{loader:()=>B(()=>import("./柱面坐标.html-11SuEEZa.js"),[]),meta:{tag:["数学"],excerpt:`<h2>表面积微元</h2>
<p>圆柱体侧面的表面积微元 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi mathvariant="bold">S</mi></mrow><annotation encoding="application/x-tex">d\\mathbf{S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">S</span></span></span></span> 的计算涉及到圆柱的表面参数化以及从中得到的法向量。</p>`,readingTime:{minutes:1.59,words:478},title:"",type:"article",s:"柱面坐标"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E7%90%83%E9%9D%A2%E5%9D%90%E6%A0%87.html",{loader:()=>B(()=>import("./球面坐标.html-HmvTkZEz.js"),[]),meta:{tag:["数学"],excerpt:`<h2>表面积微元</h2>
<p>面积元素 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>S</mi><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup><mi>sin</mi><mo>⁡</mo><mi>ϕ</mi><mtext> </mtext><mi>d</mi><mi>θ</mi><mtext> </mtext><mi>d</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">dS = R^2 \\sin \\phi \\, d\\theta \\, d\\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">ϕ</span></span></span></span> 是球体表面在球坐标系中的面积元素。这个公式是从球坐标的参数化和表面积公式中得出的。具体的推导步骤如下：</p>`,readingTime:{minutes:1.06,words:319},title:"",type:"article",s:"球面坐标"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E8%8E%B1%E5%B8%83%E5%B0%BC%E5%85%B9%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./莱布尼兹公式.html-BUhSs1T7.js"),[]),meta:{tag:["数学","公式"],excerpt:`<p>莱布尼茨公式是一个关于积分运算的公式，它在处理含参数的积分时非常有用。对于重积分中的莱布尼茨公式，通常是指对参数的导数和积分的交换问题。</p>
<p>考虑一个含有参数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 的双重积分：</p>`,readingTime:{minutes:1.01,words:302},title:"",type:"article",s:"莱布尼兹公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E8%B4%A8%E5%BF%83.html",{loader:()=>B(()=>import("./质心.html-QK3gTQJR.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<p>质心（或称为形心）是一个物体的质量分布的“平均位置”。对于形状规则、密度均匀的物体，质心通常位于物体的几何中心。然而，对于形状不规则或密度不均匀的物体，质心的计算需要通过重积分来完成。</p>
<h3>1. 质心的定义</h3>
<p>假设一个三维物体<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>的质量密度函数为<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\rho(x, y, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ρ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span>，质心<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>y</mi><mo>ˉ</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\bar{x}, \\bar{y}, \\bar{z})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>的坐标可以通过以下公式计算：</p>`,readingTime:{minutes:4.23,words:1269},title:"",type:"article",s:"质心"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E9%87%8D%E7%A7%AF%E5%88%86/%E8%BD%AC%E5%8A%A8%E6%83%AF%E9%87%8F.html",{loader:()=>B(()=>import("./转动惯量.html-B3O3A9A0.js"),[]),meta:{author:["GPT-4"],tag:["数学"],excerpt:`<h3>转动惯量说明文档</h3>
<p>转动惯量是描述物体绕某一轴旋转时惯性大小的物理量，通常用<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span>表示。对于质量分布不均匀或形状复杂的物体，转动惯量需要通过积分计算。</p>`,readingTime:{minutes:1.85,words:556},title:"",type:"article",s:"转动惯量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/---%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C---.html",{loader:()=>B(()=>import("./---假设检验---.html-CBzNyqhs.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html" target="_blank">假设检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E6%AD%A3%E6%80%81%E6%80%BB%E4%BD%93%E5%9D%87%E5%80%BC%E7%9A%84%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html" target="_blank">正态总体均值的假设检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E6%AD%A3%E6%80%81%E6%80%BB%E4%BD%93%E6%96%B9%E5%B7%AE%E7%9A%84%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html" target="_blank">正态总体方差的假设检验</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">置信区间与假设检验之间的关系</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">样本容量的选取</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E5%88%86%E5%B8%83%E6%8B%9F%E5%90%88%E6%A3%80%E9%AA%8C.html" target="_blank">分布拟合检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E7%A7%A9%E5%92%8C%E6%A3%80%E9%AA%8C.html" target="_blank">秩和检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%E9%97%AE%E9%A2%98%E7%9A%84p%E5%80%BC%E6%A3%80%E9%AA%8C%E6%B3%95.html" target="_blank">假设检验问题的p值检验法</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2Ft%E6%A3%80%E9%AA%8C.html" target="_blank">t检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2Fz%E6%A3%80%E9%AA%8C.html" target="_blank">z检验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%2FF%E6%A3%80%E9%AA%8C.html" target="_blank">F检验</a></li>
</ul>`,readingTime:{minutes:1.78,words:535},title:"文档-所有文档",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/F%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./F检验.html-r-XNgTmM.js"),[]),meta:{tag:["数学"],excerpt:`<p>F检验是一种用于比较两个样本的方差是否有显著差异的统计方法，主要用于方差分析（ANOVA）和方差齐性检验。</p>

<div class="callout" data-callout="tip">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-flame"><path d="M8.5 14.5A2.5 2.5 0 0 0 11 12c0-1.38-.5-2-1-3-1.072-2.143-.224-4.054 2-6 .5 2.5 2 4.9 4 6.5 2 1.6 3 3.5 3 5.5a7 7 0 1 1-14 0c0-1.153.433-2.294 1-3a2.5 2.5 0 0 0 2.5 2.5z"/></svg>
</div>
<div class="callout-title-inner">Tip</div>
</div>
<div class="callout-content"><p>F检验是非对称的, 所以双边检测不能简单地采用对称的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\\alpha/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord">/2</span></span></span></span> 划分置信区间</p>
</div></div>`,readingTime:{minutes:1.39,words:416},title:"",type:"article",s:"F检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/t%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./t检验.html-pkd5iAKQ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>t检验是一种用于比较样本均值和已知总体均值或两个样本均值之间差异的统计方法，特别适用于样本量较小（通常<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&lt;</mo><mn>30</mn></mrow><annotation encoding="application/x-tex">n&lt;30</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">30</span></span></span></span>）且总体标准差未知的情况。</p>',readingTime:{minutes:2.57,words:771},title:"",type:"article",s:"t检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/z%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./z检验.html-BoBR6A6d.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>z检验是一种用于检验样本均值与已知总体均值之间是否存在显著差异的统计方法. z检验特别适用于样本量较大 (通常 n &gt; 30) 且总体标准差已知的情况, 或者样本量足够大使得样本方差近似总体方差</p>
<h2>条件</h2>
<ul>
<li>样本量较大 (通常 n &gt; 30) .</li>
<li>总体标准差已知.</li>
<li>样本数据近似正态分布.</li>
</ul>
<h2>步骤</h2>
<h3>设定假设</h3>
<p>设定零假设 (H₀) 和备择假设 (H₁) :</p>
<ul>
<li>零假设 (H₀) :  <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu = \\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>备择假设 (H₁)  (双侧) :  <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo mathvariant="normal">≠</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu \\neq \\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>备择假设 (H₁)  (单侧) :  <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>&gt;</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu &gt; \\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>&lt;</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu &lt; \\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>`,readingTime:{minutes:1.61,words:484},title:"",type:"article",s:"z检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./假设检验.html-DCsLh2zI.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>假设检验是一种统计方法，用于确定一个假设关于总体的陈述是否成立。</p>
<h2>步骤</h2>
<h3>提出假设</h3>
<ul>
<li><strong>原假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：通常表示没有显著差异或效果。比如，某药物对疾病没有影响。</li>
<li><strong>备择假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">H_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：表示存在显著差异或效果。比如，某药物对疾病有影响。</li>
</ul>`,readingTime:{minutes:5.01,words:1503},title:"",type:"article",s:"假设检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%E9%97%AE%E9%A2%98%E7%9A%84p%E5%80%BC%E6%A3%80%E9%AA%8C%E6%B3%95.html",{loader:()=>B(()=>import("./假设检验问题的p值检验法.html-Bv9F2Ed0.js"),[]),meta:{tag:["数学"],excerpt:'<p><strong>p值检验法</strong>是假设检验中的一种常见方法，它主要用于决定是否拒绝原假设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 。以下是p值检验法的核心内容和步骤。</p>',readingTime:{minutes:2.12,words:637},title:"",type:"article",s:"假设检验问题的p值检验法"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E5%88%86%E5%B8%83%E6%8B%9F%E5%90%88%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./分布拟合检验.html-FMDQleUC.js"),[]),meta:{tag:["数学"],excerpt:`<h3>分布拟合检验概述</h3>
<p>分布拟合检验是统计分析中的一种方法，用于评估样本数据是否来自某个特定的理论分布。这种检验在各种领域非常重要，如质量控制、金融分析、环境科学等，可以帮助研究人员判断所收集的数据是否符合预期的分布模型。</p>
<h3>常用的分布拟合检验方法</h3>
<ol>
<li>
<p><strong>Kolmogorov-Smirnov检验（K-S检验）</strong></p>
<ul>
<li>目的：检验一个样本的分布是否符合某个指定的分布。</li>
<li>原理：计算样本的经验累积分布函数（ECDF）与指定理论分布的累积分布函数（CDF）之间的最大差异。</li>
<li>适用：适用于连续分布的检验。</li>
</ul>
</li>
<li>
<p><strong>Anderson-Darling检验</strong></p>
<ul>
<li>目的：类似于K-S检验，但对尾部差异给予更多的权重。</li>
<li>原理：基于样本数据的经验分布函数与特定理论分布的累计分布函数之间差异的加权积分。</li>
<li>适用：尤其适用于需要重点考虑分布尾部的情况。</li>
</ul>
</li>
<li>
<p><strong>Chi-square（卡方）拟合优度检验</strong></p>
<ul>
<li>目的：检验分类数据的观测频率是否符合预期频率。</li>
<li>原理：比较观测频数与预期频数之间的差异。</li>
<li>适用：适用于离散分布，如二项分布、泊松分布等。</li>
</ul>
</li>
<li>
<p><strong>Shapiro-Wilk检验</strong></p>
<ul>
<li>目的：测试数据集是否来自正态分布。</li>
<li>原理：检验样本数据的排序值与正态分布预期值的相关性。</li>
<li>适用：适用于小样本数据的正态性检验。</li>
</ul>
</li>
</ol>`,readingTime:{minutes:2.45,words:735},title:"",type:"article",s:"分布拟合检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E6%98%BE%E8%91%97%E6%80%A7%E6%B0%B4%E5%B9%B3.html",{loader:()=>B(()=>import("./显著性水平.html-DUaAEQlz.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>显著性水平用来衡量在假设检验中犯第一类错误（即误将真零假设拒绝）的概率。显著性水平通常用希腊字母 α（alpha）表示。以下是显著性水平的主要要点：</p>
<ol>
<li>
<p><strong>定义</strong>：</p>
<ul>
<li>显著性水平是预先设定的一个阈值，决定了在假设检验中拒绝零假设的标准。</li>
<li>典型的显著性水平有0.05、0.01和0.10，对应于5%、1%和10%的犯第一类错误的概率。</li>
</ul>
</li>
<li>
<p><strong>第一类错误</strong>：</p>
<ul>
<li>第一类错误是指在实际情况中零假设为真时，错误地拒绝了零假设。</li>
<li>例如，假设我们在进行药物有效性检验中，零假设是药物无效。第一类错误即为药物实际无效，但我们在统计上却认为药物有效。</li>
</ul>
</li>
<li>
<p><strong>设定显著性水平</strong>：</p>
<ul>
<li>研究者在进行假设检验之前，根据研究领域、风险承担能力和实验设计等因素选择适当的显著性水平。</li>
<li>显著性水平越低，犯第一类错误的概率越低，但也可能增加第二类错误（即错过真实效应）的概率。</li>
</ul>
</li>
<li>
<p><strong>p值与显著性水平</strong>：</p>
<ul>
<li>在实际的统计检验中，计算出的p值（概率值）用于与预设的显著性水平进行比较。</li>
<li>如果p值小于或等于显著性水平，则拒绝零假设，认为结果在统计上显著。</li>
<li>如果p值大于显著性水平，则不拒绝零假设，认为没有足够证据表明结果在统计上显著。</li>
</ul>
</li>
<li>
<p><strong>例子</strong>：</p>
<ul>
<li>假设进行一个双尾t检验，显著性水平设为0.05。若计算出的p值为0.03，则因为0.03 &lt; 0.05，我们拒绝零假设，认为结果有统计显著性。</li>
<li>反之，如果p值为0.08，因为0.08 &gt; 0.05，我们不拒绝零假设，认为结果没有统计显著性。</li>
</ul>
</li>
</ol>`,readingTime:{minutes:1.93,words:579},title:"",type:"article",s:"显著性水平"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E6%AD%A3%E6%80%81%E6%80%BB%E4%BD%93%E5%9D%87%E5%80%BC%E7%9A%84%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./正态总体均值的假设检验.html-CJwThBfI.js"),[]),meta:{tag:["数学"],excerpt:`<p>在统计学中，对正态总体均值进行假设检验是一种常用的方法，用以判断样本数据是否支持对总体均值的某个特定假设。这种假设检验在科学研究、工业生产、经济分析等领域有着广泛的应用。</p>
<h4>基本概念</h4>
<ol>
<li>
<p><strong>假设</strong>：</p>
<ul>
<li><strong>零假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：通常为研究者希望证伪的假设，如认为总体均值等于某个特定值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</li>
<li><strong>备择假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">H_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：与零假设相对，如认为总体均值不等于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，或大于、小于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\\mu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</li>
</ul>
</li>
<li>
<p><strong>错误类型</strong>：</p>
<ul>
<li><strong>第一类错误</strong>：错误地拒绝零假设（即零假设实际为真时拒绝之）。</li>
<li><strong>第二类错误</strong>：错误地接受零假设（即零假设实际为假时接受之）。</li>
</ul>
</li>
<li>
<p><strong>显著性水平（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>）</strong>：犯第一类错误的最大可容忍概率，常用的值有0.05（5%）和0.01（1%）。</p>
</li>
<li>
<p><strong>P值</strong>：在零假设为真的条件下，观察到当前样本统计量或更极端情况的概率。</p>
</li>
</ol>`,readingTime:{minutes:2.48,words:745},title:"",type:"article",s:"正态总体均值的假设检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E6%AD%A3%E6%80%81%E6%80%BB%E4%BD%93%E6%96%B9%E5%B7%AE%E7%9A%84%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./正态总体方差的假设检验.html-B2VNdMbM.js"),[]),meta:{tag:["数学"],excerpt:`<p>对正态总体方差的假设检验是另一种常见的统计方法，用于判断样本方差是否能提供关于总体方差的某个特定假设的支持。这种检验在质量控制、风险管理和科学研究等领域中尤为重要，特别是当数据的波动性或一致性是关注的焦点时。</p>
<h4>基本概念</h4>
<ol>
<li>
<p><strong>假设</strong>：</p>
<ul>
<li><strong>零假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：通常认为总体方差等于某个特定值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mn>0</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\\sigma_0^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span></span>。</li>
<li><strong>备择假设（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">H_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</strong>：与零假设相对，如认为总体方差不等于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mn>0</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\\sigma_0^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span></span>，或大于、小于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mn>0</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\\sigma_0^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span></span>。</li>
</ul>
</li>
<li>
<p><strong>统计量</strong>：</p>
</li>
</ol>`,readingTime:{minutes:2.7,words:810},title:"",type:"article",s:"正态总体方差的假设检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C/%E7%A7%A9%E5%92%8C%E6%A3%80%E9%AA%8C.html",{loader:()=>B(()=>import("./秩和检验.html-DKgxkvRi.js"),[]),meta:{tag:["数学"],excerpt:`<h3>秩和检验概述</h3>
<p>秩和检验（Rank Sum Test），通常指的是曼-惠特尼U检验（Mann-Whitney U Test），是一种用于检验两个独立样本是否来自同一总体的非参数统计方法。在统计学中，非参数方法不依赖于数据的具体分布形式，适用于样本数据不满足正态分布的情况。</p>
<h3>工作原理</h3>
<p>秩和检验的基本思想是将两组数据合并后进行排序，每个数据点在排序中的位置称为“秩”（Rank）。检验的目的是比较两组数据在秩次上的差异是否显著，以此判断两个独立样本的分布是否有显著差异。</p>
<h3>实施步骤</h3>
<ol>
<li><strong>合并与排序</strong>：将两个样本合并为一个数据集，按照数值大小进行排序。</li>
<li><strong>计算秩和</strong>：分别为两个样本中的每个数据分配秩次，然后计算每个样本的秩和。</li>
<li><strong>统计量计算</strong>：利用秩和计算统计量，如Mann-Whitney U统计量。</li>
<li><strong>显著性检验</strong>：通过查表或计算P值，判断统计量的显著性，从而判断两组数据是否有显著差异。</li>
</ol>`,readingTime:{minutes:2.88,words:863},title:"",type:"article",s:"秩和检验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/---%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83---.html",{loader:()=>B(()=>import("./---抽样分布---.html-Stfbg2OL.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%AC%E7%BB%9F%E8%AE%A1%E9%87%8F.html" target="_blank">样本统计量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E5%8D%A1%E6%96%B9%E5%88%86%E5%B8%83.html" target="_blank">卡方分布</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2Ft%E5%88%86%E5%B8%83.html" target="_blank">t分布</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2FF%E5%88%86%E5%B8%83.html" target="_blank">F分布</a></li>
</ul>`,readingTime:{minutes:.09,words:28},title:"",type:"article",s:"---抽样分布---"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/F%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./F分布.html-BAWK372E.js"),[]),meta:{tag:["数学"],excerpt:`<h3>F分布的定义</h3>
<p>F分布是统计学中用来比较两个样本方差是否显著不同的一种概率分布，通常在方差分析（ANOVA）及回归分析等领域中广泛应用。这个分布由统计学家罗纳德·费希尔（Ronald Fisher）首先引入。</p>
<h3>数学表达</h3>
<p>F分布是两个卡方分布的比值的分布，其中每个卡方分布都除以了其自由度。设随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 服从自由度为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的卡方分布，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> 服从自由度为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的卡方分布，那么变量</p>`,readingTime:{minutes:1.32,words:397},title:"",type:"article",s:"F分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/t%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./t分布.html-ChBlClaV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布广泛应用于小样本数据分析中的假设检验和置信区间的建立。</p>`,readingTime:{minutes:1.42,words:426},title:"",type:"article",s:"t分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E5%8D%A1%E6%96%B9%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./卡方分布.html-DGJgrPdA.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>简介</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布是一种常用的连续概率分布，广泛应用于统计推断中，尤其是在方差分析、假设检验和置信区间估计中。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 个独立<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E6%A0%87%E5%87%86%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83.html" target="_blank">标准正态分布</a>的平方和构成，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 是自由度。</p>`,readingTime:{minutes:1.92,words:577},title:"",type:"article",s:"卡方分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%AC.html",{loader:()=>B(()=>import("./样本.html-C9Ga9lG1.js"),[]),meta:{tag:["数学"],excerpt:'<p>从总体中随机抽取的部分数据。样本用 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X = \\{x_1, x_2, \\ldots, x_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>表示，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>为第i个样本的值, 样本大小为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>。</p>',readingTime:{minutes:.16,words:47},title:"",type:"article",s:"样本"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%ACK%E9%98%B6%E4%B8%AD%E5%BF%83%E8%B7%9D.html",{loader:()=>B(()=>import("./样本K阶中心距.html-BF74HF3v.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>μ</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\\mu_k = \\frac{1}{n} \\sum_{i=1}^n (x_i - \\bar{x})^k
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.07,words:21},title:"",type:"article",s:"样本K阶中心距"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%ACK%E9%98%B6%E5%8E%9F%E7%82%B9%E8%B7%9D.html",{loader:()=>B(()=>import("./样本K阶原点距.html-BVhU-LCZ.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>m</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">m_k = \\frac{1}{n} \\sum_{i=1}^n x_i^k
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.06,words:18},title:"",type:"article",s:"样本K阶原点距"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%AC%E5%B9%B3%E5%9D%87%E5%80%BC.html",{loader:()=>B(()=>import("./样本平均值.html-CA6FMf2w.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\bar{x} = \\frac{1}{n} \\sum_{i=1}^n x_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.06,words:19},title:"",type:"article",s:"样本平均值"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%AC%E6%96%B9%E5%B7%AE.html",{loader:()=>B(()=>import("./样本方差.html-CpaWlmfB.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 = \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\bar{x})^2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.08,words:23},title:"",type:"article",s:"样本方差"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%AC%E6%A0%87%E5%87%86%E5%B7%AE.html",{loader:()=>B(()=>import("./样本标准差.html-B6BKd87w.js"),[]),meta:{tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>S</mi><mo>=</mo><msqrt><msup><mi>S</mi><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">S = \\sqrt{S^2} = \\sqrt{\\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\bar{x})^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0361em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0039em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9639em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0361em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1568em;vertical-align:-1.2777em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791em;"><span class="svg-align" style="top:-5.1168em;"><span class="pstrut" style="height:5.1168em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391em;"><span class="pstrut" style="height:5.1168em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.1968em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="3.1968em" viewBox="0 0 400000 3196" preserveAspectRatio="xMinYMin slice"><path d="M702 80H40000040
H742v3062l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1
h-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170
c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667
219 661 l218 661zM702 80H400000v40H742z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.09,words:26},title:"",type:"article",s:"样本标准差"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83/%E6%A0%B7%E6%9C%AC%E7%BB%9F%E8%AE%A1%E9%87%8F.html",{loader:()=>B(()=>import("./样本统计量.html-wux7ZPwK.js"),[]),meta:{tag:["数学"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%AC.html" target="_blank">样本</a></p>
<p>样本从总体中来, 构成一个子集, 对其也可以计算样本统计量. 与总体的统计量稍有不同, 尤其是方差</p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%AC%E5%B9%B3%E5%9D%87%E5%80%BC.html" target="_blank">样本平均值</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%AC%E6%96%B9%E5%B7%AE.html" target="_blank">样本方差</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%AC%E6%A0%87%E5%87%86%E5%B7%AE.html" target="_blank">样本标准差</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%ACK%E9%98%B6%E5%8E%9F%E7%82%B9%E8%B7%9D.html" target="_blank">样本K阶原点距</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%2F%E6%A0%B7%E6%9C%ACK%E9%98%B6%E4%B8%AD%E5%BF%83%E8%B7%9D.html" target="_blank">样本K阶中心距</a></p>`,readingTime:{minutes:.27,words:80},title:"",type:"article",s:"样本统计量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/---%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1---.html",{loader:()=>B(()=>import("./---参数估计---.html-c9IpnUM_.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E7%82%B9%E4%BC%B0%E8%AE%A1.html" target="_blank">点估计</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E4%BC%B0%E8%AE%A1%E9%87%8F%E7%9A%84%E8%AF%84%E9%80%89%E6%A0%87%E5%87%86.html" target="_blank">估计量的评选标准</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E5%8C%BA%E9%97%B4%E4%BC%B0%E8%AE%A1.html" target="_blank">区间估计</a></li>
<li>正态总体均值与方差的区间估计</li>
<li>0-1分布参数的区间估计</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E5%8D%95%E4%BE%A7%E7%BD%AE%E4%BF%A1%E5%8C%BA%E9%97%B4.html" target="_blank">单侧置信区间</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2Fbootstrap%E6%96%B9%E6%B3%95.html" target="_blank">bootstrap方法</a></li>
</ul>`,readingTime:{minutes:.32,words:97},title:"文档-所有文档",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/bootstrap%E6%96%B9%E6%B3%95.html",{loader:()=>B(()=>import("./bootstrap方法.html-DCirUman.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ul>
<li>非参数bootstrap方法</li>
<li>参数bootstrap方法</li>
</ul>
`,readingTime:{minutes:.08,words:24},title:"",type:"article",s:"bootstrap方法"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E4%BC%B0%E8%AE%A1%E9%87%8F%E7%9A%84%E8%AF%84%E9%80%89%E6%A0%87%E5%87%86.html",{loader:()=>B(()=>import("./估计量的评选标准.html-DLvUYNg9.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>对同一参数, 不同的估计方式求出的估计量可能不相同, 我们自然会问, 采用哪个估计量更好? 以下是几个衡量标准</p>
<h1>无偏性</h1>
<p>所有样本期望均=总体期望</p>
<h2>不偏估计量 (Unbiased Estimator)</h2>
<p>不偏估计量是指其期望值等于被估计的总体参数的估计量。也就是说，如果估计量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\\hat{\\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> 是不偏的，那么其期望值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(\\hat{\\theta})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 应等于总体参数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span>。</p>`,readingTime:{minutes:.64,words:191},title:"无偏性",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E4%BC%BC%E7%84%B6%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./似然函数.html-CtuuXSNe.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>在概率和统计学中，尤度是一个函数，在给定参数下，表示观测数据出现的<strong>概率</strong>。与概率不同的是，尤度函数将参数视为变量，而观测数据是已知的。尤度函数通常用于参数估计，尤其是在<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1.html" target="_blank">最大似然估计</a>中。</p>
<p>尤度函数是基于误差项（如正态分布的误差）的概率密度函数，而没有误差项时，模型变成了一个确定性的线性方程组，无法定义概率分布，也就没有尤度函数可言。在这种确定性模型中，我们可以通过直接解方程组来确定。如果线性方程组没有一致的解，这意味着数据本身存在矛盾或测量错误。</p>`,readingTime:{minutes:3.63,words:1090},title:"似然函数和对数似然函数",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E5%8C%BA%E9%97%B4%E4%BC%B0%E8%AE%A1.html",{loader:()=>B(()=>import("./区间估计.html-DKMV0pZw.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`
<p>区间估计是统计推断中用于估计总体参数的一种方法，它提供了一个范围（区间）而不是单一的点估计值。区间估计不仅提供了参数估计值，还给出了估计值的不确定性信息。区间估计的结果通常表现为一个包含总体参数的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E7%BD%AE%E4%BF%A1%E5%8C%BA%E9%97%B4.html" target="_blank">置信区间</a>。</p>
<h2>公式</h2>
<p>对于总体均值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的置信区间，假设样本均值为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>X</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\\bar{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span>，样本标准误为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">SE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>，则置信区间可以表示为：</p>`,readingTime:{minutes:2.12,words:636},title:"区间估计",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E5%8D%95%E4%BE%A7%E7%BD%AE%E4%BF%A1%E5%8C%BA%E9%97%B4.html",{loader:()=>B(()=>import("./单侧置信区间.html-DDpyx6Hs.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>实际问题中我们关心的是未知参数的上限或下限</p>
<p>单侧置信区间（one-sided confidence interval）是一种只在一个方向上给出界限的置信区间，用于估计参数时提供偏向性的置信区间。与双侧置信区间不同，单侧置信区间只关注一个方向的尾部，提供对参数的上限或下限的估计。以下是单侧置信区间的详细解释和计算步骤：</p>
<h3>定义</h3>
<p>单侧置信区间有两种形式：</p>
<ol>
<li><strong>左侧单侧置信区间</strong>（left-sided confidence interval）：给出参数的上限估计。</li>
<li><strong>右侧单侧置信区间</strong>（right-sided confidence interval）：给出参数的下限估计。</li>
</ol>`,readingTime:{minutes:2.81,words:844},title:"相关内容",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E6%80%BB%E4%BD%93%E4%B8%8E%E6%A0%B7%E6%9C%AC.html",{loader:()=>B(()=>import("./总体与样本.html-VQklFxR_.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`
<p>在区间推定中，存在总体和样本两种不同的分布情况。</p>
<ol>
<li>总体通常服从正态分布（或假定服从正态分布）</li>
<li>样本统计量的分布，尤其是样本均值的分布，在总体标准差未知且样本量较小时，服从t分布。</li>
</ol>
<h3>总体分布</h3>
<p>总体分布是描述整个总体的概率分布。在很多情况下，我们假设总体服从正态分布。这是因为根据中心极限定理，当样本量足够大时，即使总体分布不是正态分布，样本均值的分布也会接近正态分布。</p>
<h3>样本分布</h3>
<p>样本分布是从总体中抽取的样本的分布情况。当样本量较小时，我们通常无法确切知道总体标准差，因此我们使用样本标准差来估计总体标准差。在这种情况下，样本均值的分布遵循t分布，而不是正态分布。</p>`,readingTime:{minutes:2.47,words:741},title:"总体与样本分布",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E6%80%BB%E4%BD%93%E5%92%8C%E6%A0%B7%E6%9C%AC%E7%9A%84%E6%9C%9F%E6%9C%9B%E4%B8%8E%E6%96%B9%E5%B7%AE.html",{loader:()=>B(()=>import("./总体和样本的期望与方差.html-qIqodSVa.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h3>总体和样本的期望与方差</h3>
<p>在统计学中，总体（population）和样本（sample）的期望（均值）和方差是描述数据集中趋势和离散程度的重要指标。以下是总体和样本的期望与方差的定义和计算公式。</p>
<h4>总体期望和方差</h4>
<p><strong>总体期望（均值）</strong>：总体的期望值是总体中所有个体数值的平均值。设总体中有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> 个数据点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">X_1, X_2, \\ldots, X_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则总体期望 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 定义为：</p>`,readingTime:{minutes:2.02,words:606},title:"",type:"article",s:"总体和样本的期望与方差"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1.html",{loader:()=>B(()=>import("./最大似然估计.html-oBSyGB48.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>最大似然估计是寻找参数使得似然函数(<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E4%BC%BC%E7%84%B6%E5%87%BD%E6%95%B0.html" target="_blank">似然函数</a>)最大的过程。通过最大化样本数据的尤度来估计总体参数。基本思想是找到使得样本数据出现的概率最大的参数值。它通常具有良好的统计性质，如一致性和渐近正态性，且在大样本下往往是最有效的估计量。</p>
<h3>步骤</h3>
<ol>
<li>构建似然函数：根据给定的概率分布，写出样本数据的联合概率密度函数或概率质量函数。</li>
<li>取对数：对似然函数取自然对数，得到对数似然函数, 这样求解往往比较方便。</li>
<li>求导数并求解：对对数似然函数求导数，找到使导数为零的参数值。</li>
</ol>`,readingTime:{minutes:1.02,words:307},title:"",type:"article",s:"最大似然估计"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E6%9C%80%E5%A4%A7%E5%90%8E%E9%AA%8C%E6%A6%82%E7%8E%87.html",{loader:()=>B(()=>import("./最大后验概率.html-CgV2l_Lb.js"),[]),meta:{tag:["数学"],excerpt:`<p>事后概率最大化(MAP)是一种估计参数的方法, 它结合了观测数据 (<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E4%BC%BC%E7%84%B6%E5%87%BD%E6%95%B0.html" target="_blank">似然函数</a>) 和先验分布来估计参数. 与<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1.html" target="_blank">最大似然估计</a> (MLE) 不同, MAP 考虑了先验信息.</p>
<p>在 MAP 中, 我们试图最大化事后概率分布:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mi mathvariant="normal">∣</mi><mtext>data</mtext><mo stretchy="false">)</mo><mo>∝</mo><mi>p</mi><mo stretchy="false">(</mo><mtext>data</mtext><mi mathvariant="normal">∣</mi><mi>θ</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(\\theta | \\text{data}) \\propto p(\\text{data} | \\theta) p(\\theta)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mord">∣</span><span class="mord text"><span class="mord">data</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord text"><span class="mord">data</span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:7.12,words:2137},title:"",type:"article",s:"最大后验概率"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E6%9E%A2%E8%BD%B4%E9%87%8F.html",{loader:()=>B(()=>import("./枢轴量.html-SEGKUvGb.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>枢轴量是一种由样本数据和未知参数共同组成的统计量，其分布在参数的不同取值下是已知的且不依赖于参数本身。<br>
统计推断中，通过其已知分布特性，可以有效地进行参数估计和假设检验。<br>
常见的枢轴量包括标准正态分布和 t 分布的统计量<br>
枢轴量的主要用途是构造置信区间和进行假设检验。<br>
由于其分布在不同参数取值下保持不变，因此可以利用其已知分布来推断未知参数的可能值范围或检验假设。</p>
<h2>特性</h2>
<ol>
<li><strong>已知分布</strong>：枢轴量的分布是已知的，通常为标准分布（例如标准正态分布或 t 分布）。</li>
<li><strong>独立于参数</strong>：枢轴量的分布不依赖于未知参数本身。</li>
</ol>`,readingTime:{minutes:2.58,words:774},title:"",type:"article",s:"枢轴量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E7%82%B9%E4%BC%B0%E8%AE%A1.html",{loader:()=>B(()=>import("./点估计.html-BLKgsobF.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>从样本数据中估计总体的参数, 找到尽可能接近总体参数的值。<br>
点估计的具体实现方式有多种，其中常见的有<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E7%9F%A9%E4%BC%B0%E8%AE%A1.html" target="_blank">矩估计</a>和<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1.html" target="_blank">最大似然估计</a>。<br>
以下是一些常见的点推定方法。</p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E7%9F%A9%E4%BC%B0%E8%AE%A1.html" target="_blank">矩估计</a></p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%2F%E6%9C%80%E5%A4%A7%E4%BC%BC%E7%84%B6%E4%BC%B0%E8%AE%A1.html" target="_blank">最大似然估计</a></p>`,readingTime:{minutes:.31,words:93},title:"",type:"article",s:"点估计"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E7%9F%A9%E4%BC%B0%E8%AE%A1.html",{loader:()=>B(()=>import("./矩估计.html-B4je6QwD.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>通过样本矩来估计总体矩参数</p>
<p>均值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 样本矩估计</p>`,readingTime:{minutes:.24,words:71},title:"",type:"article",s:"矩估计"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1/%E7%BD%AE%E4%BF%A1%E5%8C%BA%E9%97%B4.html",{loader:()=>B(()=>import("./置信区间.html-DvRC4fvN.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在一定置信水平下，包含总体参数的区间。<br>
置信水平（通常用 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 表示，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 是显著性水平）表示对区间包含真实总体参数的信心程度。例如，95% 的置信区间表示有 95% 的把握该区间包含真实的总体参数。</p>`,readingTime:{minutes:.76,words:227},title:"",type:"article",s:"置信区间"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%96%B9%E5%B7%AE%E5%88%86%E6%9E%90%E5%8F%8A%E5%9B%9E%E5%BD%92/---%E6%96%B9%E5%B7%AE%E5%88%86%E6%9E%90%E5%8F%8A%E5%9B%9E%E5%BD%92---.html",{loader:()=>B(()=>import("./---方差分析及回归---.html-DRv_IlKa.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li>单因素试验的方差分析</li>
<li>双因素试验的方差分析</li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%96%B9%E5%B7%AE%E5%88%86%E6%9E%90%E5%8F%8A%E5%9B%9E%E5%BD%92%2F%E4%B8%80%E5%85%83%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92.html" target="_blank">一元线性回归</a></li>
<li>多元线性回归</li>
</ul>
`,readingTime:{minutes:.16,words:47},title:"",type:"article",s:"---方差分析及回归---"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%96%B9%E5%B7%AE%E5%88%86%E6%9E%90%E5%8F%8A%E5%9B%9E%E5%BD%92/%E4%B8%80%E5%85%83%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92.html",{loader:()=>B(()=>import("./一元线性回归.html-2049ctvM.js"),[]),meta:{tag:["数学"],excerpt:`<p>一元线性回归是一种统计方法，用于分析两个连续变量之间的线性关系。它通过拟合一条直线来最小化预测值与实际值之间的误差。具体来说，一元线性回归模型可以描述如下：</p>
<h3>一元线性回归模型</h3>
<p>一元线性回归模型的形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> y = \\beta_0 + \\beta_1 x + \\epsilon 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p>`,readingTime:{minutes:2.33,words:698},title:"",type:"article",s:"一元线性回归"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/---%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86---.html",{loader:()=>B(()=>import("./---极限定理---.html-B-wMrAy7.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86%2F%E5%A4%A7%E6%95%B0%E5%AE%9A%E5%BE%8B.html" target="_blank">大数定律</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86%2F%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86.html" target="_blank">中心极限定理</a></li>
</ul>
`,readingTime:{minutes:.08,words:23},title:"",type:"article",s:"---极限定理---"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./中心极限定理.html-DjidH7pI.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>中心极限定理 (Central Limit Theorem, CLT) 是概率论中的一个基本定理。它描述了在一定条件下，大量独立随机变量的和的分布趋近于正态分布。该定理为统计学中的许多重要结果提供了理论基础，特别是在进行抽样和推断时。</p>
<h2>定理内容</h2>
<p>假设我们有一组独立同分布的随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1, X_2, \\ldots, X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，每个随机变量具有相同的数学期望 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">E(X_i) = \\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 和方差 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">Var(X_i) = \\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。定义这组随机变量的和为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p>`,readingTime:{minutes:2.08,words:625},title:"",type:"article",s:"中心极限定理"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/%E5%A4%A7%E6%95%B0%E5%AE%9A%E5%BE%8B.html",{loader:()=>B(()=>import("./大数定律.html-DTEK0rBW.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>大量实验证明, 随机事件A的频率<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_n(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>当重复实验的次数n增大到一定程度时, 总能呈现出稳定性, 稳定在某个常数附近, 频率的稳定性是概率定义的客观基础. 它表明在大量独立同分布的随机变量的平均值将收敛于其期望值。</p>',readingTime:{minutes:.46,words:139},title:"",type:"article",s:"大数定律"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/%E5%BC%B1%E5%A4%A7%E6%95%B0%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./弱大数定理.html-BQg3nTCy.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>设<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1, X_2, ... , X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立, 服从同一分布的随机变量序列, 且有数学期望<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">E(X_k)=\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> , 则序列</p>',readingTime:{minutes:.28,words:83},title:"",type:"article",s:"弱大数定理"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86/%E5%BC%BA%E5%A4%A7%E6%95%B0%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./强大数定理.html-7Nzlh_Ui.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1, X_2, \\ldots, X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一列独立同分布的随机变量，其数学期望为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>。那么有：</p>',readingTime:{minutes:.45,words:135},title:"",type:"article",s:"强大数定理"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/---%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80---.html",{loader:()=>B(()=>import("./---概率论基础---.html-D9chV2Pj.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E9%9A%8F%E6%9C%BA%E8%AF%95%E9%AA%8C.html" target="_blank">随机试验</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E6%A0%B7%E6%9C%AC%E7%A9%BA%E9%97%B4.html" target="_blank">样本空间</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E9%9A%8F%E6%9C%BA%E4%BA%8B%E4%BB%B6.html" target="_blank">随机事件</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E9%A2%91%E7%8E%87%E4%B8%8E%E6%A6%82%E7%8E%87.html" target="_blank">频率与概率</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E5%8F%A4%E5%85%B8%E6%A6%82%E5%9E%8B.html" target="_blank">古典概型</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E6%9D%A1%E4%BB%B6%E6%A6%82%E7%8E%87.html" target="_blank">条件概率</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E7%8B%AC%E7%AB%8B%E6%80%A7.html" target="_blank">独立性</a></li>
</ol>`,readingTime:{minutes:.28,words:85},title:"目录",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E4%BA%8B%E4%BB%B6%E5%85%B3%E7%B3%BB%E8%BF%90%E7%AE%97.html",{loader:()=>B(()=>import("./事件关系运算.html-DrcdoEHZ.js"),[]),meta:{tag:["数学"],excerpt:`<p>事件关系运算是随机事件之间的基本运算。<br>
对于事件<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span>, 事件关系运算包括：</p>`,readingTime:{minutes:.59,words:176},title:"",type:"article",s:"事件关系运算"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E5%85%88%E9%AA%8C%E6%A6%82%E7%8E%87.html",{loader:()=>B(()=>import("./先验概率.html-Bj-KylfI.js"),[]),meta:{tag:["数学"],excerpt:`<p>先验概率反映了我们在观察到数据之前对参数的知识或信念。它基于以往的经验、先前的研究结果或专家意见等非实验数据源。</p>
`,readingTime:{minutes:.21,words:64},title:"",type:"article",s:"先验概率"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E5%85%A8%E6%A6%82%E7%8E%87%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./全概率公式.html-DsRNf6Ao.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学","公式"],excerpt:`<h2>定义</h2>
<p>全概率公式用于计算一个事件的总概率，通过一组互斥且完备的子事件来表示。<br>
如果将样本空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> 划分为一组互斥且完备的事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1, A_2, \\ldots, A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，那么任意一个子事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 的总概率，可以通过计算 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 在每一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 上的条件概率并加权求和得到。</p>`,readingTime:{minutes:4.67,words:1401},title:"",type:"article",s:"全概率公式"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E5%8F%A4%E5%85%B8%E6%A6%82%E5%9E%8B.html",{loader:()=>B(()=>import("./古典概型.html-5VwB4NNb.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>古典概型是概率论中的一种模型，假设所有基本事件（样本空间中的元素）是等可能发生的。在这种情况下，某事件 A 的概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span> 可以通过公式计算</p>',readingTime:{minutes:.44,words:133},title:"",type:"article",s:"古典概型"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E5%90%8E%E9%AA%8C%E6%A6%82%E7%8E%87.html",{loader:()=>B(()=>import("./后验概率.html-CaQE2s6M.js"),[]),meta:{tag:["数学"],excerpt:`<p>后验概率是在已知某些证据或数据的情况下，更新或修正某一事件发生的概率。后验概率的计算基于<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E8%B4%9D%E5%8F%B6%E6%96%AF%E5%AE%9A%E7%90%86.html" target="_blank">贝叶斯定理</a>。贝叶斯定理提供了一种在已知条件下，通过现有数据更新初始假设的方法。后验概率也可被视为一种<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80%2F%E6%9D%A1%E4%BB%B6%E6%A6%82%E7%8E%87.html" target="_blank">条件概率</a>，但它强调在有新证据或数据的情况下对原有假设的更新。<br>
贝叶斯定理的公式如下：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">P(A|B) = \\frac{P(B|A) \\cdot P(A)}{P(B)}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.85,words:254},title:"",type:"article",s:"后验概率"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E6%9D%A1%E4%BB%B6%E6%A6%82%E7%8E%87.html",{loader:()=>B(()=>import("./条件概率.html-SFO1k-bj.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在事件B发生的条件下, 发生事件A的概率, 通过A和B的概率计算, 可以使用集合概念来理解.</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">P(A|B)=\\dfrac{P(A\\cap B)}{P(B)}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:.67,words:202},title:"",type:"article",s:"条件概率"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E6%A0%B7%E6%9C%AC%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./样本空间.html-np6lwOu7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>样本空间是随机试验所有可能结果的集合。用符号 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 表示。</p>',readingTime:{minutes:.28,words:85},title:"",type:"article",s:"样本空间"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E7%8B%AC%E7%AB%8B%E6%80%A7.html",{loader:()=>B(()=>import("./独立性.html-CcV93QtC.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html" target="_blank">离散随机变量的独立性</a></p>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html" target="_blank">连续随机变量的独立性</a></p>
<p>在这两种情况下，独立性的本质是相同的：一个变量的取值不影响另一个变量的概率分布。这是概率论和统计推断中的一个基本概念。</p>
<h2>性质</h2>
<p>若<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>,<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> 独立, 则<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E[X,Y]=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E[X]=E[Y]=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p>`,readingTime:{minutes:.38,words:115},title:"",type:"article",s:"独立性"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E8%B4%9D%E5%8F%B6%E6%96%AF%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./贝叶斯定理.html-BW4kLnCr.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p>贝叶斯定理描述了如何利用现有证据更新某一事件的概率。</p>
<p>对于一个构成样本空间划分的事件集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{A_1, A_2, \\ldots, A_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 和任意事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span>，贝叶斯定理的表达式为：</p>`,readingTime:{minutes:10.34,words:3103},title:"",type:"article",s:"贝叶斯定理"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E8%B4%9D%E7%89%B9%E6%9C%97%E6%82%96%E8%AE%BA.html",{loader:()=>B(()=>import("./贝特朗悖论.html-B72_GlFn.js"),[]),meta:{tag:["数学"],excerpt:`
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<div class="callout-title-inner">Question</div>
</div>
<div class="callout-content"><p>在一个圆上，随机取三点（三点独立、在圆周上均匀分布），这三点组成的三角形是锐角三角形的概率是多少？</p>
</div></div>`,readingTime:{minutes:6.63,words:1988},title:"",type:"article",s:"贝特朗悖论"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E9%9A%8F%E6%9C%BA%E4%BA%8B%E4%BB%B6.html",{loader:()=>B(()=>import("./随机事件.html-Dd1hEBPT.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>随机事件是样本空间的子集，即随机试验中可能发生的结果的集合。事件可以分为单个事件和复合事件。</p>
<p>例如，在掷骰子的试验中，事件“掷出偶数”可以表示为集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">{2,4,6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span></span></span></span></span>。</p>`,readingTime:{minutes:.27,words:80},title:"",type:"article",s:"随机事件"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E9%9A%8F%E6%9C%BA%E8%AF%95%E9%AA%8C.html",{loader:()=>B(()=>import("./随机试验.html-fhiftfXA.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>随机试验是指在相同条件下可以重复进行的试验，其结果是不确定的。例如，掷一枚硬币、抛掷一个骰子、抽取一张扑克牌等都是随机试验。</p>
`,readingTime:{minutes:.22,words:67},title:"",type:"article",s:"随机试验"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E6%A6%82%E7%8E%87%E8%AE%BA%E5%9F%BA%E7%A1%80/%E9%A2%91%E7%8E%87%E4%B8%8E%E6%A6%82%E7%8E%87.html",{loader:()=>B(()=>import("./频率与概率.html-BGpJBdk5.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>频率是指在多次重复试验中某一事件发生的次数与试验总次数的比值。概率是对随机事件发生可能性的度量，通常用 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span> 表示。</p>',readingTime:{minutes:.28,words:85},title:"",type:"article",s:"频率与概率"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/---%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F---.html",{loader:()=>B(()=>import("./---随机变量---.html-CbXcCZY9.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html" target="_blank">随机变量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html" target="_blank">离散随机变量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html" target="_blank">连续随机变量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0.html" target="_blank">分布函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E5%87%BD%E6%95%B0.html" target="_blank">随机变量的函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E4%BA%8C%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html" target="_blank">二维随机变量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E8%BE%B9%E7%BC%98%E5%88%86%E5%B8%83.html" target="_blank">边缘分布</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E6%9D%A1%E4%BB%B6%E5%88%86%E5%B8%83.html" target="_blank">条件分布</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html" target="_blank">离散随机变量的独立性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2F%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html" target="_blank">连续随机变量的独立性</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">两个随机变量的函数分布</a></li>
</ol>`,readingTime:{minutes:.44,words:133},title:"目录",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E4%BA%8C%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html",{loader:()=>B(()=>import("./二维随机变量.html-BnH4g_BH.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>二维随机变量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，由随机变量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X=X(e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>和<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y=Y(e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>确定，为了方便，我们给他起个名字, 叫<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，则</p>',readingTime:{minutes:1.98,words:593},title:"",type:"article",s:"二维随机变量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E4%BA%8C%E9%A1%B9%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./二项分布.html-luFCnCdG.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<details class="callout" data-callout="info" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">Info</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p>设实验<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>只有两个结果, 将<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>独立重复<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>次, 称为<strong><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>重伯努利实验</strong></p>
<ul>
<li>独立: 各次实验结果互不影响</li>
<li>重复: 指每次实验中<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">P(A)=p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span>保持不变</li>
</ul>
</div></details>`,readingTime:{minutes:.51,words:152},title:"",type:"article",s:"二项分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./伯努利分布.html-o9hQ-Dr7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">Info</div>
</div>
<div class="callout-content"><p>随机变量X只能取0, 1两个值, 分布规律为</p>
</div></div>`,readingTime:{minutes:.2,words:59},title:"",type:"article",s:"伯努利分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./分布函数.html-75vq9TO9.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>的分布函数</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.33,words:99},title:"定义",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E6%9D%A1%E4%BB%B6%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./条件分布.html-C8vmT-4W.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>设(X,Y)是二维离散随机变量, 随机变量Y 在条件{X =x}下的条件概率分布是指当已知X 的取值为某个特定值x之时，Y 的概率分布。</p>
<p>如果Y 在条件{X =x}下的条件概率分布是连续分布，那么其密度函数称作Y 在条件{X =x}下的条件概率密度函数（条件分布密度、条件密度函数）。</p>
`,readingTime:{minutes:.44,words:132},title:"",type:"article",s:"条件分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E6%A0%87%E5%87%86%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./标准正态分布.html-chkfwjIu.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>标准正态分布是正态分布的一种特殊形式，其均值为0，标准差为1。在统计学和概率论中，标准正态分布通常用 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span></span></span></span> 表示。以下是标准正态分布的详细介绍：</p>',readingTime:{minutes:2.43,words:728},title:"",type:"article",s:"标准正态分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./正态分布.html-BiACDf-_.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>正态分布，也称为高斯分布，是连续概率分布的一种，它在统计学和自然科学中应用广泛。正态分布的概率密度函数(PDF)具有对称的钟形曲线形状，由两个参数完全确定：均值(μ)和标准差(σ)。</p>
<h3>定义</h3>
<p>正态分布的概率密度函数：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>μ</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">[</mo><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">f(x | \\mu, \\sigma) = \\frac{1}{\\sqrt{2\\pi}\\sigma} \\exp\\left[{-\\dfrac{(x - \\mu)^2}{2\\sigma^2}}\\right]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4411em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.2028em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1328em;"><span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p>`,readingTime:{minutes:.91,words:272},title:"",type:"article",s:"正态分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E6%B3%8A%E6%9D%BE%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./泊松分布.html-DL070Hq0.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">{</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>=</mo><mfrac><mrow><msup><mi>λ</mi><mi>k</mi></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi></mrow></msup></mrow><mrow><mi>k</mi><mo stretchy="false">!</mo></mrow></mfrac><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mtext> </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P\\{X=k\\}=\\frac{\\lambda^ke^{ -\\lambda }}{k!},(k=0,1,2,\\dots)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2121em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5261em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.56,words:168},title:"泊松定理",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html",{loader:()=>B(()=>import("./离散随机变量.html-NM0p_gYE.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>离散随机变量X的分布可以用表格形式表示</p>
<table>
<thead>
<tr>
<th style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span></th>
<th style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th>
<th style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th>
<th style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em;"></span><span class="mord">...</span></span></span></span></th>
<th style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">p_{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em;"></span><span class="mord">...</span></span></span></span></td>
<td style="text-align:left"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">p_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
</tr>
</tbody>
</table>`,readingTime:{minutes:.26,words:79},title:"",type:"article",s:"离散随机变量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html",{loader:()=>B(()=>import("./离散随机变量的独立性.html-22D8XJ4Q.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>离散型随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> 被认为是独立的，如果对于所有可能的值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>，它们的联合概率分布满足以下条件：</p>',readingTime:{minutes:.22,words:67},title:"",type:"article",s:"离散随机变量的独立性"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E8%BE%B9%E7%BC%98%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./边缘分布.html-B9ApmzxP.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>联合概率分布可以通过对另一个随机变量积分或求和得到边缘分布：</p>
<h2>离散</h2>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mi>y</mi></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"/><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mi>x</mi></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">  P(X = x) = \\sum_y f(x, y), \\quad P(Y = y) = \\sum_x f(x, y).
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4361em;vertical-align:-1.3861em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.9em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3861em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3em;vertical-align:-1.25em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.9em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p>`,readingTime:{minutes:.43,words:128},title:"",type:"article",s:"边缘分布"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html",{loader:()=>B(()=>import("./连续随机变量.html-DN8xEDPh.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>随机变量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span></span></span></span> 根据法则<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span>可以在定义域内取到连续的无数个值. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>代表<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">X=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>的可能性</p>`,readingTime:{minutes:.59,words:176},title:"定义",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E7%8B%AC%E7%AB%8B%E6%80%A7.html",{loader:()=>B(()=>import("./连续随机变量的独立性.html-C8UNMV5-.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于连续型随机变量，定义类似，但使用概率密度函数而不是概率。如果两个连续型随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> 的联合概率密度函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_{X,Y}(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0785em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.2222em;">Y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 可以表示为各自边缘概率密度函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_X(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0785em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.2222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 的乘积，则这两个变量是独立的：</p>',readingTime:{minutes:.36,words:108},title:"",type:"article",s:"连续随机变量的独立性"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E9%80%80%E5%8C%96%E5%88%86%E5%B8%83.html",{loader:()=>B(()=>import("./退化分布.html-0A45-jVb.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>在数理统计中，<strong>退化分布</strong>（或<strong>确定性分布</strong>）是指只有一种值的分布，是一种绝对事件的分布。</p>

<details class="callout" data-callout="example" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-list"><line x1="8" x2="21" y1="6" y2="6"/><line x1="8" x2="21" y1="12" y2="12"/><line x1="8" x2="21" y1="18" y2="18"/><line x1="3" x2="3.01" y1="6" y2="6"/><line x1="3" x2="3.01" y1="12" y2="12"/><line x1="3" x2="3.01" y1="18" y2="18"/></svg>
</div>
<div class="callout-title-inner">Example</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p>一个六面数值均相等的骰子；一枚正反双面一模一样的硬币。尽管它并不会随机出现数字，这种分布满足随机变量的定义，因此被认为是“退化”的。<br>
它的累积分布函数是：</p>
</div></details>`,readingTime:{minutes:.45,words:134},title:"Wikipedia",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F.html",{loader:()=>B(()=>import("./随机变量.html-DmWyWdRu.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`<p>我们希望引入一个法则, 使随机试验的结果可以用数来表示, 将随机试验的每个结果与实数对应起来, 这就是随机变量</p>

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>设随机试验的样本空间为<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mi>e</mi><mo separator="true">,</mo><mi>X</mi><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S={e},X=X(e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">e</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>是定义在样本空间S上的实值单值函数, 称<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X=X(e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>为随机变量</p>
</div></div>`,readingTime:{minutes:.89,words:266},title:"",type:"article",s:"随机变量"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E7%9A%84%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./随机变量的函数.html-WliErFtr.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p><strong>随机变量的函数</strong>：<br>
如果我们有一个随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>，那么随机变量的函数是指形如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span></span></span></span> 的表达式，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span></span></span></span> 是一个普通的数学函数。例如，如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 是一个随机变量，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>X</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(X) = X^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mi>X</mi></msup></mrow><annotation encoding="application/x-tex">g(X) = e^X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0785em;">X</span></span></span></span></span></span></span></span></span></span></span> 都是随机变量的函数。</p>`,readingTime:{minutes:.89,words:266},title:"",type:"article",s:"随机变量的函数"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/---%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0---.html",{loader:()=>B(()=>import("./---随机变量函数---.html-s8UW8T1q.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E6%9C%9F%E6%9C%9B.html" target="_blank">期望</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E6%96%B9%E5%B7%AE.html" target="_blank">方差</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E6%A0%87%E5%87%86%E5%81%8F%E5%B7%AE.html" target="_blank">标准偏差</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E7%9F%A9%E7%94%9F%E6%88%90%E5%87%BD%E6%95%B0.html" target="_blank">矩生成函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E5%8D%8F%E6%96%B9%E5%B7%AE.html" target="_blank">协方差</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0%2F%E7%9B%B8%E5%85%B3%E7%B3%BB%E6%95%B0.html" target="_blank">相关系数</a></li>
</ul>`,readingTime:{minutes:.11,words:33},title:"",type:"article",s:"---随机变量函数---"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E5%81%8F%E5%BA%A6.html",{loader:()=>B(()=>import("./偏度.html-CVl1RCQU.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p><strong>偏度</strong>是统计学中用于衡量<strong>随机变量概率分布不对称性</strong>的数字特征。它描述了分布偏离对称性的程度和方向。</p>
<p>对于一个随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>，若其均值为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span>，标准差为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span></span></span></span>，则偏度 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\\gamma_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 定义为三阶标准中心矩：</p>`,readingTime:{minutes:2.58,words:773},title:"",type:"article",s:"偏度"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E5%8D%8F%E6%96%B9%E5%B7%AE.html",{loader:()=>B(()=>import("./协方差.html-C_gT71QH.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`

<div class="callout" data-callout="note">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">离散随机变量的协方差</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.51,words:154},title:"定义",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E6%96%B9%E5%B7%AE.html",{loader:()=>B(()=>import("./方差.html-BycYGpm4.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h3>连续</h3>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">Var[X]=E[(X-E[X])^{2}]=E[X^{2}]-(E(X))^{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.54,words:161},title:"",type:"article",s:"方差"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E6%9C%9F%E6%9C%9B.html",{loader:()=>B(()=>import("./期望.html-CU-mLCNT.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">E[X]=\\sum_{i=1}^{n}x_{i}p_{i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.14,words:42},title:"性质",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E6%9C%9F%E6%9C%9B%E7%9A%84%E6%9C%9F%E6%9C%9B.html",{loader:()=>B(()=>import("./期望的期望.html-CWbS6f-s.js"),[]),meta:{tag:["数学"],excerpt:'<p>计算 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E[E[X]]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]]</span></span></span></span> 时，首先需要理解两个概念：条件期望和全概率期望。设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 是一个随机变量，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E[X]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 的期望。</p>',readingTime:{minutes:2.56,words:767},title:"",type:"article",s:"期望的期望"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E6%A0%87%E5%87%86%E5%81%8F%E5%B7%AE.html",{loader:()=>B(()=>import("./标准偏差.html-BZ35sO9Q.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>σ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><msqrt><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msqrt><mo>=</mo><msqrt><mrow><mi>E</mi><mo stretchy="false">[</mo><msup><mi>X</mi><mn>2</mn></msup><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\\sigma[X]=\\sqrt{  Var[X] }=\\sqrt{ E[X^{2}]-(E(X))^{2} }
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.2561em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9839em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">]</span></span></span><span style="top:-2.9439em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
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M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2561em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.2561em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9839em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.9439em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
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M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2561em;"><span></span></span></span></span></span></span></span></span></span></p>
</div></div>`,readingTime:{minutes:.12,words:37},title:"",type:"article",s:"标准偏差"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E7%9B%B8%E5%85%B3%E7%B3%BB%E6%95%B0.html",{loader:()=>B(()=>import("./相关系数.html-B3H0QHc6.js"),[]),meta:{author:"Cyletix",tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.17,words:51},title:"",type:"article",s:"相关系数"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E7%9F%A9%E7%94%9F%E6%88%90%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./矩生成函数.html-ByCU_lCu.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>顾名思义, 就是可以用来生成原点矩和中心距的函数</p>
<h2>定义</h2>
<h3>离散</h3>

<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p></p>
</div></div>`,readingTime:{minutes:.46,words:139},title:"",type:"article",s:"矩生成函数"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%87%BD%E6%95%B0/%E9%AB%98%E6%96%AF%E5%88%86%E5%B8%83%E7%9A%84%E7%9F%A9%E7%94%9F%E6%88%90%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./高斯分布的矩生成函数.html-DO-v0UG0.js"),[]),meta:{tag:["数学"],excerpt:'<p>高斯分布的矩生成函数 (Moment Generating Function, MGF) 可以通过其定义来计算。对于随机变量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>，其矩生成函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_X(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0785em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 定义为：</p>',readingTime:{minutes:3.61,words:1084},title:"",type:"article",s:"高斯分布的矩生成函数"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/---%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B---.html",{loader:()=>B(()=>import("./---随机过程---.html-D8_J9CFv.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E6%A6%82%E5%BF%B5.html" target="_blank">随机过程的概念</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">随机过程的统计描述</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">泊松过程及维纳过程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E5%B9%B3%E7%A8%B3%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E6%A6%82%E5%BF%B5.html" target="_blank">平稳随机过程的概念</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">各态历经性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E7%9B%B8%E5%85%B3%E5%87%BD%E6%95%B0.html" target="_blank">相关函数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E5%B9%B3%E7%A8%B3%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%8A%9F%E7%8E%87%E8%B0%B1%E5%AF%86%E5%BA%A6.html" target="_blank">平稳随机过程的功率谱密度</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E8%BF%87%E7%A8%8B.html" target="_blank">马尔可夫过程</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E5%A4%9A%E6%AD%A5%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%E7%9A%84%E7%A1%AE%E5%AE%9A.html" target="_blank">多步转移概率的确定</a></li>
<li><a class="route-link nolebase-route-link-invalid" href="#" target="_target">遍历性</a></li>
</ul>`,readingTime:{minutes:.29,words:86},title:"",type:"article",s:"---随机过程---"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E5%A4%9A%E6%AD%A5%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%E7%9A%84%E7%A1%AE%E5%AE%9A.html",{loader:()=>B(()=>import("./多步转移概率的确定.html-Bi9XpFtI.js"),[]),meta:{tag:["数学"],excerpt:`<h2>多步转移概率的确定</h2>
<h3>引言</h3>
<p>在马尔可夫过程和其他随机过程的研究中，转移概率是一个重要的概念。多步转移概率（multi-step transition probability）描述了系统从一个状态转移到另一个状态所需的多个步骤的概率。在实际应用中，这种概率常用于系统建模、预测和决策分析等领域。</p>
<h3>定义</h3>
<p>设有马尔可夫链 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其状态空间为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>，转移概率由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 表示，即从状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 转移到状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> 的概率。多步转移概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">P^{(n)}_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4578em;vertical-align:-0.413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span></span></span></span> 定义为在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 步内，从状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 到状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> 的转移概率：</p>`,readingTime:{minutes:1.74,words:521},title:"",type:"article",s:"多步转移概率的确定"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E5%B9%B3%E7%A8%B3%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%8A%9F%E7%8E%87%E8%B0%B1%E5%AF%86%E5%BA%A6.html",{loader:()=>B(()=>import("./平稳随机过程的功率谱密度.html-BdxBqWgA.js"),[]),meta:{author:["PaulSun"],tag:["数学"],excerpt:'<p>考虑宽平稳随机过程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mrow><mo fence="true">(</mo><mi>t</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X\\left(t\\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">t</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 的 Fourier 展开</p>',readingTime:{minutes:3.41,words:1024},title:"功率谱密度函数的获得",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E5%B9%B3%E7%A8%B3%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E6%A6%82%E5%BF%B5.html",{loader:()=>B(()=>import("./平稳随机过程的概念.html-8cEZRRCV.js"),[]),meta:{author:["PaulSun"],tag:["数学"],excerpt:`<p>如果随机过程的某一种性质不随下角标（时间）变化而变化，则称随机过程具有该性质的平稳性。</p>
<p>众多平稳性是根据相关函数来建立的，所以有必要先介绍相关函数的定义。</p>
<h3>相关函数</h3>
<p>相关函数是两个时刻的函数，其为两时刻随机变量的相关性，写作</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>R</mi><mrow><mo fence="true">(</mo><mi>t</mi><mo separator="true">,</mo><mi>s</mi><mo fence="true">)</mo></mrow><mo>=</mo><mi mathvariant="normal">E</mi><mrow><mo fence="true">[</mo><mi>X</mi><mrow><mo fence="true">(</mo><mi>t</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mi>X</mi><mrow><mo fence="true">(</mo><mi>s</mi><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">R\\left(t,s\\right)=\\mathrm E\\left[X\\left(t\\right),X\\left(s\\right)\\right]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">s</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">t</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">s</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span></span></p>`,readingTime:{minutes:3.29,words:986},title:"",type:"article",s:"平稳随机过程的概念"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E7%9B%B8%E5%85%B3%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./相关函数.html-BaMldQ0S.js"),[]),meta:{author:["PaulSun"],tag:["数学"],excerpt:'<p>相关函数我们主要研究的是宽平稳随机过程的相关函数，即能被记作 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mrow><mo fence="true">(</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">R\\left(t-s\\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">s</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mrow><mo fence="true">(</mo><mi>τ</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">R\\left(\\tau\\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 的相关函数，这样的相关函数具有五条基本性质和正定性。</p>',readingTime:{minutes:3.17,words:952},title:"基本性质",type:"article"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E6%A6%82%E5%BF%B5.html",{loader:()=>B(()=>import("./随机过程的概念.html-DuG6u051.js"),[]),meta:{tag:["数学"],excerpt:`<p>随机过程被认为是概率论的&quot;动力学&quot;部分。意思是说，他的研究对象是随时间演变的随机现象。对于这种现象，一般来说，人们已经不能用随机变量或多为随即变量来合理地表达，而需要用一组(无限多个)随机变量来描述。</p>
<h2>示例</h2>
<ul>
<li>热噪声电压</li>
</ul>
<hr>
<h2>GPT-4o</h2>
<p>随机过程用来描述一个随时间或空间演变的随机现象。随机过程就是一组随机变量的集合，这些随机变量依赖于某个参数（通常是时间或空间）。随机过程常用于概率论、统计学以及许多工程和科学领域。</p>
<h3>定义</h3>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span></span></span></span> 是一个参数集合（如时间或空间），<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 是定义在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span></span></span></span> 上的随机变量，则随机过程可以表示为：</p>`,readingTime:{minutes:2.41,words:722},title:"",type:"article",s:"随机过程的概念"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E8%BF%87%E7%A8%8B.html",{loader:()=>B(()=>import("./马尔可夫过程.html-CbcY2WD5.js"),[]),meta:{tag:["数学"],excerpt:'<p>在一个时间演变过程中，由时间<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>系统或过程所处的状态，可以决定系统或过程在时刻<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>&gt;</mo><msub><mi>t</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t&gt;t_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6542em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>所处的状态，而无需借助于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>前的历史信息。未来的状态仅依赖于当前状态，而不依赖于过去状态。</p>',readingTime:{minutes:3.15,words:945},title:"",type:"article",s:"马尔可夫过程"}}],["/%E6%A6%82%E7%8E%87%E8%AE%BA/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E9%93%BE.html",{loader:()=>B(()=>import("./马尔可夫链.html-Cywkk_Lc.js"),[]),meta:{tag:["数学"],excerpt:'<p>当<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E6%A6%82%E7%8E%87%E8%AE%BA%2F%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%2F%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E8%BF%87%E7%A8%8B.html" target="_blank">马尔可夫过程</a>的状态空间和时间都离散时，称为马尔可夫链。假设有一个有限的状态集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>s</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>s</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S = \\{s_1, s_2, \\ldots, s_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>，其转移概率矩阵为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>s</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>X</mi><mi>t</mi></msub><mo>=</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_{ij} = P(X_{t+1} = s_j | X_t = s_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 表示从状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 转移到状态 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">s_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的概率。</p>',readingTime:{minutes:.52,words:155},title:"",type:"article",s:"马尔可夫链"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/---%E5%90%91%E9%87%8F%E7%BB%84---.html",{loader:()=>B(()=>import("./---向量组---.html-D9rnE1fH.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F%E7%BB%84%E5%8F%8A%E5%85%B6%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88.html" target="_blank">向量组及其线性组合</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F%E7%BB%84%E7%9A%84%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7.html" target="_blank">向量组的线性相关性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F%E7%BB%84%E7%9A%84%E7%A7%A9.html" target="_blank">向量组的秩</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84.html" target="_blank">线性方程组的解的结构</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a></li>
</ul>`,readingTime:{minutes:.3,words:90},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E4%B8%80%E6%AC%A1%E7%8B%AC%E7%AB%8B.html",{loader:()=>B(()=>import("./一次独立.html-DgM3bgFQ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p><strong>一次独立</strong>（即线性无关）是指一个向量组中，任意一个向量不能由其他向量的线性组合表示。换句话说，如果一个向量组中的每个向量的线性组合为零向量时，所有系数都必须为零，那么这个向量组就是线性无关的。</p>
<p>假设我们有一个向量组 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi mathvariant="bold">v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">v</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi mathvariant="bold">v</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{\\mathbf{v}_1, \\mathbf{v}_2, \\ldots, \\mathbf{v}_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>，并且我们知道它们两两之间的内积。要证明这个向量组是线性无关的，可以利用这些内积来构造一个格拉姆矩阵，然后验证这个格拉姆矩阵是否正定。</p>`,readingTime:{minutes:2.11,words:633},title:"",type:"article",s:"一次独立"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E5%90%91%E9%87%8F.html",{loader:()=>B(()=>import("./向量.html-pkYvA3D0.js"),[]),meta:{author:["Cyletix","Zacharia2"],tag:["数学"],excerpt:`<p>理论数学中向量的定义为任何在称为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>的代数结构中的元素。一般地，同时满足具有大小和方向两个性质的几何对象即可认为是向量.</p>
<p>向量可以映射到空间中的箭头、有序的数字列表<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\\left[{\\begin{array}{c}2\\\\1\\end{array}}\\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></p>`,readingTime:{minutes:2.54,words:762},title:"长度",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E5%90%91%E9%87%8F%E7%9A%84%E6%80%A7%E8%B4%A8.html",{loader:()=>B(()=>import("./向量的性质.html-DtynqSu0.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%86%85%E7%A7%AF.html%23%25E5%2586%2585%25E7%25A7%25AF" target="_blank">内积</a></p>
<h1>长度</h1>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E8%8C%83%E6%95%B0.html" target="_blank">范数</a></p>
<h1>向量正交性</h1>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E6%80%A7.html" target="_blank">正交性</a></p>
`,readingTime:{minutes:.12,words:35},title:"内积",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E5%90%91%E9%87%8F%E7%BB%84%E5%8F%8A%E5%85%B6%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88.html",{loader:()=>B(()=>import("./向量组及其线性组合.html-DpPEUkmV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>个实数或复数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,...,<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>所组成的有序数组称为<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>维向量, 这<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>个数称为该向量的<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>个分量</p>
</div></div>`,readingTime:{minutes:2,words:600},title:"",type:"article",s:"向量组及其线性组合"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E5%90%91%E9%87%8F%E7%BB%84%E7%9A%84%E7%A7%A9.html",{loader:()=>B(()=>import("./向量组的秩.html-T-zeoqa7.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>向量组的秩是这组向量中极大线性无关组所含向量的个数。</p>
<p>若把向量组作为矩阵的列向量排成矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，则向量组的秩就等于矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的秩。因此可以通过<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95.html" target="_blank">高斯消元法</a>或<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E7%9F%A9%E9%98%B5%E7%9A%84%E7%A7%A9.html" target="_blank">矩阵的秩</a>的相关结论来计算。</p>`,readingTime:{minutes:.52,words:156},title:"",type:"article",s:"向量组的秩"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E5%90%91%E9%87%8F%E7%BB%84%E7%9A%84%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7.html",{loader:()=>B(()=>import("./向量组的线性相关性.html-ClGPriiJ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h3>向量组的线性相关性概念</h3>
<h4>定义</h4>
<p>在数学中，特别是线性代数领域，向量组的线性相关性是衡量一组向量是否能通过线性组合以某种方式表示其中某个向量的性质。</p>
<ul>
<li><strong>线性相关</strong>：如果一个向量组中的某个向量可以表示为其他向量的线性组合，那么这组向量被称为线性相关。形式上，向量组 (\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_n) 线性相关意味着存在一组不全为零的系数 (c_1, c_2, \\dots, c_n) 使得：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>n</mi></msub><mo>=</mo><mover accent="true"><mn>0</mn><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">c_1 \\vec{v}_1 + c_2 \\vec{v}_2 + \\cdots + c_n \\vec{v}_n = \\vec{0}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.864em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9274em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9274em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">0</span></span><span style="top:-3.2134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span></span></p>
</li>
<li><strong>线性无关</strong>：如果上述等式只在所有系数 (c_i) 都为零的情况下成立，则称向量组 (\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_n) 线性无关。这意味着没有任何一个向量可以被表示为其他向量的线性组合。</li>
</ul>`,readingTime:{minutes:2.17,words:652},title:"",type:"article",s:"向量组的线性相关性"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E6%9E%81%E5%B0%8F%E7%9B%B8%E5%85%B3%E5%AD%90%E9%9B%86.html",{loader:()=>B(()=>import("./极小相关子集.html-5-M41qbO.js"),[]),meta:{tag:["数学"],excerpt:`<p>极小相关子集是指满足以下两个条件的<strong>列向量子集</strong>：</p>
<ol>
<li>子集本身是线性相关的。</li>
<li>子集的任何真子集都是线性无关的。</li>
</ol>
<p>换句话说，这类子集是最小的线性相关子集，一旦去掉其中任意一个向量，剩下的向量就会变成线性无关。极小相关子集在线性代数中提供了一种有效的方法来分析矩阵的线性相关性，帮助确定向量集合中的最小冗余结构，进而简化计算和理解线性系统的性质。</p>
<h2>定义</h2>
<p>设矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A \\in \\mathbb{R}^{n \\times m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span></span> 的列向量为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>a</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">a_1, a_2, \\dots, a_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。定义子集 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo>⊆</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">J \\subseteq \\{1, 2, \\dots, m\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">}</span></span></span></span>，对应的子矩阵为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mclose">]</span></span></span></span>。</p>`,readingTime:{minutes:2.52,words:756},title:"",type:"article",s:"极小相关子集"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E6%A0%BC%E6%8B%89%E5%A7%86%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./格拉姆矩阵.html-BsSnPADq.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>格拉姆矩阵是一个对称的<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>矩阵，定义为：</p>',readingTime:{minutes:.35,words:106},title:"",type:"article",s:"格拉姆矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E6%AD%A3%E4%BA%A4%E5%90%91%E9%87%8F%E7%BB%84.html",{loader:()=>B(()=>import("./正交向量组.html-BsJtxkXm.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>正交向量组是指在内积空间中互相<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E6%80%A7.html" target="_blank">正交</a>的向量集合</p>
<h1>性质</h1>
<h3>1. 内积为零</h3>
<p>在一个正交向量组中，任意两个不同的向量的内积为零。设<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>v</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{ v_1, v_2, \\ldots, v_n \\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>是一个正交向量组，则对于任意<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo mathvariant="normal">≠</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \\neq j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span>，有</p>`,readingTime:{minutes:1.69,words:506},title:"性质",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./线性空间.html-COcylMNm.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>线性空间（或向量空间）是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%2F%E9%9B%86%E5%90%88.html" target="_blank">集合</a>结构在代数意义上的扩展。<br>
其中的元素称为<strong>向量</strong>，它们可以相加并能被来自数域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span> 的标量缩放。<br>
这一结构保证运算后仍留在同一集合中，是线性代数与解析几何的共同基础。</p>`,readingTime:{minutes:2.23,words:670},title:"",type:"article",s:"线性空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E5%90%91%E9%87%8F%E7%BB%84/%E8%A7%84%E8%8C%83%E6%AD%A3%E4%BA%A4%E5%9F%BA.html",{loader:()=>B(()=>import("./规范正交基.html-DAPk-V_n.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>规范正交基是指在一个内积空间中，由一组规范正交向量构成的基。这组向量不仅两两正交，而且每个向量的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E8%8C%83%E6%95%B0.html" target="_blank">范数</a>均为1。规范正交基使得向量的表示、投影和变换都变得更加简洁和易于计算。</p>
<h3>定义</h3>
<p>在内积空间<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>中，设<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>e</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{ e_1, e_2, \\ldots, e_n \\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>是一组向量。如果满足以下条件：</p>`,readingTime:{minutes:1.76,words:527},title:"",type:"article",s:"规范正交基"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/---%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2---.html",{loader:()=>B(()=>import("./---相似变换---.html-C-J4HitC.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E5%90%91%E9%87%8F%E7%9A%84%E6%80%A7%E8%B4%A8.html" target="_blank">向量的性质</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%89%B9%E5%BE%81%E5%80%BC.html" target="_blank">特征值</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F.html" target="_blank">特征向量</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5.html" target="_blank">相似矩阵</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E5%AF%B9%E8%A7%92%E5%8C%96.html" target="_blank">对角化</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E4%BA%8C%E6%AC%A1%E5%9E%8B.html" target="_blank">二次型</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E6%AD%A3%E5%AE%9A%E4%BA%8C%E6%AC%A1%E5%9E%8B.html" target="_blank">正定二次型</a></li>
</ul>`,readingTime:{minutes:.28,words:84},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/Jordan%E6%A0%87%E5%87%86%E5%9E%8B.html",{loader:()=>B(()=>import("./Jordan标准型.html-apPdPPxR.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>Jordan标准型通过<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2.html" target="_blank">相似变换</a>将矩阵化简为一种特殊的准对角形式，每个对角块称为一个Jordan块。<br>
Jordan块是围绕一个特征值的准对角矩阵，对角线上是相同的特征值，而对角线上方可能有若干个1，其他位置为0。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>λ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>λ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>λ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow/></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>λ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>λ</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\\begin{bmatrix}
\\lambda &amp; 1 &amp; 0 &amp; \\cdots &amp; 0 &amp; 0\\\\
0 &amp; \\lambda &amp; 1 &amp; \\cdots &amp; 0 &amp; 0\\\\
0 &amp; 0 &amp; \\lambda &amp; \\cdots &amp; 0 &amp; 0\\\\
\\vdots &amp; \\vdots &amp; \\vdots &amp;  &amp; \\vdots &amp; \\vdots\\\\
0 &amp; 0 &amp; 0 &amp; \\cdots &amp; \\lambda &amp; 1\\\\
0 &amp; 0 &amp; 0 &amp; \\cdots &amp; 0 &amp; \\lambda\\\\
\\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.86em;vertical-align:-3.68em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1499em;"><span style="top:-6.1499em;"><span class="pstrut" style="height:9.8em;"></span><span style="width:0.667em;height:7.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="7.8em" viewBox="0 0 667 7800"><path d="M403 1759 V84 H666 V0 H319 V1759 v4200 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v4200 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6501em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-7.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-7.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-7.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-6.84em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.58em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.38em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.18em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-7.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-0.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.18em;"><span style="top:-7.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-0.3675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.68em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1499em;"><span style="top:-6.1499em;"><span class="pstrut" style="height:9.8em;"></span><span style="width:0.667em;height:7.8em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="7.8em" viewBox="0 0 667 7800"><path d="M347 1759 V0 H0 V84 H263 V1759 v4200 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v4200 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6501em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.97,words:591},title:"Jordan 标准型的求解过程",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E4%BA%8C%E6%AC%A1%E5%9E%8B.html",{loader:()=>B(()=>import("./二次型.html-Cg5IJ9in.js"),[]),meta:{author:["Cyletix","DeepSeekR1"],tag:["数学"],excerpt:`<h2>背景</h2>
<p>在解析几何中,为了便于研究二次曲线</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mi>y</mi><mo>+</mo><mi>c</mi><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">ax^2+bxy+cy^2=1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p>`,readingTime:{minutes:1.72,words:516},title:"",type:"article",s:"二次型"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E4%BB%A3%E6%95%B0%E9%87%8D%E6%95%B0.html",{loader:()=>B(()=>import("./代数重数.html-DnTMOQ-j.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>特征值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> 的代数重数，是指它作为矩阵<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%89%B9%E5%BE%81%E5%A4%9A%E9%A1%B9%E5%BC%8F.html" target="_blank">特征多项式</a>的根出现的次数。</p>',readingTime:{minutes:.2,words:60},title:"",type:"article",s:"代数重数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%87%A0%E4%BD%95%E9%87%8D%E6%95%B0.html",{loader:()=>B(()=>import("./几何重数.html-B4wTHpmG.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>特征值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> 的几何重数，是其特征空间</p>',readingTime:{minutes:.24,words:71},title:"",type:"article",s:"几何重数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%8D%8A%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./半正定矩阵.html-BDsbNdjX.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p>半正定矩阵是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5.html" target="_blank">正定矩阵</a>的一个补充：</p>
<ol>
<li>对称性：矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是对称的，即 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A = A^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>。</li>
<li><strong>所有特征值均非负</strong>：矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的所有特征值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 均满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\lambda_i \\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。</li>
</ol>`,readingTime:{minutes:1.05,words:314},title:"",type:"article",s:"半正定矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%A4%8D%E7%89%B9%E5%BE%81%E6%A0%B9.html",{loader:()=>B(()=>import("./复特征根.html-dkBq36bK.js"),[]),meta:{author:["nex3z"],tag:["数学"],excerpt:'<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 矩阵的特征方程是一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次多项式，若考虑复数根，该方程恰好有 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个根（重根按重复次数计算）。研究复特征值可以揭示矩阵中隐藏的信息，通常与周期、振动、旋转等问题相关。</p>',readingTime:{minutes:2.95,words:884},title:"",type:"article",s:"复特征根"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./对称矩阵.html-C1Rd6WO6.js"),[]),meta:{tag:["数学"],excerpt:`<h2>定义</h2>
<p>对称矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A = A^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>, 即矩阵关于主对角线对称.</p>`,readingTime:{minutes:1.21,words:362},title:"",type:"article",s:"对称矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%AF%B9%E8%A7%92%E5%8C%96.html",{loader:()=>B(()=>import("./对角化.html-BO_v7Syw.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 通过<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%E7%9F%A9%E9%98%B5.html" target="_blank">相似变换矩阵</a><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>转化为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E5%AF%B9%E8%A7%92%E7%9F%A9%E9%98%B5.html" target="_blank">对角矩阵</a><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 的过程</p>`,readingTime:{minutes:1.69,words:507},title:"定义",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E5%AF%B9%E8%A7%92%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./对角矩阵.html-DXgYeDcu.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对角矩阵是一个主对角线之外的元素均为零的矩阵。通过<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2.html" target="_blank">相似变换</a><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>P</mi><mo>=</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">P^{-1}AP = D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>得到对角矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>. 其一般形式可以表示为：</p>',readingTime:{minutes:.84,words:251},title:"",type:"article",s:"对角矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%83%AF%E6%80%A7%E6%8C%87%E6%95%B0.html",{loader:()=>B(()=>import("./惯性指数.html-uxRUbRfC.js"),[]),meta:{author:["GPT-4o"],tag:["数学"],excerpt:`<p>惯性指数（Inertia Index）是二次型矩阵的一个重要特性，在研究二次型或二次曲面时常被用来描述其性质。它由 <strong>正惯性指数</strong> 和 <strong>负惯性指数</strong> 构成，定义如下：</p>
<h3>1. 二次型与惯性指数</h3>
<p>一个二次型 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1, x_2, \\dots, x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的矩阵表示为对称矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>：</p>`,readingTime:{minutes:2.26,words:677},title:"",type:"article",s:"惯性指数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%96%BD%E5%AF%86%E7%89%B9%E6%AD%A3%E4%BA%A4%E5%8C%96.html",{loader:()=>B(()=>import("./施密特正交化.html-BCRhcD5M.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>施密特正交化在<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">向量空间</a>中将一组线性无关的向量转化为一组正交向量</p>
<h2>步骤</h2>
<p>设我们有一组线性无关的向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>v</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{v_1, v_2, \\ldots, v_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>。<br>
施密特正交化过程如下：</p>`,readingTime:{minutes:1.52,words:456},title:"",type:"article",s:"施密特正交化"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%A0%87%E5%87%86%E5%9E%8B.html",{loader:()=>B(()=>import("./标准型.html-C5yo61cL.js"),[]),meta:{tag:["数学"],excerpt:`<p>标准型是指通过适当的变换，将二次型化为对角形式，从而简化其表示和计算。</p>
`,readingTime:{minutes:.15,words:46},title:"",type:"article",s:"标准型"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%AD%A3%E4%BA%A4%E5%AF%B9%E8%A7%92%E5%8C%96.html",{loader:()=>B(()=>import("./正交对角化.html-BuadWGM0.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>如果一个矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是对称矩阵，那么存在一个正交矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 使得：</p>`,readingTime:{minutes:4.28,words:1283},title:"",type:"article",s:"正交对角化"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%AD%A3%E5%AE%9A%E4%BA%8C%E6%AC%A1%E5%9E%8B.html",{loader:()=>B(()=>import("./正定二次型.html-DAW9icMp.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>正定二次型是指一个<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E4%BA%8C%E6%AC%A1%E5%9E%8B.html" target="_blank">二次型</a> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Q(x) = x^T A x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span></span></span></span>, 其中矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5.html" target="_blank">正定矩阵</a>. 这意味着对于所有非零向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>, 二次型 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Q(x) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 正定矩阵的特征值全为正数, 且其行列式也大于零.</p>',readingTime:{minutes:2.92,words:876},title:"",type:"article",s:"正定二次型"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./正定矩阵.html-CCcr57iJ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>正定矩阵是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2F%25E6%2588%2596%25E8%2580%2585%25E6%259B%25B4%25E4%25B8%2580%25E8%2588%25AC%25E7%259A%2584%255B%255B%25E5%258E%2584%25E7%25B1%25B3%25E7%259F%25A9%25E9%2598%25B5%255D%255D">[对称矩阵]</a>的一种特殊形式, 满足：</p>
<ol>
<li>对称性：矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是对称的，即 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A = A^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>。</li>
<li><strong>所有特征值均为正</strong>：矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的所有特征值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 都大于零。<br>
正定矩阵的性质类似复数中的正实数。与正定矩阵相对应的线性算子是对称正定双线性形式（复域中则对应厄米正定双线性形式）</li>
</ol>`,readingTime:{minutes:3.7,words:1111},title:"",type:"article",s:"正定矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%89%B9%E5%BE%81%E5%80%BC.html",{loader:()=>B(()=>import("./特征值.html-CK-j1nIV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>特征值和特征向量描述了线性变换中的不变量.<br>
当对向量进行线性变换时, 这个线性变换不改变向量的方向, 仅改变向量的长度, 这等效于一个标量与向量数乘. 这个标量就是特征值, 描述了线性变换在特定向量上的缩放因子</p>
<h2>定义</h2>
<p>对于一个给定的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，如果存在一个非零向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span></span> 和一个标量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span>，使得矩阵方程</p>`,readingTime:{minutes:2.93,words:879},title:"",type:"article",s:"特征值"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F.html",{loader:()=>B(()=>import("./特征向量.html-C2Sdie_h.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>对于一个给定的方阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，它的特征向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 经过这个矩阵的线性变换之后，得到的新向量仍然与原来的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 保持在同一条直线上，但其长度或方向也许会改变。</p>`,readingTime:{minutes:1.02,words:305},title:"Wikipedia",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%89%B9%E5%BE%81%E5%A4%9A%E9%A1%B9%E5%BC%8F.html",{loader:()=>B(()=>import("./特征多项式.html-CbEM1z_F.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的特征多项式是关于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> 的多项式</p>',readingTime:{minutes:.2,words:59},title:"",type:"article",s:"特征多项式"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./相似变换.html-iJje6d0k.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A, B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 阶矩阵，若存在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 阶可逆矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow/><mi>P</mi></mrow><annotation encoding="application/x-tex">{} P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>，使得：</p>',readingTime:{minutes:.25,words:76},title:"",type:"article",s:"相似变换"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%E7%9A%84%E4%B8%80%E8%88%AC%E6%AD%A5%E9%AA%A4.html",{loader:()=>B(()=>import("./相似变换的一般步骤.html-nPuhqz0_.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ol>
<li>计算<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%89%B9%E5%BE%81%E5%80%BC.html" target="_blank">特征值</a></li>
<li>计算<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F.html" target="_blank">特征向量</a></li>
<li>判断<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E5%AF%B9%E8%A7%92%E5%8C%96.html%23%25E6%259D%25A1%25E4%25BB%25B6" target="_blank">对角化</a></li>
<li>如果可以对角化
<ol>
<li>构造<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%E7%9F%A9%E9%98%B5.html" target="_blank">相似变换矩阵</a>P</li>
<li>相似变换得到<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E5%AF%B9%E8%A7%92%E7%9F%A9%E9%98%B5.html" target="_blank">对角矩阵</a>D</li>
</ol>
</li>
<li>如果不能对角化
<ol>
<li>构造<a class="route-link nolebase-route-link-invalid" href="#" target="_target">Jordan块</a></li>
<li>求解<a class="route-link nolebase-route-link-invalid" href="#" target="_target">广义特征向量</a></li>
<li>构造<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%E7%9F%A9%E9%98%B5.html" target="_blank">相似变换矩阵</a>P</li>
<li>相似变换得到<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2FJordan%E6%A0%87%E5%87%86%E5%9E%8B.html" target="_blank">Jordan标准型</a><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span></span></span></span></li>
</ol>
</li>
</ol>`,readingTime:{minutes:1.99,words:596},title:"示例",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./相似变换矩阵.html-Bfofj8sQ.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>相似变换矩阵是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2.html" target="_blank">相似变换</a>中出现的可逆矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>。若</p>',readingTime:{minutes:.34,words:101},title:"",type:"article",s:"相似变换矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2/%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./相似矩阵.html-DyT7z8gV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>相似矩阵描述了矩阵间某种等价关系。两个矩阵如果表示相同的线性变换但在不同的基下，则称它们是相似的。</p>
<h2>定义</h2>
<p>如果存在一个可逆矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>，使得矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>和<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span>满足</p>`,readingTime:{minutes:1.64,words:491},title:"",type:"article",s:"相似矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/---%E7%9F%A9%E9%98%B5---.html",{loader:()=>B(()=>import("./---矩阵---.html-DdeWAX1N.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E7%9F%A9%E9%98%B5.html" target="_blank">矩阵</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E7%9F%A9%E9%98%B5%E8%BF%90%E7%AE%97.html" target="_blank">矩阵运算</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E9%80%86%E7%9F%A9%E9%98%B5.html" target="_blank">逆矩阵</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E7%9F%A9%E9%98%B5%E5%88%86%E5%9D%97%E6%B3%95.html" target="_blank">矩阵分块法</a></li>
</ul>`,readingTime:{minutes:.22,words:67},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/LUD.html",{loader:()=>B(()=>import("./LUD.html-Cscr7dcw.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>LUD</h2>
<p>LU分解是矩阵分解的一种，将一个矩阵分解为一个下三角矩阵和一个上三角矩阵的乘积，有时需要再乘上一个置换矩阵。LU分解可以被视为<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95.html" target="_blank">高斯消元法</a>的矩阵形式。</p>
<h2>定义</h2>
<p>对于方阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的 LU 分解是将它分解成一个下三角矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 与上三角矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">U</span></span></span></span> 的乘积，也就是</p>`,readingTime:{minutes:1.25,words:376},title:"",type:"article",s:"LUD"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/QRD.html",{loader:()=>B(()=>import("./QRD.html-COHKCWG4.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>QRD</h2>
<p>QR 分解是把矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 分解为一个正交矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 与一个上三角矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span> 的乘积:</p>`,readingTime:{minutes:.64,words:193},title:"",type:"article",s:"QRD"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/SVD.html",{loader:()=>B(()=>import("./SVD.html-CzINxC_-.js"),[]),meta:{author:["Cyletix"],tag:["数学","算法","信号处理"],excerpt:`<p>奇异值分解是一种矩阵分解方法，将一个矩阵分解为三个矩阵的乘积。</p>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的矩阵，若记 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mi mathvariant="normal">rank</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r=\\operatorname{rank}(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>，则其紧致奇异值分解可写为：</p>`,readingTime:{minutes:1.33,words:399},title:"",type:"article",s:"SVD"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E4%BB%A3%E6%95%B0%E4%BD%99%E5%AD%90%E5%BC%8F.html",{loader:()=>B(()=>import("./代数余子式.html-DhWzh2PY.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的每个元素 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，其代数余子式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 定义为位置系数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{i+j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span></span></span></span></span></span></span></span></span> 与对应<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E4%BD%99%E5%AD%90%E5%BC%8F.html" target="_blank">余子式</a> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的乘积。</p>',readingTime:{minutes:.64,words:192},title:"",type:"article",s:"代数余子式"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./伴随矩阵.html-Dvx7xaj1.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\\tilde{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9202em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">A</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span>表示<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>的伴随矩阵, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\\tilde{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9202em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">A</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> 的计算步骤如下：</p>',readingTime:{minutes:.31,words:93},title:"",type:"article",s:"伴随矩阵"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5%E6%B3%95.html",{loader:()=>B(()=>import("./伴随矩阵法.html-CMROXl4B.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的方阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，逆矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 可以通过以下公式计算得到：</p>',readingTime:{minutes:.24,words:71},title:"",type:"article",s:"伴随矩阵法"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E4%BD%99%E5%AD%90%E5%BC%8F.html",{loader:()=>B(()=>import("./余子式.html-CBkU-q39.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，任取 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 行与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 列，它们交叉得到一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k \\times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 子矩阵；这个子矩阵的行列式称为一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 阶余子式。</p>',readingTime:{minutes:.31,words:94},title:"",type:"article",s:"余子式"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E5%85%B1%E8%BD%AD%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./共轭矩阵.html-BPazIhVV.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>当A为复矩阵时,用<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>a</mi><mo>ˉ</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\\bar{a}_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8539em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>表示<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>的共轭复数. 共轭矩阵记为</p>',readingTime:{minutes:.21,words:64},title:"性质",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E5%AD%90%E5%BC%8F.html",{loader:()=>B(()=>import("./子式.html-BUoxplAH.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{m\\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span>的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span>阶子式是 任取 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span>行和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span>列相交得到的<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">k^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>个交点 组成的子方阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的行列式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>M</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\det(M_{ij})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。</p>',readingTime:{minutes:.22,words:66},title:"",type:"article",s:"子式"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E5%AE%9E%E6%95%B0%E4%B8%8E%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95.html",{loader:()=>B(()=>import("./实数与矩阵乘法.html-OvMlfNP2.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>λ</mi><mi>A</mi><mo>=</mo><mi>A</mi><mi>λ</mi><mo>=</mo><mo stretchy="false">[</mo><mi>λ</mi><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\\lambda A=A\\lambda =[\\lambda a_{ij}]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">[</span><span class="mord mathnormal">λ</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>`,readingTime:{minutes:.06,words:19},title:"",type:"article",s:"实数与矩阵乘法"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./矩阵.html-D5uTvsOS.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>矩阵的形式是将代数元素按行和列组合在一起</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.72,words:515},title:"简介",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95.html",{loader:()=>B(()=>import("./矩阵乘法.html-BZdc518C.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>矩阵乘法是一种根据两个矩阵得到第三个矩阵的二元运算，第三个矩阵即前两者的乘积，称为矩阵积. 矩阵可以用来表示<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84.html" target="_blank">线性映射</a>，矩阵积则可以用来表示线性映射的复合。</p>
<h1>定义</h1>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的矩阵，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">m \\times p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 的矩阵，则它们的矩阵积 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">n \\times p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 的矩阵。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 中每一行的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个元素都与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 中对应列的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个元素对应相乘，这些乘积的和就是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 中的一个元素。</p>`,readingTime:{minutes:1.77,words:531},title:"矩阵乘法",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E7%9F%A9%E9%98%B5%E5%88%86%E5%9D%97%E6%B3%95.html",{loader:()=>B(()=>import("./矩阵分块法.html-CLzSCGur.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h3>分块矩阵乘法</h3>
<p>分块矩阵乘法是一种矩阵计算的等价表示形式，其本质是基于矩阵的分块表示后，仍遵循矩阵乘法规则（即考虑到交叉项的贡献）。假设矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 分块为：</p>`,readingTime:{minutes:1.3,words:389},title:"",type:"article",s:"矩阵分块法"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E7%9F%A9%E9%98%B5%E5%8A%A0%E6%B3%95.html",{loader:()=>B(()=>import("./矩阵加法.html-HsDXC1Kk.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A+B=[a_{ij}+b_{ij}]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>`,readingTime:{minutes:.06,words:18},title:"",type:"article",s:"矩阵加法"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E7%9F%A9%E9%98%B5%E8%BF%90%E7%AE%97.html",{loader:()=>B(()=>import("./矩阵运算.html-na17PDaF.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E7%9F%A9%E9%98%B5%E5%8A%A0%E6%B3%95.html" target="_blank">矩阵加法</a><br>
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<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E5%88%9D%E7%AD%89%E8%A1%8C%E5%8F%98%E6%8D%A2.html%23%25E5%2588%259D%25E7%25AD%2589%25E8%25A1%258C%25E5%258F%2598%25E6%258D%25A2" target="_blank">初等行变换</a><br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9F%A9%E9%98%B5%2F%E5%85%B1%E8%BD%AD%E7%9F%A9%E9%98%B5.html" target="_blank">共轭矩阵</a></p>`,readingTime:{minutes:.17,words:52},title:"矩阵分解",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E8%BD%AC%E7%BD%AE.html",{loader:()=>B(()=>import("./转置.html-DYeP_AUk.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>的转置是另一个矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>, 由下列等价动作建立：</p>`,readingTime:{minutes:1.25,words:375},title:"转置",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%9F%A9%E9%98%B5/%E9%80%86%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./逆矩阵.html-BnFmUdqF.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>对于<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>阶矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>，如果存在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>阶矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span>，使得:</p>',readingTime:{minutes:.56,words:167},title:"性质",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/---%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84---.html",{loader:()=>B(()=>import("./---线性方程组---.html-D6OtrkQd.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E5%88%9D%E7%AD%89%E8%A1%8C%E5%8F%98%E6%8D%A2.html" target="_blank">初等行变换</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E7%9F%A9%E9%98%B5%E7%9A%84%E7%A7%A9.html" target="_blank">矩阵的秩</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E8%A7%A3.html" target="_blank">线性方程组的解</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%2F%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95.html" target="_blank">高斯消元法</a></li>
</ul>`,readingTime:{minutes:1.33,words:399},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E4%B8%BB%E5%85%83.html",{loader:()=>B(()=>import("./主元.html-BXbmpAh7.js"),[]),meta:{tag:["数学"],excerpt:`<p>在矩阵的行阶梯形或简化行阶梯形中，“主元”（pivot）指的是每一行中最左边的非零元素。更具体地说：</p>
<ul>
<li><strong>行阶梯形矩阵：</strong> 满足以下条件的矩阵：
<ul>
<li>所有非零行（即至少包含一个非零元素的行）都在所有零行（即所有元素都为零的行）的上面。</li>
<li>每个非零行的第一个非零元素（即该行的最左边非零元素）所在的列位于其上一行的第一个非零元素所在列的右侧。</li>
<li>所有主元下面的元素都是零。</li>
</ul>
</li>
<li><strong>简化行阶梯形矩阵：</strong> 是行阶梯形矩阵的特例，它还满足以下条件：
<ul>
<li>所有主元都是 1。</li>
<li>所有主元上方的元素都是零。</li>
</ul>
</li>
</ul>`,readingTime:{minutes:2.45,words:734},title:"",type:"article",s:"主元"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E5%88%9D%E7%AD%89%E8%A1%8C%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./初等行变换.html-Cm1T9em6.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>矩阵初等变换是对矩阵进行的基本操作, 可以将矩阵变换成不同的形式, 但不会改变矩阵的秩. 初等变换主要用于求解线性方程组、计算行列式、求逆矩阵和进行矩阵分解.</p>

<div class="callout" data-callout="tip">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-flame"><path d="M8.5 14.5A2.5 2.5 0 0 0 11 12c0-1.38-.5-2-1-3-1.072-2.143-.224-4.054 2-6 .5 2.5 2 4.9 4 6.5 2 1.6 3 3.5 3 5.5a7 7 0 1 1-14 0c0-1.153.433-2.294 1-3a2.5 2.5 0 0 0 2.5 2.5z"/></svg>
</div>
<div class="callout-title-inner">Tip</div>
</div>
<div class="callout-content"><p>在求解线性方程组中默认指代<strong>初等行变换</strong>.</p>
</div></div>`,readingTime:{minutes:1.82,words:545},title:"简介",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E6%AD%A3%E8%A7%84%E6%96%B9%E7%A8%8B.html",{loader:()=>B(()=>import("./正规方程.html-L2XJc_7j.js"),[]),meta:{tag:["数学"],excerpt:'<p>对于求 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mi>A</mi><mi>x</mi><mo>−</mo><mi>b</mi><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\\|Ax-b\\|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord">∥</span></span></span></span> 的最小值或使之最小的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的问题，常见的方法是通过最小二乘法来解决。这里主要涉及到线性代数和数值分析中的一些基本概念和计算方法。</p>',readingTime:{minutes:3.27,words:980},title:"",type:"article",s:"正规方程"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E7%9F%A9%E9%98%B5%E7%9A%84%E7%A7%A9.html",{loader:()=>B(()=>import("./矩阵的秩.html-D8z_f_Ar.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的秩是它的行向量(或列向量)中最大线性无关集的向量个数。<br>
<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的秩记作 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>k</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">rank(A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0315em;">ank</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span>。</p>`,readingTime:{minutes:.8,words:241},title:"推论",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E8%A7%A3.html",{loader:()=>B(()=>import("./线性方程组的解.html-B7wYGuhs.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>线性方程组是由多个线性方程组成的方程组，通常可以用矩阵方法来求解。一个线性方程组的解是找到变量的值，使得每个方程同时成立。线性方程组的一般形式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>a</mi><mn>11</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>12</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>a</mi><mn>21</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>22</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>a</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><msub><mi>b</mi><mi>m</mi></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\\begin{cases}
a_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n = b_1 \\\\
a_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2 \\\\
\\vdots \\\\
a_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n = b_m \\\\
\\end{cases}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.252em;vertical-align:-2.876em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-1.366em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-1.358em;"><span class="pstrut" style="height:3.216em;"></span><span style="height:1.216em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="1.216em" style="width:0.8889em" viewBox="0 0 888.89 1216" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V1216 H384z M384 0 H504 V1216 H384z"/></svg></span></span><span style="top:-3.216em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.358em;"><span class="pstrut" style="height:3.216em;"></span><span style="height:1.216em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="1.216em" style="width:0.8889em" viewBox="0 0 888.89 1216" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V1216 H384z M384 0 H504 V1216 H384z"/></svg></span></span><span style="top:-5.566em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.376em;"><span style="top:-6.0555em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6155em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.6835em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.2435em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">mn</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.876em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:2,words:599},title:"",type:"article",s:"线性方程组的解"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84.html",{loader:()=>B(()=>import("./线性方程组的解的结构.html-CACCxQXo.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>根据rank判断, 线性方程组的解结构可以分为三种情况:</p>
<ul>
<li>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\\text{rank}(A) = \\text{rank}(A|b) = n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>, 有唯一解</li>
<li>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\\text{rank}(A) = \\text{rank}(A|b) &lt; n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>, 有无穷个解</li>
<li>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mtext>rank</mtext><mo stretchy="false">(</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\text{rank}(A) &lt; \\text{rank}(A|b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">rank</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>, 无解</li>
</ul>`,readingTime:{minutes:.26,words:77},title:"",type:"article",s:"线性方程组的解的结构"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E8%A1%8C%E6%9C%80%E7%AE%80%E5%BD%A2.html",{loader:()=>B(()=>import("./行最简形.html-aIULCP9W.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p><strong>行最简形式</strong>（Reduced Row Echelon Form, RREF）是一种特殊的行梯形形式，其中每个主元是1，并且是其所在列的唯一非零元素。这实际上是此矩阵中可能存在的最大单位矩阵,</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>RREF</mtext><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>I</mi><mi>r</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>F</mi><mrow><mi>r</mi><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>O</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo>×</mo><mi>r</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>O</mi><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\\text{RREF}(C) = \\begin{bmatrix}
I_r &amp; F_{r\\times (n-r)} \\\\
O_{(m-r)\\times r} &amp; O_{(m-r)\\times (n-r)}
\\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">RREF</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mclose mtight">)</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mbin mtight">×</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mclose mtight">)</span><span class="mbin mtight">×</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p>`,readingTime:{minutes:2.41,words:724},title:"",type:"article",s:"行最简形"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95.html",{loader:()=>B(()=>import("./高斯消元法.html-BM6VSbW9.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>主要用于</p>
<ol>
<li><a href="#%E6%B1%82%E8%A7%A3%E9%80%86%E7%9F%A9%E9%98%B5">求解逆矩阵</a></li>
<li><a href="#%E6%B1%82%E8%A7%A3%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84">求解线性方程组</a></li>
<li><a href="#%E8%AE%A1%E7%AE%97%E8%A1%8C%E6%9C%80%E7%AE%80%E5%BD%A2">计算行最简形</a></li>
</ol>
<p>这几种应用的本质都是通过初等行变换进行矩阵化简, 只是增广矩阵的构造和最终目标有所不同.</p>`,readingTime:{minutes:2.79,words:837},title:"高斯消元法",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/---%E8%A1%8C%E5%88%97%E5%BC%8F---.html",{loader:()=>B(()=>import("./---行列式---.html-CP7MhrGZ.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E8%A1%8C%E5%88%97%E5%BC%8F.html" target="_blank">行列式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E5%85%A8%E6%8E%92%E5%88%97.html" target="_blank">全排列</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E9%80%86%E5%BA%8F%E6%95%B0.html" target="_blank">逆序数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E5%AF%B9%E6%8D%A2.html" target="_blank">对换</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E8%A1%8C%E5%88%97%E5%BC%8F%E6%80%A7%E8%B4%A8.html" target="_blank">行列式性质</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%B1%95%E5%BC%80%E6%B3%95.html" target="_blank">拉普拉斯展开法</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E8%A1%8C%E5%88%97%E5%BC%8F%2F%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99.html" target="_blank">克拉默法则</a></li>
</ul>`,readingTime:{minutes:.27,words:81},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./克拉默法则.html-Dc0bR5sX.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>克拉默法则是一种解线性方程组的技巧, 利用行列式来表达未知数的解析解。<br>
对于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个未知数的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 元线性方程组:</p>`,readingTime:{minutes:1.49,words:448},title:"",type:"article",s:"克拉默法则"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E5%85%A8%E6%8E%92%E5%88%97.html",{loader:()=>B(()=>import("./全排列.html-A38jVwUc.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>在行列式的 Leibniz 展开式中, 所谓全排列是指数字 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1,2,\\dots,n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 的所有排列。每个排列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span></span></span></span> 对应一项</p>`,readingTime:{minutes:.64,words:193},title:"定义",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E5%AF%B9%E6%8D%A2.html",{loader:()=>B(()=>import("./对换.html-DUkX7Qht.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在排列中,将任意两个元素对调,其尔的元察不动,这种作出新排列的手续叫做对换。<br>
在排列中, 将任意两个元素对调, 其余元素保持不动, 这种得到新排列的操作叫做对换。<br>
将相邻两个元素对换, 叫做相邻对换。</p>
<p>定理 1：一个排列中的任意两个元素对换, 排列的奇偶性改变。<br>
定理 2：n阶行列式也可定义为</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo>=</mo><mo>∑</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>t</mi></msup><msub><mi>a</mi><mrow><mn>1</mn><msub><mi>p</mi><mn>1</mn></msub></mrow></msub><msub><mi>a</mi><mrow><mn>2</mn><msub><mi>p</mi><mn>2</mn></msub></mrow></msub><mo>⋯</mo><msub><mi>a</mi><mrow><mi>n</mi><msub><mi>p</mi><mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">D=\\sum (-1)^t a_{1p_1}a_{2p_2}\\cdots a_{np_n}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.56,words:169},title:"",type:"article",s:"对换"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%B1%95%E5%BC%80%E6%B3%95.html",{loader:()=>B(()=>import("./拉普拉斯展开法.html-Ba1hAca4.js"),[]),meta:{author:["Cyletix","Claude3"],tag:["数学"],excerpt:`<p>拉普拉斯展开法可以将行列式的计算化为低阶行列式的运算。</p>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 阶行列式:</p>`,readingTime:{minutes:1.67,words:501},title:"相关内容",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E8%8C%83%E5%BE%B7%E8%92%99%E5%BE%B7%E8%A1%8C%E5%88%97%E5%BC%8F.html",{loader:()=>B(()=>import("./范德蒙德行列式.html-DaZfwNje.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>范德蒙德行列式</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>3</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"/></mpadded></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>n</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mi>n</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow></mrow><annotation encoding="application/x-tex">V(x_1, x_2, \\ldots, x_n) = \\begin{vmatrix}
1 &amp; x_1 &amp; x_1^2 &amp; \\cdots &amp; x_1^{n-1} \\\\
1 &amp; x_2 &amp; x_2^2 &amp; \\cdots &amp; x_2^{n-1} \\\\
1 &amp; x_3 &amp; x_3^2 &amp; \\cdots &amp; x_3^{n-1} \\\\
\\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\
1 &amp; x_n &amp; x_n^2 &amp; \\cdots &amp; x_n^{n-1}
\\end{vmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:6.7027em;vertical-align:-3.1014em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5499em;"><span style="top:-5.5499em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.333em;height:6.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="6.6em" viewBox="0 0 333 6600"><path d="M145 15 v585 v5400 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-5400 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v5400 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0501em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6014em;"><span style="top:-6.4346em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2204em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-4.0061em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.1461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-0.9461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1014em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6014em;"><span style="top:-6.4346em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.2204em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0061em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-0.9461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1014em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6014em;"><span style="top:-6.4346em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.2204em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0061em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-0.9461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1014em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6014em;"><span style="top:-6.2471em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.0329em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.8186em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-1.9586em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-0.7586em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1014em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6014em;"><span style="top:-6.4346em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.4337em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.2204em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.4337em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0061em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.4337em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-0.9461em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1014em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5499em;"><span style="top:-5.5499em;"><span class="pstrut" style="height:8.6em;"></span><span style="width:0.333em;height:6.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="6.6em" viewBox="0 0 333 6600"><path d="M145 15 v585 v5400 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-5400 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v5400 v585 h43z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.0501em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:3.74,words:1122},title:"推广",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E8%A1%8C%E5%88%97%E5%BC%8F.html",{loader:()=>B(()=>import("./行列式.html-DFVLgU_U.js"),[]),meta:{author:["Cyletix","Zacharia2"],tag:["数学"],excerpt:`<p>行列式是一个数，它与一个方阵相关联，并用于描述该方阵的一些性质。<br>
给定一个 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的方阵A，行列式可以表示为det(A)或|A|。<br>
行列式的定义可以通过递归地将方阵拆分为更小的子方阵来得到。</p>`,readingTime:{minutes:4.08,words:1225},title:"",type:"article",s:"行列式"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E8%A1%8C%E5%88%97%E5%BC%8F%E6%80%A7%E8%B4%A8.html",{loader:()=>B(()=>import("./行列式性质.html-DC9JzKLX.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>方阵的行列式具有以下性质:</p>',readingTime:{minutes:.29,words:86},title:"",type:"article",s:"行列式性质"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%8C%E5%88%97%E5%BC%8F/%E9%80%86%E5%BA%8F%E6%95%B0.html",{loader:()=>B(()=>import("./逆序数.html-LRSbhBKW.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>逆序数</mtext><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">[</mo><msub><mi>σ</mi><mi>i</mi></msub><mo>&gt;</mo><msub><mi>σ</mi><mi>j</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\\text{逆序数} = \\sum_{i=1}^{n-1} \\sum_{j=i+1}^{n} [\\sigma_i &gt; \\sigma_j]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">逆序数</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.2149em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>`,readingTime:{minutes:.08,words:25},title:"",type:"article",s:"逆序数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%A5%E5%85%85/---%E8%A1%A5%E5%85%85---.html",{loader:()=>B(()=>import("./---补充---.html-DIOtDlsG.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`<p>目前整理的线性代数内容已经非常全面和系统，涵盖了许多重要主题和细节。不过，根据遇到的一些额外内容，补充以确保目录更加完整：</p>
<h1>文档-所有文档</h1>
<div class="language-dataview line-numbers-mode" data-highlighter="shiki" data-ext="dataview" style="--shiki-light:#383A42;--shiki-dark:#abb2bf;--shiki-light-bg:#FAFAFA;--shiki-dark-bg:#282c34"><pre class="shiki shiki-themes one-light one-dark-pro vp-code"><code class="language-dataview"><span class="line"><span>LIST</span></span>
<span class="line"><span>WHERE file.folder=this.file.folder</span></span>
<span class="line"><span>AND !contains(file.name,this.file.name)</span></span></code></pre>
<div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div>`,readingTime:{minutes:.28,words:85},title:"文档-所有文档",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%A5%E5%85%85/Matroid%E7%9A%84%E5%9F%BA.html",{loader:()=>B(()=>import("./Matroid的基.html-BP1Uw5Uv.js"),[]),meta:{tag:["数学"],excerpt:`<p>(<strong>超纲内容</strong>)<br>
拟阵的基是一个最大独立集，即一个独立集且不被包含在任何其他独立集中的集合。</p>
<h2>示例</h2>
<p>在二维欧几里得平面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\mathbb{R}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 上的拟阵中，考虑以下独立集：</p>`,readingTime:{minutes:1.98,words:594},title:"",type:"article",s:"Matroid的基"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E8%A1%A5%E5%85%85/%E5%B7%A1%E5%9B%9E%E8%A1%8C%E5%88%97.html",{loader:()=>B(()=>import("./巡回行列.html-BLPi5mvF.js"),[]),meta:{tag:["数学"],excerpt:`
<div class="callout" data-callout="info">
<div class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-info"><circle cx="12" cy="12" r="10"/><path d="M12 16v-4"/><path d="M12 8h.01"/></svg>
</div>
<div class="callout-title-inner">定义</div>
</div>
<div class="callout-content"><p>在线性代数中，<strong>循环矩阵</strong>是一种特殊形式的 Toeplitz矩阵，它的列向量的每个元素都是前一个列向量各元素依次右移一个位置得到的结果。由于可以用离散傅立叶变换快速解循环矩阵，所以在数值分析中有重要的应用。</p>
</div></div>`,readingTime:{minutes:.95,words:285},title:"",type:"article",s:"巡回行列"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/---%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4---.html",{loader:()=>B(()=>import("./---线性空间---.html-mqLqkenr.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E7%BB%B4%E6%95%B0.html" target="_blank">维数</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%9F%BA.html" target="_blank">基</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%9D%90%E6%A0%87.html" target="_blank">坐标</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%9F%BA%E5%8F%98%E6%8D%A2.html" target="_blank">基变换</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%9D%90%E6%A0%87%E5%8F%98%E6%8D%A2.html" target="_blank">坐标变换</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2.html" target="_blank">线性变换</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%AD%90%E7%A9%BA%E9%97%B4.html" target="_blank">子空间</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%86%85%E7%A7%AF.html" target="_blank">内积</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E6%80%A7.html" target="_blank">正交性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E6%8A%95%E5%BD%B1.html" target="_blank">正交投影</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E8%A1%A5%E7%A9%BA%E9%97%B4.html" target="_blank">正交补空间</a></li>
</ol>`,readingTime:{minutes:.41,words:123},title:"目录",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E4%B8%89%E7%BB%B4%E5%9D%90%E6%A0%87%E7%B3%BB.html",{loader:()=>B(()=>import("./三维坐标系.html-CUf2C8O4.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在三维坐标系统中，通常采用的是右手坐标系作为标准的坐标方向规定，这是在多数工程和科学领域中广泛接受的约定。右手坐标系的规定如下：</p>
<p>右手定则确定了右手坐标系：<br>
<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%8F%B3%E6%89%8B%E5%AE%9A%E5%88%99.html" target="_blank">右手定则</a></p>
`,readingTime:{minutes:.29,words:86},title:"",type:"article",s:"三维坐标系"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%83%8F%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./像空间.html-Z7Eho0ar.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>给定一个线性映射 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f: V \\rightarrow W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> 是向量空间，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 的像空间是所有向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span> 的集合，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 取遍 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 中的所有向量。<br>
像空间可以表示为：</p>`,readingTime:{minutes:1.3,words:389},title:"",type:"article",s:"像空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./内积空间.html-yz37VK5O.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>内积空间是增添了内积运算的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">向量空间</a>，它推广了原来欧几里德空间的点积，而从比较一般的角度看待向量的“夹角”、“长度”还有正交性。</p>
<p>内积空间有时也叫做准希尔伯特空间（pre-Hilbert space），因为由内积定义的距离完备化之后就会得到一个希尔伯特空间。</p>
<p>在早期的著作中，本条目所定义的内积空间被称作酉空间，但这些著作里的“内积空间”反而指的是有限维欧几里德空间或可数维的<a class="route-link nolebase-route-link-invalid" href="#" target="_target">Lp空间</a>。</p>`,readingTime:{minutes:.59,words:176},title:"",type:"article",s:"内积空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%8F%B3%E6%89%8B%E5%AE%9A%E5%88%99.html",{loader:()=>B(()=>import("./右手定则.html-DT7TLaGO.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<ol>
<li>食指与手掌平行</li>
<li>中指与手掌垂直</li>
<li>拇指垂直前两者</li>
</ol>
<p>以上规则确定了右手坐标系:</p>
<ul>
<li>拇指指向X轴正方向</li>
<li>食指指向Y轴正方向</li>
<li>中指指向Z轴正方向</li>
</ul>
<p>这个原则还有一个旋向的表示版本:<br>
右手四指握拳的旋向为x轴正方向到y轴正方向旋转, 这个旋向确定的右手拇指方向为z轴方向.<br>
如果旋向确定的方向与z轴相反, 则说明坐标系为左手坐标系</p>
`,readingTime:{minutes:.52,words:156},title:"",type:"article",s:"右手定则"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%9D%90%E6%A0%87.html",{loader:()=>B(()=>import("./坐标.html-D_SbMBk9.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h4>Introduction</h4>
<p>在线性代数里，坐标是向量相对于一组基的表示系数。</p>
<h4>Definition</h4>
<p>设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 是一个向量空间，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>e</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B=\\{e_1,e_2,\\ldots,e_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 的一组基。若向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v\\in V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 可以唯一写成</p>`,readingTime:{minutes:.78,words:234},title:"",type:"article",s:"坐标"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%9D%90%E6%A0%87%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./坐标变换.html-i8y5_7da.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h4>Introduction</h4>
<p>在线性代数中，坐标变换是指同一个向量在不同基下的坐标表示之间的转换。</p>
<h4>Definition</h4>
<p>设同一个向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 在旧基下的坐标为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>v</mi><msub><mo stretchy="false">]</mo><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">[v]_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，在新基下的坐标为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>v</mi><msub><mo stretchy="false">]</mo><msup><mi>B</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></msub></mrow><annotation encoding="application/x-tex">[v]_{B&#x27;}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。若新基向量在旧基下的坐标按列组成矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>，则有：</p>`,readingTime:{minutes:.86,words:259},title:"",type:"article",s:"坐标变换"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%9F%BA.html",{loader:()=>B(()=>import("./基.html-St3UlNyx.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>基指的是向量空间中的一个向量集合, 满足以下两个条件:</p>
<ol>
<li><strong>线性独立</strong>: 基中的向量之间互相独立, 即没有任何一个向量可以表示为其他向量的线性组合. 用公式表示, 如果有向量集合<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>v</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{v_1, v_2, \\dots, v_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>, 那么对于标量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>c</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">c_1, c_2, \\dots, c_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, 如果</li>
</ol>`,readingTime:{minutes:1.31,words:394},title:"",type:"article",s:"基"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%9F%BA%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./基变换.html-Ccp4Qgsm.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h4>Introduction</h4>
<p>基变换是指在向量空间中，从一个基变换到另一个基的过程。</p>
<h4>Definition</h4>
<p>给定两个基 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>e</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B=\\{e_1, e_2, \\ldots, e_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>B</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mo stretchy="false">{</mo><msubsup><mi>e</mi><mn>1</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo><msubsup><mi>e</mi><mn>2</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msubsup><mi>e</mi><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B&#x27;=\\{e&#x27;_1, e&#x27;_2, \\ldots, e&#x27;_n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>，同一向量在这两个基下会有不同的坐标表示。基变换矩阵用于把一种坐标表示转换成另一种坐标表示。</p>`,readingTime:{minutes:1.03,words:309},title:"",type:"article",s:"基变换"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%AD%90%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./子空间.html-CdfHA-oY.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>子空间是原向量空间的一个子集,  <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo>⊂</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">W \\subset V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> ，它本身也构成一个向量空间。</p>`,readingTime:{minutes:.98,words:294},title:"",type:"article",s:"子空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./对偶空间.html-B-lwiJQ2.js"),[]),meta:{tag:["数学"],excerpt:`<h3>对偶空间的定义</h3>
<p>在数学中，特别是线性代数和泛函分析里，一个向量空间的对偶空间是由其上的所有线性泛函（线性映射到其标量域）组成的向量空间。如果有一个向量空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>，其定义在域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span> 上（例如实数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>或复数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span>），那么 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 的对偶空间，通常表示为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>V</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>，包含所有从 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span> 的线性映射。</p>`,readingTime:{minutes:1.35,words:405},title:"",type:"article",s:"对偶空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./希尔伯特空间.html-DmpVGe0T.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在数学里，希尔伯特空间即完备的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4.html" target="_blank">内积空间</a>，也就是一个带有<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%86%85%E7%A7%AF.html" target="_blank">内积</a>的完备向量空间。内积的构造推广了欧几里得空间的距离和角的概念；完备则确保了其上所有的柯西序列会收敛到此空间里的一点，从而微积分中的许多概念都可以推广到希尔伯特空间中。</p>
<p>希尔伯特空间为基于任意正交系上的多项式表示的<a class="route-link nolebase-route-link-invalid" href="#" target="_target">傅立叶级数</a>和<a class="route-link nolebase-route-link-invalid" href="#" target="_target">傅立叶变换</a>提供了一种有效的表述方式，而这也是泛函分析的核心概念之一。另外希尔伯特空间也是量子力学的重要数学基础之一。</p>`,readingTime:{minutes:3.48,words:1045},title:"",type:"article",s:"希尔伯特空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%97%8B%E8%BD%AC%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./旋转变换.html-DjXQfNQE.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>二维直角坐标的函数从原点旋转<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span>°的旋转变换为: 对于函数上每点<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>右乘<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>:</p>',readingTime:{minutes:.97,words:291},title:"三维",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%97%8B%E8%BD%AC%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./旋转矩阵.html-sSP4vn53.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>旋转矩阵是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E6%9D%8E%E7%BE%A4%E6%9D%8E%E4%BB%A3%E6%95%B0%2F%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5.html" target="_blank">正交矩阵</a>的一种特殊形式，用于描述二维平面上的旋转变换。</p>
<h2>二维空间</h2>
<p>在二维空间中，旋转可以用一个单一的角 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 定义。作为约定，正角表示逆时针旋转。把笛卡尔坐标的列向量关于原点逆时针旋转 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 的矩阵是:</p>`,readingTime:{minutes:2.91,words:874},title:"性质",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0.html",{loader:()=>B(()=>import("./正交函数.html-YCygVnFR.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`
<details class="callout" data-callout="ai" data-callout-fold="+" open>
<summary class="callout-title">
<div class="callout-title-icon">
<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-pencil"><path d="M17 3a2.85 2.83 0 1 1 4 4L7.5 20.5 2 22l1.5-5.5Z"/><path d="m15 5 4 4"/></svg>
</div>
<div class="callout-title-inner">ChatGPT</div>
<div class="callout-fold"></div>
</summary>
<div class="callout-content"><p></p>
<p>正交函数是指在某个特定的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4.html" target="_blank">内积空间</a>中，两个函数之间的内积为零。换句话说，正交函数是在内积空间中相互垂直的函数。如果两个函数正交，意味着它们在某种程度上互不相关。<br>
在实数域上，两个函数 f(x) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>是正交的，如果它们的乘积在某个区间上满足以下条件：</p>
</div></details>`,readingTime:{minutes:.91,words:272},title:"",type:"article",s:"正交函数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%AD%A3%E4%BA%A4%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./正交变换.html-BOznA0Dr.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>在线性代数中，正交变换是线性变换的一种。如果对于任意向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 其内积等于正交变换后之向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span> 之内积，则称之为正交变换。</p>`,readingTime:{minutes:.65,words:195},title:"Wikipedia",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%AD%A3%E4%BA%A4%E6%80%A7.html",{loader:()=>B(()=>import("./正交性.html-BAGp2-6y.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在带有<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%2F%E5%86%85%E7%A7%AF.html" target="_blank">内积</a>的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>中，正交指的是两个向量在内积意义下“垂直”。如果两个向量的内积为零，则称这两个向量正交。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">⟨</mo><mover accent="true"><mi>a</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>b</mi><mo>⃗</mo></mover><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\langle \\vec{a}, \\vec{b}\\rangle=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2274em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p>`,readingTime:{minutes:.42,words:126},title:"",type:"article",s:"正交性"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%AD%A3%E4%BA%A4%E6%8A%95%E5%BD%B1.html",{loader:()=>B(()=>import("./正交投影.html-DYbW9fvC.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h4>Introduction</h4>
<p>正交投影是将向量垂直投影到一个子空间的过程。</p>
<h4>Definition</h4>
<p>若 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>w</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>w</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>w</mi><mi>k</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{w_1, w_2, \\ldots, w_k\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是子空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> 的一组两两正交基，则向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> 上的正交投影 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>proj</mtext><mi>W</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\text{proj}_W(v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">proj</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2342em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">W</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span> 为：</p>`,readingTime:{minutes:.89,words:267},title:"",type:"article",s:"正交投影"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E6%AD%A3%E4%BA%A4%E8%A1%A5%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./正交补空间.html-Bvy5Hdch.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<p>正交补空间（Orthogonal Complement）是线性代数和向量空间理论中的一个重要概念，主要用于描述向量空间中某个子空间相对于整个空间的正交关系。</p>
<h2>定义</h2>
<p>给定一个向量空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 和它的一个子空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> 的正交补空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>W</mi><mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">W^\\perp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⊥</span></span></span></span></span></span></span></span></span></span></span> 是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 中所有与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> 中的每个向量都正交的向量构成的集合。用数学语言表达就是：</p>`,readingTime:{minutes:2.16,words:648},title:"",type:"article",s:"正交补空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E7%9B%B4%E5%92%8C.html",{loader:()=>B(()=>import("./直和.html-Dfb3OvJR.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>直和用于描述两个或多个较小的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2F%25E5%25A6%2582%255B%255B%25E7%25BE%25A4%255D%255D%25E3%2580%2581%255B%255B%25E7%258E%25AF%255D%255D%25E3%2580%2581%255B%255B%25E7%25BA%25BF%25E6%2580%25A7%25E7%25A9%25BA%25E9%2597%25B4%257C%25E5%2590%2591%25E9%2587%258F%25E7%25A9%25BA%25E9%2597%25B4%255D%255D%25E7%25AD%2589">[代数结构]</a>组合成一个更大结构的方法，同时保持原有结构的独立性。</p>
<h3>向量空间的直和</h3>
<p>直和最常见的应用是在向量空间中。如果有两个向量空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span>，它们的直和记为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>⊕</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V \\oplus W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span>，定义为所有可能的有序对 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo separator="true">,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v, w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mclose">)</span></span></span></span> 的集合，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \\in V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span>。这个新的向量空间的加法和标量乘法定义为：</p>`,readingTime:{minutes:1.42,words:426},title:"",type:"article",s:"直和"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E7%BA%BF%E6%80%A7.html",{loader:()=>B(()=>import("./线性.html-6i7QQJqS.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>线性描述的是一种简单直接的变换的性质:</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo stretchy="false">[</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mi>T</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T[ax + by] = a T(x) + b T(y)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:.26,words:78},title:"",type:"article",s:"线性"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./线性变换.html-BH3IBz2T.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>线性变换是满足<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E7%BA%BF%E6%80%A7.html" target="_blank">线性</a>性质的变换. 也是从一个向量空间到另一个向量空间的函数.</p>
<h2>定义</h2>
<p>一个线性变换：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">T: V \\to W
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span></span></p>`,readingTime:{minutes:.9,words:270},title:"",type:"article",s:"线性变换"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E7%BA%BF%E6%80%A7%E7%8B%AC%E7%AB%8B.html",{loader:()=>B(()=>import("./线性独立.html-k3j8h-0o.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>两个向量独立（线性无关）指的是这两个向量之间不存在非零的线性组合使其等于零向量。具体来说，设有两个向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">v</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\\mathbf{v}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">v</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\\mathbf{v}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.016em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，它们线性独立的定义如下：</p>',readingTime:{minutes:4.68,words:1404},title:"两个以上的情况",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E7%BB%B4%E6%95%B0.html",{loader:()=>B(()=>import("./维数.html-C6yN4YuW.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:`<h4>Introduction</h4>
<p>维数（Dimension）是数学中描述向量空间或几何体大小的一个重要概念。</p>
<h4>Definition</h4>
<p>对于有限维向量空间，维数指的是任一组基所含向量的个数。若一个向量空间的一组基由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个向量组成，则该向量空间的维数为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>。</p>`,readingTime:{minutes:.8,words:240},title:"",type:"article",s:"维数"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E8%8C%83%E6%95%B0.html",{loader:()=>B(()=>import("./范数.html-D8Rr6-ND.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>向量的<strong>大小</strong>也称<strong>模长</strong>、<strong>长度</strong>(Magnitude)表示<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E5%90%91%E9%87%8F%E7%BB%84%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4.html" target="_blank">线性空间</a>中从原点到该向量所指向的点的直线距离.</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">∥</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mo fence="true">∥</mo></mrow><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\\left\\|{\\vec {v}}\\right\\|=\\sqrt{ \\sum_{i=1}^n v_{i}^2 }
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∥</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∥</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1568em;vertical-align:-1.2777em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791em;"><span class="svg-align" style="top:-5.1168em;"><span class="pstrut" style="height:5.1168em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.8391em;"><span class="pstrut" style="height:5.1168em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.1968em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="3.1968em" viewBox="0 0 400000 3196" preserveAspectRatio="xMinYMin slice"><path d="M702 80H40000040
H742v3062l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1
h-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170
c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667
219 661 l218 661zM702 80H400000v40H742z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:.43,words:130},title:"性质",type:"article"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E9%9B%B6%E5%8C%96%E5%BA%A6.html",{loader:()=>B(()=>import("./零化度.html-IMaRqaTq.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>零化度(Nullity)是矩阵<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E9%9B%B6%E7%A9%BA%E9%97%B4.html" target="_blank">零空间</a>的维数, 即所有使得<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Ax = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>的向量集合的维数.<br>
零化度反映了线性方程组<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Ax = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>的解空间的自由度.</p>`,readingTime:{minutes:.82,words:245},title:"",type:"article",s:"零化度"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E9%9B%B6%E7%A9%BA%E9%97%B4.html",{loader:()=>B(()=>import("./零空间.html-Bk2AKmqf.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是矩阵, 它的零空间是所有满足 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Ax=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的向量构成的线性<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E5%AD%90%E7%A9%BA%E9%97%B4.html" target="_blank">子空间</a>。这个子空间的维度称为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 的<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FNullity">[零化度]</a>。</p>`,readingTime:{minutes:2.19,words:658},title:"",type:"article",s:"零空间"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E9%9B%B6%E7%A9%BA%E9%97%B4%E7%9A%84%E8%AE%A1%E7%AE%97%E8%BF%87%E7%A8%8B.html",{loader:()=>B(()=>import("./零空间的计算过程.html-DX8QALJ9.js"),[]),meta:{tag:["数学","计算"],excerpt:`<h2>问题描述</h2>
<p>给定一个矩阵：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 2 &amp; 4 &amp; 6 \\\\ 3 &amp; 6 &amp; 9 \\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:1.08,words:325},title:"",type:"article",s:"零空间的计算过程"}}],["/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/%E9%BD%90%E6%AC%A1.html",{loader:()=>B(()=>import("./齐次.html-qLp38uRN.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`
<p>齐次是描述数学对象(多项式, 函数, 微分方程, 线性方程组)的一种性质, 反映了一种在尺度变换下的不变性或对称性。</p>
<h1>齐次变换</h1>
<p>对于所有的向量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span>和任意标量<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>, 齐次变换<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span></span></span></span>满足: $$<br>
T(c\\vec{v}) = cT(\\vec{v})$$^1</p>`,readingTime:{minutes:2.01,words:602},title:"简介",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02-B-%E5%95%8F1.html",{loader:()=>B(()=>import("./R02-B-問1.html-nONqVAG-.js"),[]),meta:{author:["Cyletix"],tag:["例题","信息论"],excerpt:`<h3>問1</h3>
<p>【問1】以下に示す定常2次マルコフ情報源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> の状態遷移図について，次の問いに答えよ．</p>`,readingTime:{minutes:16.47,words:4942},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02-B-%E5%95%8F2.html",{loader:()=>B(()=>import("./R02-B-問2.html-CJxcKEuc.js"),[]),meta:{author:["Cyletix"],tag:["信息论","例题","過去問"],excerpt:`
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> を，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{0, 1\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> に値をとる確率変数とする．パラメータ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo separator="true">,</mo><mi>β</mi><mo separator="true">,</mo><mi>γ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\\alpha, \\beta, \\gamma \\in [0, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0528em;">β</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> に対して，確率を次のように定める：</p>`,readingTime:{minutes:4.19,words:1256},title:"問2",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02-C-%E5%95%8F1.html",{loader:()=>B(()=>import("./R02-C-問1.html-BLzJPZJ_.js"),[]),meta:{author:["Cyletix"],tag:["自动机","過去問","例题"],excerpt:`<h2>問1</h2>
<h3>日本語原文</h3>
<p><strong>【問1】</strong><br>
決定性有限オートマトン <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mi>P</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><msub><mi>δ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_1 = (P, \\Sigma, δ_1, p_1, F_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> を考える．ただし，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><msub><mi>δ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P, \\Sigma, δ_1, p_1, F_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> はそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の状態集合，アルファベット，遷移関数，初期状態，最終状態の集合を表す．</p>`,readingTime:{minutes:7.32,words:2196},title:"",type:"article",s:"R02-C-問1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02-C-%E5%95%8F2.html",{loader:()=>B(()=>import("./R02-C-問2.html-CZ7gCHM0.js"),[]),meta:{author:["Cyletix"],tag:["自动机","過去問","例题"],excerpt:`<h3>日文</h3>
<p><strong>【問2】</strong><br>
アルファベット <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\Sigma = \\{a, b\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">}</span></span></span></span> 上の文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> に対し，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> の長さを <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|w|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord">∣</span></span></span></span> と表す．また，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">1 \\le i \\le |w|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7955em;vertical-align:-0.136em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord">∣</span></span></span></span> に対して <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">w[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> は <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> の <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 番目の文字を表す．<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> の逆文字列を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mi>R</mi></msup></mrow><annotation encoding="application/x-tex">w^R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0077em;">R</span></span></span></span></span></span></span></span></span></span></span> と表す．<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>y</mi><mi mathvariant="normal">∣</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|x| = |y| \\ge 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> を満たす <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> 上の文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> に対して，</p>`,readingTime:{minutes:20.06,words:6018},title:"第一部分：",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./R02-数学-向量分析.html-C3JWUyHk.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`<h2>3. ベクトル解析 Vector analysis</h2>
<p>在直角坐标系中，沿 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x, y, z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 轴方向的单位矢量分别为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo separator="true">,</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">i, j, k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span>，回答以下问题：</p>`,readingTime:{minutes:4.82,words:1446},title:"",type:"article",s:"R02-数学-向量分析"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R02infait.html",{loader:()=>B(()=>import("./R02infait.html-kqryGyAu.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p>6选3</p>
<h2>1. 線形代数 Linear algebra</h2>
<p>已知数列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">a_0, a_1, a_2, \\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span>，定义为：</p>`,readingTime:{minutes:4.04,words:1212},title:"数学",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-B-%E5%95%8F1.html",{loader:()=>B(()=>import("./R03-B-問1.html-Cxjb2nIL.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p>入力アルファベットと出力アルファベットがともに <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{1, 2, 3, 4\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mclose">}</span></span></span></span> である無記憶な通信路 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(y|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> の通信路行列が</p>`,readingTime:{minutes:4.42,words:1327},title:"問1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-B-%E5%95%8F2.html",{loader:()=>B(()=>import("./R03-B-問2.html-rfKx9FfT.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p>定常無記憶情報源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><msub><mi>X</mi><mn>2</mn></msub><mo>…</mo></mrow><annotation encoding="application/x-tex">X_1X_2 \\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span> を考える。この情報源のアルファベットを有限集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\\chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">χ</span></span></span></span> とし、各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は確率分布 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> に従うものとする。任意に固定された <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\varepsilon &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> に対し、系列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>χ</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(x_1, x_2, \\dots, x_n) \\in \\chi^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8588em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> が</p>`,readingTime:{minutes:4.5,words:1351},title:"問2",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-C-%E5%95%8F1.html",{loader:()=>B(()=>import("./R03-C-問1.html-DQ-L33hc.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`<h2>問1</h2>
<p>状態数2の決定性有限オートマトンを考える。決定性有限オートマトン <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mo stretchy="false">(</mo><mi>K</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>δ</mi><mo separator="true">,</mo><msub><mi>q</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M = (K, \\Sigma, \\delta, q_0, F)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mclose">)</span></span></span></span> において、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>q</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>q</mi><mn>1</mn></msub><mo stretchy="false">}</mo><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">}</mo><mo separator="true">,</mo><mi>F</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>q</mi><mn>0</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">K = \\{q_0, q_1\\}, \\Sigma = \\{a, b\\}, F = \\{q_0\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> とする。ただし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>δ</mi><mo separator="true">,</mo><msub><mi>q</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">K, \\Sigma, \\delta, q_0, F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span></span></span></span> はそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> の状態集合、アルファベット、遷移関数、初期状態、最終状態の集合を表し、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> は文字である。なお <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><mi>r</mi><mo>∈</mo><mi>K</mi><mo separator="true">,</mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\delta(p, x) = r \\ (p, r \\in K, x \\in \\Sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Σ</span><span class="mclose">)</span></span></span></span> は、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> が文字 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> を読むことで状態 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> から状態 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> に移ることを表す。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span></span></span></span> によって受理される言語を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(M)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="mclose">)</span></span></span></span> で表す。<br>
<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> 上の正規表現 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(a, b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> が表す言語を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\\|e(a, b)\\|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord">∥</span></span></span></span> で表す。次の各問いに答えよ。</p>`,readingTime:{minutes:16.89,words:5068},title:"GPT-4o",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-C-%E5%95%8F2.html",{loader:()=>B(()=>import("./R03-C-問2.html-DB38D93F.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`<h2>問2</h2>
<p>文脈自由文法を4つ組 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>P</mi><mo separator="true">,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N, \\Sigma, P, S)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mclose">)</span></span></span></span> で表す。ただし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>P</mi><mo separator="true">,</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">N, \\Sigma, P, S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> をそれぞれ非終端記号の集合、終端記号の集合、生成規則の集合、開始記号とする。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> 上の文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> と終端記号 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">x \\in \\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> に対して、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>w</mi><msub><mi mathvariant="normal">∣</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">|w|_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> 中の <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> の出現回数とする。以下の各問いに答えよ。</p>`,readingTime:{minutes:4.67,words:1401},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./R03-数学-向量分析.html-lO8ZfoBE.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p>直交座標系において、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 軸方向の単位ベクトルをそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">i</mi></mrow><annotation encoding="application/x-tex">\\mathbf{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">i</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">j</mi></mrow><annotation encoding="application/x-tex">\\mathbf{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbf">j</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\\mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">k</span></span></span></span> とする。次の各問に答えよ。</p>`,readingTime:{minutes:3.62,words:1087},title:"ベクトル解析",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-%E6%95%B0%E5%AD%A6-%E5%A4%8D%E7%B4%A0%E6%95%B0.html",{loader:()=>B(()=>import("./R03-数学-复素数.html-mnF6KQoF.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p>次の各問に答えよ。</p>
<ol>
<li>複素関数</li>
</ol>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(z) = \\frac{1}{z(z - 2)^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:5.36,words:1607},title:"複素関数論",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03-%E6%95%B0%E5%AD%A6-%E6%A6%82%E7%8E%87%E7%BB%9F%E8%AE%A1.html",{loader:()=>B(()=>import("./R03-数学-概率统计.html-BE2AqTe9.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`<h2>確率・統計</h2>
<p>実数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> は <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 &lt; p &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> を満たすものとする。確率変数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span></span></span></span> は独立に同一の確率関数</p>`,readingTime:{minutes:1.17,words:351},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R03ist.html",{loader:()=>B(()=>import("./R03ist.html-BX5N1C71.js"),[]),meta:{author:["Cyletix"],tag:["過去問","例题","数学"],excerpt:`
<h2>線形代数</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 実行列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A \\in \\mathbb{R}^{n \\times m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span></span> の第 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> 列 (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">j = 1, 2, \\dots, m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span></span></span></span>) を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a_j \\in \\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> とする。<br>
各部分集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>J</mi><mo>⊆</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">J \\subseteq \\{1, 2, \\dots, m\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">}</span></span></span></span> について、その要素数を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>J</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|J|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mord">∣</span></span></span></span> で表し、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>j</mi><mo>∈</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_j \\ (j \\in J)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mclose">)</span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> に関する昇順で左から並べて得られる <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> の部分行列を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi mathvariant="normal">∣</mi><mi>J</mi><mi mathvariant="normal">∣</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A[J] \\in \\mathbb{R}^{n \\times |J|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0962em;">J</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.0962em;">J</span><span class="mord mtight">∣</span></span></span></span></span></span></span></span></span></span></span></span> で表す。<br>
このとき、以下の問いに答えよ。</p>`,readingTime:{minutes:1.27,words:381},title:"数学",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-B-%E5%95%8F1.html",{loader:()=>B(()=>import("./R04-B-問1.html-eHaVwvNR.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問1</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> を正の整数とする．入力アルファベットが <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">}</mo><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X = \\{0, 1\\}^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span>, 出力アルファベットが <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">}</mo><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">Y = \\{0, 1\\}^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span> の無記憶な通信路 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>Y</mi><mi mathvariant="normal">∣</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(Y|X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mclose">)</span></span></span></span> を</p>`,readingTime:{minutes:2.77,words:831},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-B-%E5%95%8F2.html",{loader:()=>B(()=>import("./R04-B-問2.html-CH5AZsSP.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問2</h2>
<p>アルファベットが <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{1, 2, 3, 4\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mclose">}</span></span></span></span> である単純マルコフ情報源の遷移確率行列が</p>`,readingTime:{minutes:6.9,words:2069},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-C-%E5%95%8F1.html",{loader:()=>B(()=>import("./R04-C-問1.html-CL0jdQz8.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h3>問1</h3>
<p>以下の状態遷移図を持つ決定性有限オートマトンを <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mo stretchy="false">(</mo><mi>K</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>δ</mi><mo separator="true">,</mo><msub><mi>q</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M = (K, \\Sigma, \\delta, q_0, F)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0715em;">K</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">F</span><span class="mclose">)</span></span></span></span> とする.</p>`,readingTime:{minutes:6.95,words:2084},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-C-%E5%95%8F2.html",{loader:()=>B(()=>import("./R04-C-問2.html-BT47jh1Z.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h3>問2</h3>
<p>文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> と文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> に対して, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">#</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\#_y(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">#</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> における <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> の部分文字列としての出現回数とする．例えば、文字列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>b</mi><mi>b</mi><mi>a</mi><mi>b</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">x=abbbabb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">abbbabb</span></span></span></span>と文字列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>b</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">y=bb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bb</span></span></span></span>について、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">#</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\\#_y(x)=3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord">#</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span>である. また、文字列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>の反転文字列を<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>R</mi></msup></mrow><annotation encoding="application/x-tex">x^R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0077em;">R</span></span></span></span></span></span></span></span></span></span></span>と表す. 以下の言語を考える:</p>`,readingTime:{minutes:1.02,words:306},title:"",type:"article",s:"R04-C-問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./R04-数学-向量分析.html-BkuP3yiT.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>ベクトル解析</h2>
<p>直交座標系において, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 軸方向の単位ベクトルをそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">i</mi></mrow><annotation encoding="application/x-tex">\\mathbf{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">i</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">j</mi></mrow><annotation encoding="application/x-tex">\\mathbf{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbf">j</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\\mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">k</span></span></span></span> とする．ベクトル場 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> を</p>`,readingTime:{minutes:6.94,words:2083},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./R04-数学-微积分.html-CZMIf8MQ.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>解析学・微積分</h2>
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">R^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 上で微分可能な実数値関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) \\ (x = (x_1, x_2, \\dots, x_m))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span> について, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_i = v_i(t) \\ (i = 1, 2, \\dots, m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> とおく．ただし, 各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span> 上で微分可能な関数とする．次の各問いに答えよ.
<ul>
<li>(a) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{df}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1076em;">df</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\partial f}{\\partial x_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3773em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.1076em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msub><mi>v</mi><mi>i</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dv_i}{dt} \\ (i = 1, 2, \\dots, m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2412em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> で表せ.</li>
<li>(b) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo separator="true">,</mo><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>t</mi><mo separator="true">,</mo><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">m = 2, f(x) = x_1^2 + x_1 x_2 + 2x_2^2, v_1(t) = \\sin t, v_2(t) = e^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> のとき, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{df}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.1076em;">df</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> を求めよ.</li>
</ul>
</li>
<li>次の微分方程式の一般解を求めよ. $$\\frac{dy}{dx} - 2xy = e<sup>{x</sup>2}$$</li>
<li>閉曲線 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> に沿った複素積分   $$\\oint_C \\frac{\\cos z}{(2z - \\pi)^3} , dz$$<br>
を求めよ．ただし, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> は円 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">|z| = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> とする.</li>
</ol>`,readingTime:{minutes:4.58,words:1374},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-%E6%95%B0%E5%AD%A6-%E6%A6%82%E7%8E%87%E7%BB%9F%E8%AE%A1.html",{loader:()=>B(()=>import("./R04-数学-概率统计.html-83LlrtcJ.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>確率・統計</h2>
<p>2以上の自然数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> に対して, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>P</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>P</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P = (P_1, \\dots, P_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> は一様ランダムに選ばれた <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{1, \\dots, n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">}</span></span></span></span> の順列とする．任意の自然数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i, j \\ (1 \\leq i &lt; j \\leq n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> に対して,</p>`,readingTime:{minutes:1.96,words:589},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html",{loader:()=>B(()=>import("./R04-数学-线性代数.html-C6e2r6Kv.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>線形代数</h2>
<p>n次元ユークリッド空間上の <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 個の点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>p</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">p_1, p_2, \\dots, p_{n+1} \\in \\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7474em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> に対し, 2点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>p</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">p_i, p_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 間のユークリッド距離を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">∥</mi><msub><mi>p</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>j</mi></msub><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">d_{i,j} = \\|p_i - p_j\\|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∥</span></span></span></span> で表す．ただし, 各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は列ベクトルである．また, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><msubsup><mi>d</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>d</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>d</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><mn>2</mn></msubsup><mtext> </mtext><mo stretchy="false">(</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_{i,j} = d_{i,n+1}^2 + d_{j,n+1}^2 - d_{i,j}^2 \\ (1 \\leq i, j \\leq n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2089em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2089em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2089em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> を添字順に並べて得られる行列を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">G = (g_{i,j}) \\in \\mathbb{R}^{n \\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span> とする．このとき以下の各問いに答えよ.</p>`,readingTime:{minutes:8.25,words:2475},title:"GPT-4o",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R04ist.html",{loader:()=>B(()=>import("./R04ist.html-D4615udg.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問","数学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR04-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html%23%25E7%25B7%259A%25E5%25BD%25A2%25E4%25BB%25A3%25E6%2595%25B0" target="_blank">R04-数学-线性代数</a></p>
<hr>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR04-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html%23%25E8%25A7%25A3%25E6%259E%2590%25E5%25AD%25A6%25E3%2583%25BB%25E5%25BE%25AE%25E7%25A9%258D%25E5%2588%2586" target="_blank">R04-数学-微积分</a></p>
<hr>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR04-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html%23%25E3%2583%2599%25E3%2582%25AF%25E3%2583%2588%25E3%2583%25AB%25E8%25A7%25A3%25E6%259E%2590" target="_blank">R04-数学-向量分析</a></p>`,readingTime:{minutes:.32,words:96},title:"数学",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-B-%E5%95%8F1.html",{loader:()=>B(()=>import("./R05-B-問1.html-Ccwv0m_p.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問1</h2>
<p>以下の各問いに答えよ。<br>
(1) 区間 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, a]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mclose">]</span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>) 上の一様分布に従う確率変数の微分エントロピーを求めよ。<br>
(2) 区間 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, a]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mclose">]</span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>) 上で定義された確率密度関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><msup><mi>a</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">p(x) = \\frac{2x}{a^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> に従う確率変数の微分エントロピーを求めよ。</p>`,readingTime:{minutes:2.4,words:719},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-B-%E5%95%8F2.html",{loader:()=>B(()=>import("./R05-B-問2.html-DyCthLYe.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問2</h2>
<p>時刻 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> の入力 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mi>t</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X_t \\in \\{0, 1\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">t = 1, 2, \\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span>) に対し、入力と独立な誤り源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>E</mi></msub></mrow><annotation encoding="application/x-tex">S_E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> から発生した記号 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Z</mi><mi>t</mi></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Z_t \\in \\{0, 1\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> が加わった値 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mi>t</mi></msub><mo>=</mo><msub><mi>X</mi><mi>t</mi></msub><mo>⊕</mo><msub><mi>Z</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">Y_t = X_t \\oplus Z_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が出力される加法的2元通信路 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> を考える。ただし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\\oplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span></span></span></span> は排他的論理和を表し、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>⊕</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \\oplus 1 = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>⊕</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \\oplus 1 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> である。誤り源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>E</mi></msub></mrow><annotation encoding="application/x-tex">S_E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">∣</mi><msub><mi>Z</mi><mi>t</mi></msub><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.25</mn></mrow><annotation encoding="application/x-tex">P(Z_{t+1} = 1 | Z_t = 0) = 0.25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.25</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">∣</mi><msub><mi>Z</mi><mi>t</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">P(Z_{t+1} = 1 | Z_t = 1) = 0.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span></span></span></span> となる定常な単純マルコフ情報源である場合について、以下の問いに答えよ。<br>
(1) 誤り源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>E</mi></msub></mrow><annotation encoding="application/x-tex">S_E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の定常確率分布を求めよ。<br>
(2) 誤り源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>E</mi></msub></mrow><annotation encoding="application/x-tex">S_E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> のエントロピーレート <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><msub><mi>S</mi><mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(S_E)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0576em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> を求めよ。<br>
(3) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>X</mi><mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X^n = (X_1, \\dots, X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>t</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">P(X_t = 1) = \\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">t = 1, 2, \\dots, n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span>) である離散無記憶情報源からの出力であり、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Z</mi><mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>Z</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z^n = (Z_1, \\dots, Z_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> が定数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>z</mi><mi>n</mi></msup><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">}</mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">z^n \\in \\{0, 1\\}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7035em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> に固定されていると仮定する。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Y</mi><mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Y</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>Y</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y^n = (Y_1, \\dots, Y_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Y</mi><mi>t</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">P(Y_t = 1) = \\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">t = 1, 2, \\dots, n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span>) である離散無記憶情報源の出力であることを示せ。<br>
(4) 通信路 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span> の通信路容量は以下の式で定義される。</p>`,readingTime:{minutes:2.42,words:727},title:"",type:"article",s:"R05-B-問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-C-%E5%95%8F1.html",{loader:()=>B(()=>import("./R05-C-問1.html-BGeE4C2k.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問1</h2>
<p>白玉が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 個、赤玉が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0197em;">l</span></span></span></span> 個入っている袋から1つずつ玉を取り出すという操作を、袋が空になるまで繰り返す。（全部で <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">n = k + l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0197em;">l</span></span></span></span> 回の操作が行われる。）このとき、取り出された順に玉の色を並べることによって得られる長さ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> の文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><mo>…</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">a = a_1 a_2 \\dots a_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を次のように定義する。各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \\in \\{1, 2, \\dots, n\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">}</span></span></span></span> に対し、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 回目の操作で白玉が取り出されたとき <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">a_i = w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span>、赤玉が取り出されたとき <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">a_i = r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> とする。このようにして得られるすべての文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> からなる集合を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>l</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{k,l}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0197em;">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> とする。すなわち、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>w</mi><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\Sigma = \\{w, r\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mclose">}</span></span></span></span> としたとき <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>l</mi></mrow></msub><mo>⊆</mo><msup><mi mathvariant="normal">Σ</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">L_{k,l} \\subseteq \\Sigma^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0197em;">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord">Σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> である。また、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 回目の操作で白玉が取り出されたとき、その時点で袋の中に残っている赤玉の個数を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、赤玉が取り出されたとき、その時点で袋の中に残っている白玉の個数を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> とし、報酬 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が与えられるとする。このようにして得られるすべての報酬の系列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><mo>…</mo><msub><mi>p</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">p = p_1 p_2 \\dots p_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> からなる集合を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>l</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{k,l}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0197em;">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> とする。例えば、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mi>l</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k = l = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0197em;">l</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> で、赤、白、白、赤、赤、白の順に玉を袋から取り出したとき，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mi>r</mi><mi>w</mi><mi>w</mi><mi>r</mi><mi>r</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">a = rwwrrw</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>322110</mn></mrow><annotation encoding="application/x-tex">p = 322110</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">322110</span></span></span></span> となるので、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>L</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a \\in L_{3,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>∈</mo><msub><mi>P</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p \\in P_{3,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> であるが、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>w</mi><mi>w</mi><mi>w</mi><mi>r</mi><mi>w</mi><mo mathvariant="normal">∉</mo><msub><mi>L</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">rwwwrw \\notin L_{3,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> の出現数は等しいはず）、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>322111</mn><mo mathvariant="normal">∉</mo><msub><mi>P</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">322111 \\notin P_{3,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">322111</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>（最後に受け取る報酬は必ず0 のはず）である。</p>`,readingTime:{minutes:6.31,words:1892},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-C-%E5%95%8F2.html",{loader:()=>B(()=>import("./R05-C-問2.html-Cmi_rwqD.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>問2</h2>
<p>文脈自由文法を４つ組 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><mi>P</mi><mo separator="true">,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N, \\Sigma, P, S)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mclose">)</span></span></span></span> で表す。ただし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> をそれぞれ非終端記号の集合、終端記号の集合、生成規則の集合、開始記号とする。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\Sigma = \\{a, b\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">}</span></span></span></span> とする。<br>
(1) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo><msup><mi>a</mi><mi>i</mi></msup><msup><mi>b</mi><mi>j</mi></msup><mi mathvariant="normal">∣</mi><mn>0</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">L_1 = \\{a^i b^j | 0 \\leq i &lt; j\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span></span></span></span></span><span class="mord">∣0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">}</span></span></span></span> とする。言語 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">L_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を生成する文脈自由文法 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mn>1</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">Σ</mi><mo separator="true">,</mo><msub><mi>P</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_1 = (N_1, \\Sigma, P_1, S)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mclose">)</span></span></span></span> の生成規則の集合 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を与えよ。ただし <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>N</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|N_1| = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> とする。<br>
(2) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">L_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を以下の条件を満たす言語とする。</p>`,readingTime:{minutes:3.89,words:1167},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./R05-数学-向量分析.html-Dhpiy2Bw.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>ベクトル解析</h2>
<p>直交座標系において、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 軸方向の単位ベクトルをそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">i</mi></mrow><annotation encoding="application/x-tex">\\mathbf{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">i</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">j</mi></mrow><annotation encoding="application/x-tex">\\mathbf{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbf">j</span></span></span></span>、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\\mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">k</span></span></span></span> とする。ベクトル場 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi><mo>=</mo><mi>y</mi><mi mathvariant="bold">i</mi><mo>−</mo><mi>x</mi><mi mathvariant="bold">j</mi><mo>+</mo><mi>z</mi><mi mathvariant="bold">k</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F} = y \\mathbf{i} - x \\mathbf{j} + z \\mathbf{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord mathbf">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathbf">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mord mathbf">k</span></span></span></span> とする。次の各問に答えよ。<br>
(1) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2 = 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> で定義される円とする。次に示す <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> および <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">C_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> に沿った線積分</p>`,readingTime:{minutes:1.76,words:527},title:"",type:"article",s:"R05-数学-向量分析"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./R05-数学-微积分.html-0xlDVQ0R.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>解析学・微積分 (Analysis and calculus)</h2>
<p>(1) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> 上の関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">f(x) = \\cos x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span> の <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 階導関数を <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{(k)}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> で表す。以下の各問いに答えよ。<br>
(a) 全ての <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \\geq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> について <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{(k)}(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> を求めよ。<br>
(b) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> の原点周りでのテイラー級数を</p>`,readingTime:{minutes:1.67,words:500},title:"解答",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-%E6%95%B0%E5%AD%A6-%E6%A6%82%E7%8E%87%E7%BB%9F%E8%AE%A1.html",{loader:()=>B(()=>import("./R05-数学-概率统计.html-72Nila2t.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>確率・統計</h2>
<p>箱の中に5枚のコイン（コイン1～コイン5）がある。箱から一様ランダムにコインを1枚選んで何度か投げる試行を考える。ただし、それぞれのコイン <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> の表が出る確率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は次の通りである。</p>`,readingTime:{minutes:1.7,words:509},title:"",type:"article",s:"R05-数学-概率统计"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html",{loader:()=>B(()=>import("./R05-数学-线性代数.html-DM5UHgAp.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>線形代数 (Linear algebra)</h2>
<p><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> の実対称行列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A = [a_{ij}]_{n \\times n} \\in \\mathbb{R}^{n \\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span> に対して、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> の各要素 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">a_{ij} \\in \\{0,1\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 \\leq i,j \\leq n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>) かつ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_{ii} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ii</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> (<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 \\leq i \\leq n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7955em;vertical-align:-0.136em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>) を満たすとする。<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> に対して、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">D = [\\delta_{ij} (\\sum_{k=1}^n a_{ik})]_{n \\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">ik</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> と定義する。ただし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\\delta_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> は、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 \\leq i,j \\leq n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> に対して、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i = j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> のとき <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\\delta_{ij} = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>、そうでないとき <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\delta_{ij} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0379em;">δ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> によって定義される。さらに、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mi>D</mi><mo>−</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">L = D - A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> と定義する。以下の各問いに答えよ。</p>`,readingTime:{minutes:1.87,words:560},title:"解答",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R05ist.html",{loader:()=>B(()=>import("./R05ist.html-CcPwZ4uk.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問","数学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR05-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html%23%25E7%25B7%259A%25E5%25BD%25A2%25E4%25BB%25A3%25E6%2595%25B0%2520%28Linear%2520algebra%29" target="_blank">R05-数学-线性代数</a></p>
<hr>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR05-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html%23%25E8%25A7%25A3%25E6%259E%2590%25E5%25AD%25A6%25E3%2583%25BB%25E5%25BE%25AE%25E7%25A9%258D%25E5%2588%2586%2520%28Analysis%2520and%2520calculus%29" target="_blank">R05-数学-微积分</a></p>`,readingTime:{minutes:.34,words:103},title:"数学",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-B-%E5%95%8F1.html",{loader:()=>B(()=>import("./R06-B-問1.html-Cn0LR1Gm.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>【問1】</h2>
<p>アルファベット <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>s</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>3</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>4</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>5</mn></msub><mo separator="true">,</mo><msub><mi>s</mi><mn>6</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{s_1, s_2, s_3, s_4, s_5, s_6\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 上の無記憶情報源 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> の符号化に関して，以下の問いに答えよ。ただし，符号語アルファベットを <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{0, 1\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span> とする。</p>`,readingTime:{minutes:2,words:601},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-B-%E5%95%8F2.html",{loader:()=>B(()=>import("./R06-B-問2.html-BqMJsPOB.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>【問2】</h2>
<p>アルファベット <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X = \\{1, 2, 3\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mclose">}</span></span></span></span> 上の確率分布 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を以下のように定める：</p>`,readingTime:{minutes:1.36,words:409},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-C-%E5%95%8F1.html",{loader:()=>B(()=>import("./R06-C-問1.html-DJOqClcP.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h3>問1</h3>
<p>アルファベット <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\Sigma = \\{a, d\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mclose">}</span></span></span></span> 上の言語 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> を以下のように定義する：</p>`,readingTime:{minutes:1.19,words:357},title:"",type:"article",s:"R06-C-問1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-C-%E5%95%8F2.html",{loader:()=>B(()=>import("./R06-C-問2.html-B2-0sgb6.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h3>問2</h3>
<p>アルファベット <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mo>→</mo><mo separator="true">,</mo><mo>←</mo><mo separator="true">,</mo><mo>↑</mo><mo separator="true">,</mo><mo>↓</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\Sigma = \\{\\rightarrow, \\leftarrow, \\uparrow, \\downarrow\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mrel">→</span></span><span class="base"><span class="strut" style="height:0.5613em;vertical-align:-0.1944em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↑</span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">}</span></span></span></span> 上の文字列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>⋯</mo><mi>a</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a = a[1]a[2]\\cdots a[n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span>（各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> に対し <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>∈</mo><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">a[i] \\in \\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span>）に対し、2次元ベクトルの系列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a) = (p[0], p[1], \\dots, p[n])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">])</span></span></span></span>（各 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> に対し <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p[i] = (x[i], y[i])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x[i], y[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> は整数）を次のように定義する：</p>`,readingTime:{minutes:2.93,words:879},title:"",type:"article",s:"R06-C-問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-%E6%95%B0%E5%AD%A6-%E5%90%91%E9%87%8F%E5%88%86%E6%9E%90.html",{loader:()=>B(()=>import("./R06-数学-向量分析.html--qVxfCng.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>ベクトル解析</h2>
<p>直交座標系において，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>軸方向の単位ベクトルをそれぞれ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">i</mi></mrow><annotation encoding="application/x-tex">\\mathbf{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">i</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">j</mi></mrow><annotation encoding="application/x-tex">\\mathbf{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbf">j</span></span></span></span> とする．次の (1)，(2) で示すベクトル場 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> と経路 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> について，線積分</p>`,readingTime:{minutes:2.97,words:892},title:"o1",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./R06-数学-微积分.html-DPxMi_EQ.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>解析学・微積分 (Analysis and Calculus)</h2>
<ol>
<li>
<p>次の積分を計算せよ：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><msup><mi>x</mi><mn>5</mn></msup><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">I = \\int_{0}^{\\infty} x^5 \\exp(-x^4) \\, dx
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>
<p>ただし、以下を証明なく用いてよい：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\\int_{-\\infty}^{\\infty} \\exp(-x^2) \\, dx = \\sqrt{\\pi}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1908em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8492em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span><span style="top:-2.8092em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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</li>
<li>
<p>次の微分方程式の一般解を求めよ：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi><mo>=</mo><mi>x</mi><mi>sinh</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{dy}{dx} + y = x \\sinh(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sinh</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>
</li>
<li>
<p>複素関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>z</mi><mn>4</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(z) = \\frac{1}{z^4 + 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> を考える。以下に答えよ：</p>
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> の極を全て求めよ。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span> を図示した半円とし、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">R &gt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> の場合、複素積分 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∫</mo><mi>C</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">\\int_C f(z) \\, dz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1608em;vertical-align:-0.3558em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.1945em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1225em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> を計算せよ。</li>
</ol>
</li>
<li>
<p><strong>积分の計算</strong><br>
次の積分を計算する：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><msup><mi>x</mi><mn>5</mn></msup><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">I = \\int_{0}^{\\infty} x^5 \\exp(-x^4) \\, dx
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p>
</li>
</ol>`,readingTime:{minutes:1.75,words:526},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-%E6%95%B0%E5%AD%A6-%E6%A6%82%E7%8E%87%E7%BB%9F%E8%AE%A1.html",{loader:()=>B(()=>import("./R06-数学-概率统计.html-DRPuvFbf.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>確率・統計</h2>
<ol>
<li>箱の中に <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 個の白球と <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 個の黒球があり、総数は <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><msub><mi>N</mi><mn>1</mn></msub><mo>+</mo><msub><mi>N</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N = N_1 + N_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> である。箱からランダムに 2 個の球を取り出すとき、両方が白球である確率が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/2</span></span></span></span> であることが分かっている。
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が奇数の場合、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の最小値を求めよ。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が偶数の場合、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の最小値を求めよ。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> の最小値を 3 通り求めよ。</li>
</ol>
</li>
</ol>`,readingTime:{minutes:.81,words:244},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html",{loader:()=>B(()=>import("./R06-数学-线性代数.html-DnSgWp35.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","過去問"],excerpt:`<h2>線形代数 (Linear Algebra)</h2>
<ol>
<li>実数行列 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A \\in \\mathbb{R}^{m \\times n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span> および実数ベクトル <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">b \\in \\mathbb{R}^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> に対して、集合を以下のように定義する：<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>S</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>∣</mo><mi>A</mi><mi>x</mi><mo>=</mo><mi>b</mi><mo stretchy="false">}</mo><mo separator="true">,</mo><mspace width="1em"/><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">S = \\{x \\in \\mathbb{R}^n \\mid Ax = b\\}, \\quad f(x) = Ax
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span></span></span></span></span></p>
次の事実を証明なく用いてよい。
<ul>
<li>ベクトル空間 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> の部分空間 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo>⊆</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">W \\subseteq V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> である条件は次の通りである：
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mn>1</mn><mo>:</mo><mn>0</mn><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">C1: 0 \\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span></li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mn>2</mn><mo>:</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>∈</mo><mi>W</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>u</mi><mo>+</mo><mi>v</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">C2: u, v \\in W \\implies u + v \\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span></li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mn>3</mn><mo>:</mo><mi>u</mi><mo>∈</mo><mi>W</mi><mo separator="true">,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>c</mi><mi>u</mi><mo>∈</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">C3: u \\in W, c \\in \\mathbb{R} \\implies cu \\in W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7129em;vertical-align:-0.024em;"></span><span class="mord mathbb">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">W</span></span></span></span><br>
以下に答えよ：</li>
</ul>
</li>
</ul>
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>24</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>24</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">A = \\begin{bmatrix} 2 &amp; 0 &amp; 2 &amp; 0 \\\\ 0 &amp; 8 &amp; 24 &amp; 24 \\\\ 0 &amp; 2 &amp; 6 &amp; 6 \\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">24</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">24</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.6em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.6em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
M347 1759 V0 H263 V1759 v0 v1759 h84z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span> の場合、写像 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> のカーネル <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\mathrm{Ker}(f)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">Ker</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mclose">)</span></span></span></span> の次元と基底を求めよ。</li>
<li>一般的に <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\mathrm{Ker}(f)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">Ker</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mclose">)</span></span></span></span> が <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> の部分空間であることを示せ。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>⊆</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S \\subseteq \\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> が部分空間の場合、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">b = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> を示せ。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>⊆</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S \\subseteq \\mathbb{R}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> が部分空0不8間かつ <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> が正方行列の場合、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> が可逆なら <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S = \\{0\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mclose">}</span></span></span></span> を示せ。</li>
</ol>
</li>
</ol>`,readingTime:{minutes:1.86,words:559},title:"DeepSeekV3",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/R06ist.html",{loader:()=>B(()=>import("./R06ist.html-CgsE3As-.js"),[]),meta:{author:["Cyletix"],tag:["例题","過去問"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR06-%E6%95%B0%E5%AD%A6-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0.html%23%25E7%25B7%259A%25E5%25BD%25A2%25E4%25BB%25A3%25E6%2595%25B0%2520%28Linear%2520Algebra%29" target="_blank">R06-数学-线性代数</a></p>
<hr>
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F%2FR06-%E6%95%B0%E5%AD%A6-%E5%BE%AE%E7%A7%AF%E5%88%86.html%23%25E8%25A7%25A3%25E6%259E%2590%25E5%25AD%25A6%25E3%2583%25BB%25E5%25BE%25AE%25E7%25A9%258D%25E5%2588%2586%2520%28Analysis%2520and%2520Calculus%29" target="_blank">R06-数学-微积分</a></p>`,readingTime:{minutes:.32,words:97},title:"数学",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E4%B9%9D%E5%B7%9E%E9%81%8E%E5%8E%BB%E5%95%8F/%E9%80%92%E6%8E%A8%E5%85%B3%E7%B3%BB%E7%9A%84%E7%BA%BF%E6%80%A7%E8%A1%A8%E7%A4%BA.html",{loader:()=>B(()=>import("./递推关系的线性表示.html-COdl5VWx.js"),[]),meta:{tag:["数学","例题"],excerpt:`<p>递推关系为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_n = a_{n-1} + a_{n-2} + 2a_{n-3}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8528em;vertical-align:-0.2083em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p>`,readingTime:{minutes:3.26,words:979},title:"",type:"article",s:"递推关系的线性表示"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%95%B0%E4%B8%80%E7%9C%9F%E9%A2%98%E6%80%BB%E7%BB%93%E6%80%A7%E6%96%87%E6%A1%A3/%E6%A6%82%E7%8E%8704-23%E7%9C%9F%E9%A2%98.html",{loader:()=>B(()=>import("./概率04-23真题.html-CWfeXuZQ.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`<h3>常见分布的期望方差：二级结论</h3>
<table>
<thead>
<tr>
<th style="text-align:center">分右类型</th>
<th style="text-align:center">分布律或概率密度</th>
<th style="text-align:center">期留</th>
<th style="text-align:center">方差</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 分布</td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">{</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>=</mo><msup><mi>p</mi><mi>k</mi></msup><msup><mi>q</mi><mrow><mn>1</mn><mo>−</mo><mi>k</mi></mrow></msup><mspace width="1em"/><mo stretchy="false">(</mo><mi>q</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_k=P\\{X=k\\}=p^k q^{1-k} \\quad(q=1-p),(k=0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(1-q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mclose">)</span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center">二项分布</td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">{</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>=</mo><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">p_k=P\\{X=k\\}=C_n^k p^k q^{n-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0961em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span> <br> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q=1-p),(k=1,2, \\ldots, n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">n p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n p(1-q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mclose">)</span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center">几何分布</td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">{</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi mathvariant="normal">p</mi><msup><mo stretchy="false">)</mo><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">p_k=P\\{X=k\\}=(1-\\mathrm{p})^{k-1} p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathrm">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">p</span></span></span></span>, <br> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(k=1,2, \\ldots, n), 0&lt;p&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1}{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3262em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow><msup><mi>p</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1-p}{p^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3783em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8972em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">p</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center"><mark>泊松分布</mark></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">{</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>=</mo><mfrac><msup><mi>λ</mi><mi>k</mi></msup><mrow><mi>k</mi><mo stretchy="false">!</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi></mrow></msup><mspace width="1em"/><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo>…</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_k=P\\{X=k\\}=\\frac{\\lambda^k}{k !} e^{-\\lambda} \\quad(i=0,1,2,3 \\ldots)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3879em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0429em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span><span class="mclose mtight">!</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.927em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center"><mark>指数分布</mark></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>λ</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo separator="true">,</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo><mi>x</mi><mo>≤</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">f(x)=\\left\\{\\begin{array}{l}\\lambda e^{-\\lambda x}, x&gt;0 \\\\ 0, x \\leq 0\\end{array}\\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4091em;vertical-align:-0.9546em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4546em;"><span style="top:-3.6054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.4054em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9546em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>λ</mi></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1}{\\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><msup><mi>λ</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\\frac{1}{\\lambda^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center"><mark>正态分布</mark></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></msup><mspace width="1em"/><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>σ</mi><mo>&gt;</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} e^{-\\frac{(x-\\mu)^2}{2 \\sigma^2}} \\quad(-\\infty&lt;x&lt;+\\infty, \\sigma&gt;0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8671em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.551em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3291em;"><span style="top:-3.4534em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.251em;"><span style="top:-2.5062em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9384em;"><span style="top:-2.9384em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.5021em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0484em;"><span style="top:-3.0484em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4938em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center"><mark>均匀分布</mark></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">f(x)=\\left\\{\\begin{array}{ll}\\frac{1}{b-a}, &amp; a&lt;x&lt;b \\\\ 0, &amp; \\text{otherwise}\\end{array}\\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4484em;vertical-align:-0.9742em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4742em;"><span style="top:-3.6291em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.3858em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9742em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4742em;"><span style="top:-3.6291em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.3858em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9742em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\\frac{a+b}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">b</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mn>12</mn></mfrac></mrow><annotation encoding="application/x-tex">\\frac{(b-a)^2}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4539em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1089em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
</tr>
<tr>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布</td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1, X_2, \\ldots X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立, 且都服从标准正态分布 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\mathrm{N}(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><br> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><msubsup><mi>X</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>X</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mi>X</mi><mi>n</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\\chi^2=X_1^2+X_2^2+\\ldots+X_n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span></td>
<td style="text-align:center"><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2 n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span></td>
</tr>
</tbody>
</table>`,readingTime:{minutes:3.89,words:1166},title:"",type:"article",s:"概率04-23真题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%95%B0%E4%B8%80%E7%9C%9F%E9%A2%98%E6%80%BB%E7%BB%93%E6%80%A7%E6%96%87%E6%A1%A3/%E7%BA%BF%E4%BB%A304-23%E7%9C%9F%E9%A2%98.html",{loader:()=>B(()=>import("./线代04-23真题.html-BO8n-h1I.js"),[]),meta:{tag:["例题"],excerpt:`<ul>
<li>共110道
<ul>
<li>9</li>
<li>19</li>
<li>14</li>
<li>20</li>
<li>26</li>
<li>22</li>
</ul>
</li>
</ul>
<h2>行列式(5+4道)</h2>
<h3>代数余子式</h3>
<h6><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98%2F%E6%95%B0%E4%B8%802021.html%23%2815%29" target="_blank">数一2021</a></h6>
<h6><a class="route-link nolebase-route-link-invalid" href="#" target="_target">数二2019#(14)</a></h6>`,readingTime:{minutes:2.52,words:757},title:"",type:"article",s:"线代04-23真题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%95%B0%E4%B8%80%E7%9C%9F%E9%A2%98%E6%80%BB%E7%BB%93%E6%80%A7%E6%96%87%E6%A1%A3/%E9%AB%98%E6%95%B004-23%E7%9C%9F%E9%A2%98.html",{loader:()=>B(()=>import("./高数04-23真题.html-Dk_NYrCW.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`<p>通过替换六级标题来完成数 &quot;(一二三)之间的切换</p>
<ul>
<li>共502道
<ul>
<li>69</li>
<li>88</li>
<li>78</li>
<li>60</li>
<li>3</li>
<li>48</li>
<li>51</li>
<li>43</li>
<li>34</li>
<li>28</li>
</ul>
</li>
</ul>
<h2>概念题</h2>
<h3>数列极限是否存在，是否收敛</h3>
<h6><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98%2F%E6%95%B0%E4%B8%802022.html%23%283%29" target="_blank">数一2022</a>√</h6>`,readingTime:{minutes:11.33,words:3400},title:"数列极限，中值定理，不等式(28道)",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802009.html",{loader:()=>B(()=>import("./数一2009.html-DaUMjhdN.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目要 求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>x</mi><mo>→</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle x \\rightarrow 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>a</mi><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f(x)=x-\\sin a x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>b</mi><mi>x</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle g(x)=x^{2} \\ln (1-b x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是等价无穷小量, 则 ()</p>`,readingTime:{minutes:28.05,words:8415},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802010.html",{loader:()=>B(()=>import("./数一2010.html-jgT2DUoo.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目要 求，把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>极限 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mo fence="true">[</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow><mi>x</mi></msup><mo>=</mo><mo stretchy="false">(</mo><mspace width="1em"/></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow \\infty}\\left[\\frac{x^{2}}{(x-a)(x+b)}\\right]^{x}=(\\quad</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4954em;vertical-align:-0.95em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.5454em;"><span style="top:-3.944em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span></span></span></span> )<br>
(A) 1 .<br>
(B) e.<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">e</mi><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\\mathrm{e}^{a-b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathrm">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">e</mi><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\\mathrm{e}^{b-a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathrm">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span></span></span></span>.</p>`,readingTime:{minutes:32.22,words:9666},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802011.html",{loader:()=>B(()=>import("./数一2011.html-qwm5ugXL.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`<h2>(1)</h2>
<p>曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><msup><mo stretchy="false">)</mo><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">y=(x-1)(x-2)^{2}(x-3)^{3}(x-4)^{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span> 的拐点是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>.<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>.<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(4,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>.</p>`,readingTime:{minutes:30.82,words:9246},title:"",type:"article",s:"数一2011"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802012.html",{loader:()=>B(()=>import("./数一2012.html-BZyegioU.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目要 求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">y=\\frac{x^{2}+x}{x^{2}-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4213em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的渐近线的条数为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) 0 .<br>
(B) 1 .<br>
(C) 2 .<br>
(D) 3 .</p>`,readingTime:{minutes:41.63,words:12490},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802013.html",{loader:()=>B(()=>import("./数一2013.html-Dbjxh7Ef.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>已知极限 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>x</mi><mo>−</mo><mi>arctan</mi><mo>⁡</mo><mi>x</mi></mrow><msup><mi>x</mi><mi>k</mi></msup></mfrac><mo>=</mo><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow 0} \\frac{x-\\arctan x}{x^{k}}=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0092em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">0</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2921em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span>, 其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">k, c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span> 为常数, 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c \\neq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mi>c</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">k=2, c=-\\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
( B ) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">k=2, c=\\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn><mo separator="true">,</mo><mi>c</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">k=3, c=-\\frac{1}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn><mo separator="true">,</mo><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">k=3, c=\\frac{1}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>`,readingTime:{minutes:35.94,words:10783},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802014.html",{loader:()=>B(()=>import("./数一2014.html-BoiLxh0D.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>下列曲线中有渐近线的是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y=x+\\sin x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span>.<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y=x^{2}+\\sin x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span>.<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mo>⁡</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">y=x+\\sin \\frac{1}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>sin</mi><mo>⁡</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">y=x^{2}+\\sin \\frac{1}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>`,readingTime:{minutes:31,words:9300},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802015.html",{loader:()=>B(()=>import("./数一2015.html-CZV8FqGK.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题,每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有 一项符合题目要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\\infty,+\\infty)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span></span> 上连续, 其 2 阶导函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{\\prime \\prime}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的图形 如右图所示,<br> <img src="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2F_assets_%2FImages%2F20241213074414.png" alt="|150" loading="lazy"><br>则曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的拐点个数为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span>(A) 0 .<br>
( B ) 1 .<br>
(C) 2 .<br>
( D) 3 .</p>`,readingTime:{minutes:29.63,words:8888},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802016.html",{loader:()=>B(()=>import("./数一2016.html-MArYBf8j.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>数一2016 若反常积分 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>a</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>b</mi></msup></mrow></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\int_{0}^{+\\infty} \\frac{1}{x^{a}(1+x)^{b}} \\mathrm{~d} x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4572em;vertical-align:-0.936em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">x</span></span></span></span> 收敛, 则( )<br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a+b&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a+b&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.</p>`,readingTime:{minutes:29.33,words:8798},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802017.html",{loader:()=>B(()=>import("./数一2017.html-BehZwgvV.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>若函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><msqrt><mi>x</mi></msqrt></mrow><mrow><mi>a</mi><mi>x</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>b</mi><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>⩽</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f(x)=\\left\\{\\begin{array}{ll}\\frac{1-\\cos \\sqrt{x}}{a x}, &amp; x&gt;0, \\\\ b, &amp; x \\leqslant 0\\end{array}\\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.598em;vertical-align:-1.049em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.549em;"><span style="top:-3.549em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.038em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4739em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mop mtight"><span class="mtight">c</span><span class="mtight">o</span><span class="mtight">s</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8059em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-2.7659em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2341em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.349em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.049em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.549em;"><span style="top:-3.549em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.349em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">⩽</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.049em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 处连续, 则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span>(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">a b=\\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
( B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">a b=-\\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
( C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a b=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">a b=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>.</p>`,readingTime:{minutes:32.68,words:9803},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802018.html",{loader:()=>B(()=>import("./数一2018.html-DLLX-KZJ.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目 要求, 把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<ul>
<li>下列函数中, 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 处不可导的是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mi>sin</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f(x)=|x| \\sin |x|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span></span></span></span>.<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mi>sin</mi><mo>⁡</mo><msqrt><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">f(x)=|x| \\sin \\sqrt{|x|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f(x)=\\cos |x|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><msqrt><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">f(x)=\\cos \\sqrt{|x|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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</ul>`,readingTime:{minutes:27.69,words:8306},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802019.html",{loader:()=>B(()=>import("./数一2019.html-BcGvjqqO.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项等合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \\rightarrow 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时,若 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x-\\tan x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">x^{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span></span></span></span></span></span></span></span></span> 是同阶无穷小,则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k=(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span>(A) 1 .(B) 2 .( C) 3 .(D) 4 .</p>`,readingTime:{minutes:36.85,words:11054},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802020.html",{loader:()=>B(()=>import("./数一2020.html-CjCI79N4.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 8 小题, 每小题 4 分, 共 32 分. 在每小题给出的四个选项中, 只有一项符合题目要 求，把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><msup><mn>0</mn><mo lspace="0em" rspace="0em">+</mo></msup></mrow><annotation encoding="application/x-tex">x \\rightarrow 0^{+}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span></span></span></span></span></span></span></span></span></span></span></span>时,下列无穷小量中最高阶的是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mrow><mo fence="true">(</mo><msup><mi mathvariant="normal">e</mi><msup><mi>t</mi><mn>2</mn></msup></msup><mo>−</mo><mn>1</mn><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi>t</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\int_{0}^{x}\\left(\\mathrm{e}^{t^{2}}-1\\right) \\mathrm{d} t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathrm">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span></span><br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mi>ln</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><msqrt><msup><mi>t</mi><mn>3</mn></msup></msqrt><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi>t</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\int_{0}^{x} \\ln \\left(1+\\sqrt{t^{3}}\\right) \\mathrm{d} t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0039em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.9639em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0361em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span></span>.<br>
( C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow></msubsup><mi>sin</mi><mo>⁡</mo><msup><mi>t</mi><mn>2</mn></msup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\int_{0}^{\\sin x} \\sin t^{2} \\mathrm{~d} t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4924em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5804em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">s</span><span class="mtight">i</span><span class="mtight">n</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi></mrow></msubsup><msqrt><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mi>t</mi></mrow></msqrt><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\int_{0}^{1-\\cos x} \\sqrt{\\sin ^{3} t} \\mathrm{~d} t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.476em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.564em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mop mtight"><span class="mtight">c</span><span class="mtight">o</span><span class="mtight">s</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1698em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span><span style="top:-3.1298em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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<p>(本题共 10 小题, 每小题 5 分, 共 50 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内. )</p>
<h2>(1)</h2>
<p>数一2021 函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>−</mo><mn>1</mn></mrow><mi>x</mi></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f(x)=\\left\\{\\begin{array}{ll}\\frac{\\mathrm{e}^{x}-1}{x}, &amp; x \\neq 0, \\\\ 1, &amp; x=0\\end{array}\\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.471em;vertical-align:-0.9855em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4855em;"><span style="top:-3.5745em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.3745em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9855em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4855em;"><span style="top:-3.5745em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.3745em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9855em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 处 ( )<br>
(A) 连续且取极大值.(B) 连续且取极小值.(C) 可导且导数等于 0 .(D) 可导且导数不为 0 .</p>`,readingTime:{minutes:36.05,words:10814},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802022.html",{loader:()=>B(()=>import("./数一2022.html-B5KDbjc0.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<p>(本题共 10 小题, 每小题 5 分, 共 50 分. 在每小题给出的四个选项中, 只有一项符合题目 要求,把所选项前的字母填在题后的括号内.</p>
<h2>(1)</h2>
<p>数一2022<br>
设 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow 1} \\frac{f(x)}{\\ln x}=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1441em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">1</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, 则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mspace width="1em"/><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\\quad)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace" style="margin-right:1em;"></span><span class="mclose">)</span></span></span></span><br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(1)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow 1} f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4671em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">1</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>.<br>
<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">C</mi><mo stretchy="false">)</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\\mathrm{C}) f^{\\prime}(1)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathrm">C</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow 1} f^{\\prime}(x)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.519em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">1</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>.<br>
数一2022<br>
答 应选 B.<br>
解 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><munderover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow/></mpadded><mpadded width="+0.6em" lspace="0.3em"><mfrac><mn>0</mn><mn>0</mn></mfrac></mpadded></munderover><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle \\lim _{x \\rightarrow 1} \\frac{f(x)}{\\ln x}=1 \\xrightarrow[]{\\frac00} \\lim _{x \\rightarrow 1} f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1441em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">1</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.335em;vertical-align:-0.122em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.213em;"><span style="top:-3.622em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="svg-align" style="top:-2.989em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
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 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z"/></svg></span></span><span style="top:-2.878em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.122em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4671em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">1</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 故选 B.<br>
这道题的核心是理解给定的极限表达式和如何从中推断函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>在<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x \\rightarrow 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>时的行为。下面是对答案的逐步解释：</p>`,readingTime:{minutes:40.31,words:12092},title:"一、选择题",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802023.html",{loader:()=>B(()=>import("./数一2023.html-BfnPt7uk.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`
<h2>(1)</h2>
<h2>曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mi>ln</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi mathvariant="normal">e</mi><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y=x \\ln \\left(\\mathrm{e}+\\frac{1}{x-1}\\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2533em;vertical-align:-0.4033em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathrm">e</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> 的斜渐近线方程为<br>
A. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">e</mi></mrow><annotation encoding="application/x-tex">y=x+\\mathrm{e}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathrm">e</span></span></span></span>.<br>
B. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi mathvariant="normal">e</mi></mfrac></mrow><annotation encoding="application/x-tex">y=x+\\frac{1}{\\mathrm{e}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.<br>
C. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>.<br>
D. <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mi mathvariant="normal">e</mi></mfrac></mrow><annotation encoding="application/x-tex">y=x-\\frac{1}{\\mathrm{e}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</h2>`,readingTime:{minutes:38.24,words:11471},title:"一、选择题: 1 10 小题, 每小题 5 分, 共 50 分. 下列每题给出的四个选项中, 只有一个选项是最符合题目要求的",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E9%AB%98%E6%95%B01%E7%9C%9F%E9%A2%98/%E6%95%B0%E4%B8%802024.html",{loader:()=>B(()=>import("./数一2024.html-DKvRwms9.js"),[]),meta:{tag:["数学","例题","真题"],excerpt:`<p>一、选择题:1 10 小题, 每小题 5 分, 共 50 分。下列每题给出的四个选项中, 只有一个选项是最符合题目要求的，请将所选项前的字母填在答题纸指定位置上。</p>
<h3>(1)</h3>
<p>已知函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mi>cos</mi><mo>⁡</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow></msubsup><msup><mi>e</mi><msup><mi>t</mi><mn>2</mn></msup></msup><mi>d</mi><mi>t</mi></mstyle></mrow><annotation encoding="application/x-tex">\\displaystyle f(x)=\\int_0^x e^{\\cos t} d t, g(x)=\\int_0^{\\sin x} e^{t^2} d t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">c</span><span class="mtight">o</span><span class="mtight">s</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4924em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5804em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">s</span><span class="mtight">i</span><span class="mtight">n</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span>, 则 ( )<br>
(A) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是奇函数, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是偶函数<br>
(B) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是偶函数, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是奇函数<br>
(C) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 均为奇函数<br>
(D) <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 与 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 均为周期函数<br>
<a href="proxy.php?url=https%3A%2F%2Fb23.tv%2FYaQkcAl" target="_blank" rel="noopener noreferrer">李艳芳真题系列|考研数学一历年真题逐题精讲（2010-2024年）[更新至最新]-哔哩哔哩</a></p>`,readingTime:{minutes:46.5,words:13949},title:"",type:"article",s:"数一2024"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r2%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r2 問題1 問2.html-TjOZ0rIM.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mn>4</mn><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>π</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msubsup><mi>d</mi><mi>θ</mi><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>a</mi><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></msubsup><mi>r</mi><mi>d</mi><mi>r</mi><msubsup><mo>∫</mo><mn>0</mn><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></msqrt></msubsup><mi>d</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">V=4\\int_{0}^{\\pi/2}d\\theta \\int_{0}^{a\\cos\\theta}rdr \\int_{0}^{\\sqrt{a^2-r^2}}dz
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6703em;vertical-align:-0.9119em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6379em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span><span class="mord mtight">/2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop mtight"><span class="mtight">c</span><span class="mtight">o</span><span class="mtight">s</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">θ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7584em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9221em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.8821em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<h2>問題1</h2>
<h3>問2</h3>
<p>求解由以下两个方程围成的体积, 其中<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a\\geq0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p>`,readingTime:{minutes:.36,words:109},title:"専門試験",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E5%95%8F%E9%A1%8C4%20%E5%95%8F%E9%A1%8C1-%E5%95%8F3.html",{loader:()=>B(()=>import("./82-sc-r3 問題4 問題1-問3.html-CNx0391q.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`
<p>要求函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>α</mi></msup></mrow><annotation encoding="application/x-tex">f(x) = (1+x)^\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span></span></span></span>的麦克劳林级数，我们可以使用二项式定理展开，当<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|x| &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>时，该级数收敛。二项式定理给出：</p>`,readingTime:{minutes:3.02,words:906},title:"(1)",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E5%95%8F%E9%A1%8C4.html",{loader:()=>B(()=>import("./82-sc-r3 問題4.html-N4KPlzrS.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>π</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{\\partial^2 u}{\\partial t^2} = \\frac{\\partial^2 u}{\\partial x^2} + \\sin(\\pi t) \\sin(\\pi x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:8.06,words:2418},title:"1. 对方程进行拉普拉斯变换",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題1 問2.html-CvhrpneP.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<h2>(1)</h2>
<p>对于极点在积分路径上的情况，直接使用留数定理求解会遇到问题。这里我们将对路径进行处理，使其不经过极点，然后再应用留数定理。具体步骤如下：</p>
<ol>
<li>
<p><strong>识别奇点：</strong><br>
积分函数为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msup><mi>e</mi><mfrac><mi>z</mi><mn>3</mn></mfrac></msup><mrow><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \\frac{e^{\\frac{z}{3}}}{z + \\pi i} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2934em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5241em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8471em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>奇点在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">z = -\\pi i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span></span>。</p>
</li>
<li>
<p><strong>路径变形：</strong><br>
积分路径 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是以 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|z + \\pi i| = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 表示的圆周。这个路径经过奇点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">z = -\\pi i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span></span>。为了处理这种情况，我们可以将路径略微变形，使其绕过奇点。在实际操作中，我们可以考虑在奇点附近形成一个小的半圆（绕过奇点），然后再应用留数定理。</p>
</li>
<li>
<p><strong>计算主值积分：</strong><br>
我们可以将积分分为两部分：主值积分和绕过奇点的小弧线积分。这里我们仅考虑主值积分的部分，因为绕过奇点的小弧线积分在极限中趋于零。</p>
</li>
<li>
<p><strong>应用留数定理：</strong><br>
留数定理告诉我们，绕路径 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 积分的结果是该路径内部所有奇点的留数之和乘以 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">2\\pi i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span></span>。</p>
<p>奇点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">z = -\\pi i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span></span> 的留数为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Res</mtext><mrow><mo fence="true">(</mo><mfrac><msup><mi>e</mi><mfrac><mi>z</mi><mn>3</mn></mfrac></msup><mrow><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi></mrow></mfrac><mo separator="true">,</mo><mi>z</mi><mo>=</mo><mo>−</mo><mi>π</mi><mi>i</mi><mo fence="true">)</mo></mrow><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>z</mi><mo>→</mo><mo>−</mo><mi>π</mi><mi>i</mi></mrow></munder><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><msup><mi>e</mi><mfrac><mi>z</mi><mn>3</mn></mfrac></msup><mrow><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><mi>i</mi></mrow><mn>3</mn></mfrac></mrow></msup></mrow><annotation encoding="application/x-tex"> \\text{Res}\\left(\\frac{e^{\\frac{z}{3}}}{z + \\pi i}, z = -\\pi i\\right) = \\lim_{z \\to -\\pi i} (z + \\pi i) \\cdot \\frac{e^{\\frac{z}{3}}}{z + \\pi i} = e^{-\\frac{\\pi i}{3}} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4741em;vertical-align:-0.95em;"></span><span class="mord text"><span class="mord">Res</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5241em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8471em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.536em;vertical-align:-0.786em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3723em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span><span class="mrel mtight">→</span><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.786em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2934em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5241em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8471em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0116em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0116em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8551em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
</li>
<li>
<p><strong>计算积分：</strong><br>
由于路径仅包含一个奇点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mo>−</mo><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">z = -\\pi i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span></span>，应用留数定理：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∮</mo><msub><mi>C</mi><mn>1</mn></msub></msub><mfrac><msup><mi>e</mi><mfrac><mi>z</mi><mn>3</mn></mfrac></msup><mrow><mi>z</mi><mo>+</mo><mi>π</mi><mi>i</mi></mrow></mfrac><mtext> </mtext><mi>d</mi><mi>z</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><mi>i</mi></mrow><mn>3</mn></mfrac></mrow></msup></mrow><annotation encoding="application/x-tex"> \\oint_{C_1} \\frac{e^{\\frac{z}{3}}}{z + \\pi i} \\, dz = 2\\pi i \\cdot e^{-\\frac{\\pi i}{3}} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5361em;vertical-align:-1.012em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∮</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.012em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5241em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8471em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0116em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0116em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8551em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
</li>
</ol>`,readingTime:{minutes:2.48,words:744},title:"",type:"article",s:"82-sc-r3 追試験 問題1 問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F1%20(1)%201.html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題3 問1 (1) 1.html-CSCROZvB.js"),[]),meta:{tag:["数学"],excerpt:`<p>问题可以用贝叶斯定理来解决。具体步骤如下：</p>
<ol>
<li>
<p>定义事件：</p>
<ul>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">H_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>: 选择了A硬币</li>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">H_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>: 选择了B硬币</li>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>: 第一次投掷结果是正面，第二次投掷结果是反面</li>
</ul>
</li>
<li>
<p>确定先验概率：</p>
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">P(H_A) = P(H_B) = 0.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span></span></span></span>，因为选择A或B硬币的概率是相等的</li>
</ul>
</li>
<li>
<p>计算每种硬币投出正反面的概率：</p>
<ul>
<li>硬币A投出正面再投出反面的概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.6</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0.6</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.6</mn><mo>×</mo><mn>0.4</mn><mo>=</mo><mn>0.24</mn></mrow><annotation encoding="application/x-tex">P(E|H_A) = 0.6 \\times (1 - 0.6) = 0.6 \\times 0.4 = 0.24</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.4</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.24</span></span></span></span></li>
<li>硬币B投出正面再投出反面的概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0.5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn><mo>×</mo><mn>0.5</mn><mo>=</mo><mn>0.25</mn></mrow><annotation encoding="application/x-tex">P(E|H_B) = 0.5 \\times (1 - 0.5) = 0.5 \\times 0.5 = 0.25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.5</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.25</span></span></span></span></li>
</ul>
</li>
<li>
<p>利用全概率公式计算事件E的总概率：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.24</mn><mo>×</mo><mn>0.5</mn><mo>+</mo><mn>0.25</mn><mo>×</mo><mn>0.5</mn><mo>=</mo><mn>0.12</mn><mo>+</mo><mn>0.125</mn><mo>=</mo><mn>0.245</mn></mrow><annotation encoding="application/x-tex">P(E) = P(E|H_A)P(H_A) + P(E|H_B)P(H_B) = 0.24 \\times 0.5 + 0.25 \\times 0.5 = 0.12 + 0.125 = 0.245
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.25</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.125</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.245</span></span></span></span></span></p>
</li>
<li>
<p>利用贝叶斯定理计算在事件E发生的情况下选择了A硬币的后验概率：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mi mathvariant="normal">∣</mi><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>0.24</mn><mo>×</mo><mn>0.5</mn></mrow><mn>0.245</mn></mfrac><mo>≈</mo><mn>0.4898</mn></mrow><annotation encoding="application/x-tex">P(H_A|E) = \\frac{P(E|H_A)P(H_A)}{P(E)} = \\frac{0.24 \\times 0.5}{0.245} \\approx 0.4898
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.245</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.4898</span></span></span></span></span></p>
</li>
</ol>`,readingTime:{minutes:3.37,words:1010},title:"",type:"article",s:"82-sc-r3 追試験 問題3 問1 (1) 1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F1%20(1).html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題3 問1 (1).html-DNRwDo7G.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p>问题可以用贝叶斯定理来解决。具体步骤如下：</p>
<ol>
<li>
<p>定义事件：</p>
<ul>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">H_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>: 选择了A硬币</li>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">H_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>: 选择了B硬币</li>
<li>事件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span>: 第一次投掷结果是正面，第二次投掷结果是反面</li>
</ul>
</li>
<li>
<p>确定先验概率：</p>
<ul>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">P(H_A) = P(H_B) = 0.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span></span></span></span>，因为选择A或B硬币的概率是相等的</li>
</ul>
</li>
<li>
<p>计算每种硬币投出正反面的概率：</p>
<ul>
<li>硬币A投出正面再投出反面的概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.6</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0.6</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.6</mn><mo>×</mo><mn>0.4</mn><mo>=</mo><mn>0.24</mn></mrow><annotation encoding="application/x-tex">P(E|H_A) = 0.6 \\times (1 - 0.6) = 0.6 \\times 0.4 = 0.24</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.4</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.24</span></span></span></span></li>
<li>硬币B投出正面再投出反面的概率 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0.5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.5</mn><mo>×</mo><mn>0.5</mn><mo>=</mo><mn>0.25</mn></mrow><annotation encoding="application/x-tex">P(E|H_B) = 0.5 \\times (1 - 0.5) = 0.5 \\times 0.5 = 0.25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.5</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.25</span></span></span></span></li>
</ul>
</li>
<li>
<p>利用全概率公式计算事件E的总概率：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.24</mn><mo>×</mo><mn>0.5</mn><mo>+</mo><mn>0.25</mn><mo>×</mo><mn>0.5</mn><mo>=</mo><mn>0.12</mn><mo>+</mo><mn>0.125</mn><mo>=</mo><mn>0.245</mn></mrow><annotation encoding="application/x-tex">P(E) = P(E|H_A)P(H_A) + P(E|H_B)P(H_B) = 0.24 \\times 0.5 + 0.25 \\times 0.5 = 0.12 + 0.125 = 0.245
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.25</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0.12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.125</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.245</span></span></span></span></span></p>
</li>
<li>
<p>利用贝叶斯定理计算在事件E发生的情况下选择了A硬币的后验概率：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mi mathvariant="normal">∣</mi><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi mathvariant="normal">∣</mi><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>H</mi><mi>A</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>0.24</mn><mo>×</mo><mn>0.5</mn></mrow><mn>0.245</mn></mfrac><mo>≈</mo><mn>0.4898</mn></mrow><annotation encoding="application/x-tex">P(H_A|E) = \\frac{P(E|H_A)P(H_A)}{P(E)} = \\frac{0.24 \\times 0.5}{0.245} \\approx 0.4898
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.245</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.4898</span></span></span></span></span></p>
</li>
</ol>`,readingTime:{minutes:3.38,words:1014},title:"",type:"article",s:"82-sc-r3 追試験 問題3 問1 (1)"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F3%20(1)%201.html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題3 問3 (1) 1.html-CS5xJMAh.js"),[]),meta:{tag:["数学"],excerpt:`<p>以下内容为CPT-4生成，其中有错误步骤，请尝试找出</p>
<hr>
<p>尤度函数是统计模型中用于表示在给定参数下观测到的数据的概率的函数. 对于线性模型 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">y = a_1 x_1 + a_2 x_2 + e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>, 其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> 是服从标准正态分布的误差项, 我们可以通过观测数据求尤度函数.</p>`,readingTime:{minutes:2.27,words:682},title:"",type:"article",s:"82-sc-r3 追試験 問題3 問3 (1) 1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F3%20(1).html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題3 問3 (1).html-HFJPcMfZ.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p>以下内容为CPT-4生成，其中有错误步骤，请尝试找出</p>
<hr>
<p>尤度函数是统计模型中用于表示在给定参数下观测到的数据的概率的函数. 对于线性模型 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">y = a_1 x_1 + a_2 x_2 + e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span>, 其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> 是服从标准正态分布的误差项, 我们可以通过观测数据求尤度函数.</p>`,readingTime:{minutes:2.29,words:686},title:"",type:"article",s:"82-sc-r3 追試験 問題3 問3 (1)"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F3%20(3).html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題3 問3 (3).html-ClfdQuhN.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p>事後確率最大化法( MAP, Maximum A Posteriori) 结合了观测数据的尤度和先验信息. 在这种情况下, 先验信息是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 - x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 服从均值为 1, 方差为 2 的正态分布.<br>
假设我们有以下观测数据 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_1, a_2) = (1, 0), (0, 1), (1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和对应的测定值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><mo separator="true">,</mo><mn>7</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y = -5, 7, 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span>. 我们需要通过 MAP 方法来估计 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.<br>
步骤如下:</p>`,readingTime:{minutes:4.41,words:1322},title:"",type:"article",s:"82-sc-r3 追試験 問題3 問3 (3)"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C4.html",{loader:()=>B(()=>import("./82-sc-r3 追試験 問題4.html-httExeWK.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<h2>题目 4</h2>
<p>考虑函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 满足以下的微分方程和初始条件：</p>`,readingTime:{minutes:5.11,words:1532},title:"",type:"article",s:"82-sc-r3 追試験 問題4"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r3.html",{loader:()=>B(()=>import("./82-sc-r3.html-Djhk8atx.js"),[]),meta:{author:["Cyletix"],tag:["例题","数学"],excerpt:`
<h2>問題1</h2>
<h3>問3</h3>
<h4>(1)</h4>
<p>次の関数のマクローリン級数を求めよ。ただし，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>と<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>は実数とする</p>`,readingTime:{minutes:3.65,words:1095},title:"専門試験",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E5%95%8F%E9%A1%8C2%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r4 問題2 問2.html-BCCYbCet.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题","机器人动力学"],excerpt:`
<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F%2F82-sc-r4.html%23%25E5%25B0%2582%25E9%2596%2580%25E8%25A9%25A6%25E9%25A8%2593%23%25E5%2595%258F%25E9%25A1%258C2%23%25E5%2595%258F2" target="_blank">82-sc-r4</a></p>
<h1>求解</h1>
<p>与えられた問題では、3次元空間で等速円運動する点の速度を、回転軸の方向ベクトル<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\\vec{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span>と角速度<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">ω</span></span></span></span>を用いて表現する行列<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span></span>を求める必要があります. この問題は、点の速度が位置ベクトルと回転軸ベクトルの外積に角速度を掛けたものに等しいことから解くことができます.<br>
在给定的问题中, 需要利用旋转轴的方向向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\\vec{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> 和角速度 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">ω</span></span></span></span> 求矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span></span></span></span></span>, 以表示在三维空间中匀速圆周运动的点的速度.  这个问题的解法是: 点的速度等于位置向量与旋转轴向量的外积乘以角速度.<br>
まず、<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>3</mn></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">\\vec{p} = (p_1, p_2, p_3)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>として、位置ベクトル<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>⃗</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\vec{x}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>との外積を考えると、次のようになります:<br>
首先, 当 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>p</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>p</mi><mn>3</mn></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">\\vec{p} = (p_1, p_2, p_3)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9084em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span> 时, 与位置向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>⃗</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\vec{x}(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 的外积为</p>`,readingTime:{minutes:9.48,words:2844},title:"题目",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E5%95%8F%E9%A1%8C2%20%E5%95%8F3.html",{loader:()=>B(()=>import("./82-sc-r4 問題2 問3.html-Bu6-NbwM.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:'<p>对于一个反对称矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span>，我们可以使用其特征根（固有值）来表示 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi>e</mi><mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y = e^X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0785em;">X</span></span></span></span></span></span></span></span></span></span></span> 的行列式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\det(Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mclose">)</span></span></span></span>。具体来说，行列式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\det(Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.2222em;">Y</span><span class="mclose">)</span></span></span></span> 可以通过 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 的特征根来表示。下面是详细的推导过程：</p>',readingTime:{minutes:4.54,words:1361},title:"",type:"article",s:"82-sc-r4 問題2 問3"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E5%95%8F%E9%A1%8C3%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r4 問題3 問2.html-CW6t5IH3.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:'<p>(1) 让我们假设总人口为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span>。</p>',readingTime:{minutes:2.35,words:705},title:"",type:"article",s:"82-sc-r4 問題3 問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F1%20(2).html",{loader:()=>B(()=>import("./82-sc-r4 追試験 問題1 問1 (2).html-C_1aAH60.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:'<p>为了计算参数方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>θ</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">x = \\theta - \\sin \\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">y = 1 - \\cos \\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>θ</mi><mo>&lt;</mo><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">0 &lt; \\theta &lt; 2\\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span> 范围内的弧长，我们需要使用弧长公式。对于参数方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x = f(\\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = g(\\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span>，弧长 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 可以表示为：</p>',readingTime:{minutes:1.51,words:454},title:"",type:"article",s:"82-sc-r4 追試験 問題1 問1 (2)"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r4 追試験 問題1 問2.html-D5nedffp.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2FOther%2F%E4%BE%8B%E9%A2%98%2F%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F%2F82-sc-r4.html%23%25E8%25BF%25BD%25E8%25A9%25A6%25E9%25A8%2593%23%25E5%2595%258F%25E9%25A1%258C1%23%25E5%2595%258F2" target="_blank">82-sc-r4</a></p>
<h2>坐标变换</h2>
<p>从笛卡尔坐标 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 到柱面坐标 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo separator="true">,</mo><mi>θ</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r, \\theta, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span></span></span></span> 的变换由以下关系给出：</p>`,readingTime:{minutes:.52,words:155},title:"",type:"article",s:"82-sc-r4 追試験 問題1 問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E8%BF%BD%E8%A9%A6%E9%A8%93%20%E5%95%8F%E9%A1%8C4.html",{loader:()=>B(()=>import("./82-sc-r4 追試験 問題4.html-IoPLXxwg.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<h3>问题 1</h3>
<p>在将 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u=g(x)h(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 代入拉普拉斯方程 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.499em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 后，可分离变量，得到:</p>`,readingTime:{minutes:13.59,words:4076},title:"",type:"article",s:"82-sc-r4 追試験 問題4"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4%20%E8%BF%BD%E8%A9%A6%E9%A8%93-%E5%95%8F%E9%A1%8C1-%E5%95%8F1%E2%80%90(2).html",{loader:()=>B(()=>import("./82-sc-r4 追試験-問題1-問1‐(2).html-BmzQ-W7I.js"),[]),meta:{author:"Cyletix",tag:["数学","例题"],excerpt:'<p>要计算由这两个参数方程定义的曲线 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x = a(\\theta - \\sin(\\theta))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">))</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = a(1 - \\cos(\\theta))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">))</span></span></span></span>，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> 的取值范围是 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>2</mn><mi>π</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, 2\\pi]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mclose">]</span></span></span></span> 的长度，我们可以使用弧长公式。这个公式是：</p>',readingTime:{minutes:.69,words:206},title:"",type:"article",s:"82-sc-r4 追試験-問題1-問1‐(2)"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r4.html",{loader:()=>B(()=>import("./82-sc-r4.html-CERY0nDE.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`
<h2>問題１</h2>
<h3>問１</h3>
<h4>(1)</h4>
<p>次の極限値を求めよ。</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>n</mi><mo>→</mo><mn>0</mn></mrow></munder><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>x</mi><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>e</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\\lim_{ n \\to 0 } \\dfrac{x-\\log_{e}(1+x)}{x^2}  
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1441em;vertical-align:-0.7171em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3829em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">0</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7171em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>`,readingTime:{minutes:6.28,words:1884},title:"専門試験",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r5%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F1.html",{loader:()=>B(()=>import("./82-sc-r5 問題1 問1.html-C-AiR-Tg.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<p>以下是GPT-4的回答，跟闹着玩似的，看看就好</p>
<p><strong>问题</strong><br>
在三维空间中，取直角坐标系 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo>−</mo><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">O-xyz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>。区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span> 由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2 \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>z</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \\leq z \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 确定，其表面为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>。向量场 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (x, y, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 给出。计算以下在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span> 上的法线面积分 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">I_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p>`,readingTime:{minutes:2.07,words:620},title:"",type:"article",s:"82-sc-r5 問題1 問1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r5%20%E5%95%8F%E9%A1%8C1%20%E5%95%8F2.html",{loader:()=>B(()=>import("./82-sc-r5 問題1 問2.html-BvR5w8vX.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`<h3>问题 2</h3>
<h4>问 1</h4>
<p>给定矩阵<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">P</span></span></span></span>和<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>如下：</p>`,readingTime:{minutes:9.17,words:2750},title:"",type:"article",s:"82-sc-r5 問題1 問2"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/82-sc-r5.html",{loader:()=>B(()=>import("./82-sc-r5.html-CZuESbHd.js"),[]),meta:{author:["Cyletix"],tag:["数学","例题"],excerpt:`
<h2>問題1</h2>
<h3>問1</h3>
<p>3次元空間に直交座標系 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo>−</mo><mi>x</mi><mi>y</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">O-xyz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> をとる。 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2\\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>z</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\\leq z \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> で定められる 領域<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.2222em;">V</span></span></span></span>の表面を<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>とする。 ベクトル場<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>が<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A= (x,y,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>で与えられる場合に次式の<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span></span></span></span>上における法線面積分<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">I_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>を計算せよ。</p>`,readingTime:{minutes:1.94,words:583},title:"専門試験",type:"article"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E4%BA%8C%E6%AC%A1%E5%9E%8B%E7%9A%84%E6%AD%A3%E4%BA%A4%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./二次型的正交变换.html-DEXHAdzW.js"),[]),meta:{author:["GPT-4"],tag:["数学","例题"],excerpt:`<h2>问题</h2>
<p>如何将<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%9B%B8%E4%BC%BC%E5%8F%98%E6%8D%A2%2F%E4%BA%8C%E6%AC%A1%E5%9E%8B.html" target="_blank">二次型</a> <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">3x^2 + 2xy + 3y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 通过<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%2F%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%2F%E6%AD%A3%E4%BA%A4%E5%8F%98%E6%8D%A2.html" target="_blank">正交变换</a>化为只含 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(x&#x27;)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(y&#x27;)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 的标准形式.</p>`,readingTime:{minutes:2.15,words:645},title:"",type:"article",s:"二次型的正交变换"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E4%BE%8B%E9%A2%98-%E9%9D%9E%E9%BD%90%E6%AC%A1%E6%B3%A2%E5%8A%A8%E6%96%B9%E7%A8%8B%E6%B1%82%E8%A7%A3.html",{loader:()=>B(()=>import("./例题-非齐次波动方程求解.html-Bol4CwpP.js"),[]),meta:{tag:["数学"],excerpt:`<p>给定的偏微分方程是：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>π</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\frac{\\partial^2 u}{\\partial t^2} = \\frac{\\partial^2 u}{\\partial x^2} + \\sin(\\pi t) \\sin(\\pi x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.0556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">u</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p>`,readingTime:{minutes:2.11,words:632},title:"",type:"article",s:"例题-非齐次波动方程求解"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F%E6%8E%A8%E5%B9%BF.html",{loader:()=>B(()=>import("./倍角公式推广.html-CnV_kGeG.js"),[]),meta:{tag:["数学","公式"],excerpt:'<p>倍角公式是三角函数的一个基础概念，用于表示某个角度的倍数的三角函数值。对于任意正整数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，我们可以通过递推或其他技巧推导出 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 倍角的正弦和余弦公式。以下是一些常见的倍角公式，包括二倍角和三倍角公式，以及一般形式的推导：</p>',readingTime:{minutes:1.5,words:450},title:"",type:"article",s:"倍角公式推广"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0%E4%BE%8B%E9%A2%98%201.html",{loader:()=>B(()=>import("./傅立叶级数例题 1.html-CBhWwGnh.js"),[]),meta:{tag:["数学"],excerpt:`<h3>傅里叶级数公式</h3>
<p>对于周期函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，周期为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">π</span></span></span></span>，可以表示为傅里叶级数形式：</p>`,readingTime:{minutes:4.13,words:1239},title:"",type:"article",s:"傅立叶级数例题 1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E5%9F%BA%E5%BA%95%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2.html",{loader:()=>B(()=>import("./傅里叶级数基底线性变换.html-BnmBDa9R.js"),[]),meta:{tag:["数学"],excerpt:`<p>以下内容为GPT4生成，但结果是错的，请尝试找出错误的步骤</p>
<hr>
<p>为了使用线性代数的方法证明由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\\sin t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\\cos t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mn>3</mn><mi>t</mi></mrow><annotation encoding="application/x-tex">\\sin 3t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mord mathnormal">t</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mn>3</mn><mi>t</mi></mrow><annotation encoding="application/x-tex">\\cos 3t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mord mathnormal">t</span></span></span></span> 构成的空间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> 中的任意元素 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span></span></span></span> 可以表示为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\\sin t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\\cos t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mi>t</mi></mrow><annotation encoding="application/x-tex">\\sin^3 t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8719em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span>，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mi>t</mi></mrow><annotation encoding="application/x-tex">\\cos^3 t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span></span></span></span> 的线性组合，我们可以考虑这些函数构成的向量空间及其基底的变换关系。</p>`,readingTime:{minutes:2.03,words:608},title:"",type:"article",s:"傅里叶级数基底线性变换"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%88%A4%E6%96%AD%E7%A7%AF%E5%88%86%E9%A1%BA%E5%BA%8F.html",{loader:()=>B(()=>import("./判断积分顺序.html-BSuSH4VA.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>要计算 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 平面和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">xz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 平面单位圆柱相交部分的体积，我们需要分析它们在空间中的几何形状。单位圆柱通常定义为底面半径为1的圆柱，延伸的高度可以根据情况变化。这里的问题是两个单位圆柱沿不同轴延伸，一个沿 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 轴（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 平面的圆柱），一个沿 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 轴（<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">xz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 平面的圆柱）。</p>',readingTime:{minutes:3.35,words:1006},title:"",type:"article",s:"判断积分顺序"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%9B%9B%E9%98%B6%E9%80%86%E7%9F%A9%E9%98%B5.html",{loader:()=>B(()=>import("./四阶逆矩阵.html-rcAG39XB.js"),[]),meta:{tag:["数学"],excerpt:'<p>给定的矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span></span></span></span> 是：</p>',readingTime:{minutes:1.53,words:459},title:"",type:"article",s:"四阶逆矩阵"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%A4%8D%E7%B3%BB%E6%95%B0%E9%9D%9E%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./复系数非齐次线性微分方程例题.html-wCbm_Um5.js"),[]),meta:{tag:["数学","例题"],excerpt:'<p>给定一个具有特定非齐次项 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>w</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(z) = R(s) \\sin(wx)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的复系数非齐次线性微分方程，其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mi>π</mi><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">R(s) = -\\frac{\\pi}{s^2 + \\pi^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0987em;vertical-align:-0.4033em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0359em;">π</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 并且 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">s = a + bi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">bi</span></span></span></span>，我们可以采用待定系数法来寻找特解。这种方法适用于非齐次项是周期函数（如正弦或余弦）的情况。</p>',readingTime:{minutes:2.06,words:617},title:"",type:"article",s:"复系数非齐次线性微分方程例题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E5%BA%94%E7%94%A8%E4%B8%89%E5%80%8D%E8%A7%92%E5%85%AC%E5%BC%8F%E8%BF%9B%E8%A1%8C%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E5%88%86%E8%A7%A3.html",{loader:()=>B(()=>import("./应用三倍角公式进行傅里叶级数分解.html-l2iCZZ8W.js"),[]),meta:{tag:["数学"],excerpt:'<p>傅立叶级数用于将周期函数表示为正弦和余弦函数的无限和。对于非线性三角函数，如 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\sin^3(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1219em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>，我们可以通过使用三角恒等式将其简化为基本的正弦和余弦函数的组合，进而写出其傅立叶级数。</p>',readingTime:{minutes:1.29,words:388},title:"",type:"article",s:"应用三倍角公式进行傅里叶级数分解"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2%E8%A7%A3%E5%86%B3%E5%AE%9E%E6%8C%87%E6%95%B0%E7%A7%AF%E5%88%86.html",{loader:()=>B(()=>import("./拉普拉斯变换解决实指数积分.html-Ci-siC88.js"),[]),meta:{tag:["数学"],excerpt:`<p>我们要求的是如下极限：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mi>τ</mi></mrow></msup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>ω</mi><mi>τ</mi><mo stretchy="false">)</mo><mi>d</mi><mi>τ</mi></mrow><annotation encoding="application/x-tex">\\lim_{t \\to \\infty}\\int_0^{t}e^{-\\tau} \\cos(\\omega \\tau)d\\tau
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4554em;vertical-align:-0.9119em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5435em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.1132em;">τ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">ω</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span></span></p>`,readingTime:{minutes:1.26,words:379},title:"",type:"article",s:"拉普拉斯变换解决实指数积分"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./最小二乘法例题.html-BAnHDkwM.js"),[]),meta:{tag:["数学"],excerpt:`<h3>修正方程式</h3>
<p>首先，我们需要修正给出的方程式，确保参数的定义和线性方程组的形式正确无误。你提供的正确的方程式为：</p>
<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>α</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>γ</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>γ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>β</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>ξ</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>η</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\\begin{bmatrix}
\\alpha &amp; \\gamma \\\\
\\gamma &amp; \\beta
\\end{bmatrix}
\\begin{bmatrix}
a \\\\
b
\\end{bmatrix}
=
\\begin{bmatrix}
\\xi \\\\
\\eta
\\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0556em;">γ</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0528em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.046em;">ξ</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">η</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p>`,readingTime:{minutes:.9,words:271},title:"",type:"article",s:"最小二乘法例题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./概率密度例题.html-Sf6PAYFS.js"),[]),meta:{tag:["数学","例题"],excerpt:'<p>確率密度関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> を持つ確率変数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">X</span></span></span></span> に対して，モーメント母関数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">M</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span><span class="mclose">)</span></span></span></span> は、変数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">θ</span></span></span></span> を用いて次式で定義されます：</p>',readingTime:{minutes:2.17,words:652},title:"",type:"article",s:"概率密度例题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E6%B1%82%E5%8F%98%E6%8D%A2%E7%9A%84%E7%BA%BF%E6%80%A7%E8%A1%A8%E7%A4%BA.html",{loader:()=>B(()=>import("./求变换的线性表示.html-Cl4JcN5h.js"),[]),meta:{tag:["数学"],excerpt:'<p>为了计算矩阵 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> 的线性表示，使得 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mi>N</mi><mo stretchy="false">[</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>3</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>4</mn></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">B = N[a_1, a_2, a_3, a_4]^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span>，我们首先需要明确向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span> 和向量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>3</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>4</mn></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">[a_1, a_2, a_3, a_4]^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.1389em;">T</span></span></span></span></span></span></span></span></span></span></span> 的关系。根据您给出的关系式：</p>',readingTime:{minutes:1.28,words:384},title:"",type:"article",s:"求变换的线性表示"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E7%AC%AC%E4%BA%8C%E7%B1%BB%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%E4%BE%8B%E9%A2%98%201.html",{loader:()=>B(()=>import("./第二类曲面积分例题 1.html-DlSCU8nT.js"),[]),meta:{tag:["数学","例题","過去問"],excerpt:'<p>当向量场从 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 变为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mn>3</mn></msup><mo separator="true">,</mo><msup><mi>z</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x^3, z^2, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 时，需要重新计算通过圆柱侧面以及上下两个平面的通量。以下是详细步骤：</p>',readingTime:{minutes:1.89,words:566},title:"",type:"article",s:"第二类曲面积分例题 1"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E7%AC%AC%E4%BA%8C%E7%B1%BB%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./第二类曲面积分例题.html-r5KGIPNA.js"),[]),meta:{tag:["数学","例题","過去問"],excerpt:'<p>在柱面坐标系下，计算圆柱侧面上的面积元素 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi mathvariant="bold">S</mi></mrow><annotation encoding="application/x-tex">d\\mathbf{S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathbf">S</span></span></span></span> 是通过表面的参数化和叉积来完成的。下面我们详细讲解计算的每一步。</p>',readingTime:{minutes:3.62,words:1085},title:"",type:"article",s:"第二类曲面积分例题"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E9%80%9A%E9%87%8F%E7%9A%84%E7%A7%AF%E5%88%86%E8%AE%A1%E7%AE%97%E7%BB%86%E8%8A%82.html",{loader:()=>B(()=>import("./通量的积分计算细节.html-DWJjiLsA.js"),[]),meta:{tag:["数学"],excerpt:`<p>对于您提出的问题，涉及到的不等式是：</p>
<ol>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2 \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，表示一个单位圆柱体。</li>
<li><span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>z</mi><mo>≤</mo><mi>h</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 \\leq z \\leq h(1 - x - y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span>，这是一个由平面 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z = h(1 - x - y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 平面界定的区域。</li>
</ol>`,readingTime:{minutes:3.97,words:1192},title:"",type:"article",s:"通量的积分计算细节"}}],["/Other/%E4%BE%8B%E9%A2%98/%E6%9D%B1%E5%B7%A5%E9%81%8E%E5%8E%BB%E5%95%8F/%E9%80%9A%E9%87%8F%E7%A7%AF%E5%88%86%E4%BE%8B%E9%A2%98.html",{loader:()=>B(()=>import("./通量积分例题.html-CM6ox2PW.js"),[]),meta:{tag:["数学"],excerpt:'<p>在直角坐标系中，考虑一个由 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\\{x, y, z\\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mclose">}</span></span></span></span> 坐标定义的三维空间。给定的区域 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span> 是由条件 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2 \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 和 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>z</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \\leq z \\leq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 定义的，它可以被看作是一个直径在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>-平面上、高为1的圆柱体。</p>',readingTime:{minutes:3.71,words:1114},title:"",type:"article",s:"通量积分例题"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/---%E5%BE%AE%E5%88%86---.html",{loader:()=>B(()=>import("./---微分---.html-HHC1DREn.js"),[]),meta:{author:["Cyletix"],tag:["数学","目录"],excerpt:`
<ol>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E5%87%BD%E6%95%B0%E7%9A%84%E5%BE%AE%E5%88%86.html" target="_blank">函数的微分</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html" target="_blank">微分中值定理</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%B4%9B%E5%BF%85%E8%BE%BE%E6%B3%95%E5%88%99.html" target="_blank">洛必达法则</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html" target="_blank">泰勒公式</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E5%8D%95%E8%B0%83%E6%80%A7.html" target="_blank">单调性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E5%87%B9%E5%87%B8%E6%80%A7.html" target="_blank">凹凸性</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%8B%90%E7%82%B9.html" target="_blank">拐点</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%9E%81%E5%80%BC.html" target="_blank">极值</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%BA%BF%E6%9B%B2%E9%9D%A2%E7%A7%AF%E5%88%86%2F%E6%9B%B2%E7%8E%87.html" target="_blank">曲率</a></li>
</ol>`,readingTime:{minutes:.31,words:93},title:"目录",type:"article"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E4%BD%99%E9%A1%B9.html",{loader:()=>B(()=>import("./余项.html-D2Zq2emd.js"),[]),meta:{author:["DeepSeekV3"],tag:["数学"],excerpt:`<p>在泰勒公式中，余项 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是其灵魂所在。它精确地描述了用一个有限项的多项式 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_n(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 去逼近一个（通常更复杂的）函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 时所产生的误差。<br>
一个完整的泰勒定理表述如下：<br>
设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的某个邻域内具有直到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 阶的导数，则对该邻域内的任意 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，有：</p>`,readingTime:{minutes:3.19,words:957},title:"",type:"article",s:"余项"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%85%AC%E5%BC%8F-%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./公式-泰勒公式.html-Bq_VkQQZ.js"),[]),meta:{tag:["数学","公式"],excerpt:'<p>函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处的 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 阶泰勒多项式为：</p>',readingTime:{minutes:.22,words:66},title:"",type:"article",s:"公式-泰勒公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%87%B9%E5%87%B8%E6%80%A7.html",{loader:()=>B(()=>import("./凹凸性.html-BPTEsuqI.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`<p>凹凸性描述了函数的图像是向上弯曲（凹）还是向下弯曲（凸）。</p>
<h2>定义</h2>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span> 上二阶可导。<br>
如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;&#x27;(x) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span> 上是凹的(Concave up)<br>
如果 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;&#x27;(x) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span> 上是凸的(Concave down)</p>`,readingTime:{minutes:.75,words:226},title:"",type:"article",s:"凹凸性"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%87%BD%E6%95%B0%E7%9A%84%E5%BE%AE%E5%88%86.html",{loader:()=>B(()=>import("./函数的微分.html-B-LEAaWY.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<h2>定义</h2>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在某区间内有定义，<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 及 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x_0+\\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span> 在定义区间内，函数从 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 到 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x_0+\\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span> 的增量 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\Delta y=f(x_0+\\Delta x)-f(x_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>. 如果函数的增量可表示为下式, 且其中 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是不依赖于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span> 的常数</p>`,readingTime:{minutes:1.53,words:458},title:"",type:"article",s:"函数的微分"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%8D%95%E8%B0%83%E6%80%A7.html",{loader:()=>B(()=>import("./单调性.html-DdcDO3p6.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学"],excerpt:`<h2>简介</h2>
<p>单调性是描述函数值随自变量在某个区间内变化趋势的核心概念. 一个函数可以是单调递增或单调递减, 也可以在不 同区间内表现出不同的单调性. 通过分析函数的单调性, 可以深入理解其图形的形状, 走向以及极值点的分布.</p>
<h2>定义</h2>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的定义域为 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>, 区间 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">I \\subseteq D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">D</span></span></span></span>.</p>`,readingTime:{minutes:2.61,words:782},title:"",type:"article",s:"单调性"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%B8%B8%E8%A7%81%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./常见麦克劳林公式.html-AM73pCxm.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学","公式"],excerpt:`<p v-pre class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo stretchy="false">!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo stretchy="false">!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo stretchy="false">!</mo></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><msup><mi>x</mi><mi>n</mi></msup><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow></mfrac><mo>+</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">e^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots + \\frac{x^n}{n!} + \\cdots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0274em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3414em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span></span></p>`,readingTime:{minutes:.66,words:197},title:"",type:"article",s:"常见麦克劳林公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./微分中值定理.html-D7LzQriE.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>描述了指定区间内函数与其导数的关系, 主要有三个</p>
<ul>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E7%BD%97%E5%B0%94%E5%AE%9A%E7%90%86.html" target="_blank">罗尔定理</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html" target="_blank">拉格朗日中值定理</a></li>
<li><a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%9F%AF%E8%A5%BF%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html" target="_blank">柯西中值定理</a></li>
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`,readingTime:{minutes:.19,words:57},title:"",type:"article",s:"微分中值定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./拉格朗日中值定理.html-DMMHvdGw.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>在某些条件下，函数在某个区间内的平均变化率等于在该区间内某一点的瞬时变化率。</p>
<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 满足:</p>`,readingTime:{minutes:.61,words:182},title:"",type:"article",s:"拉格朗日中值定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%8B%90%E7%82%B9.html",{loader:()=>B(()=>import("./拐点.html-B9jkZN-E.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:`<p>拐点是函数凹凸性的转变点。<br>
点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> 是函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的拐点当且仅当：</p>`,readingTime:{minutes:.23,words:69},title:"",type:"article",s:"拐点"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%9E%81%E5%80%BC.html",{loader:()=>B(()=>import("./极值.html-xdmNLchl.js"),[]),meta:{author:["Cyletix","GPT-4"],tag:["数学"],excerpt:'<p>极值包括极大值和极小值。函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在某点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">x=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> 处取得极大值或极小值，这时函数值 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> 分别称为极大值或极小值。</p>',readingTime:{minutes:.68,words:203},title:"",type:"article",s:"极值"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%9F%AF%E8%A5%BF%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./柯西中值定理.html-BmSUQW-3.js"),[]),meta:{author:"kevin-yyy",tag:["数学"],excerpt:'<p>如果函数<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>及<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>满足</p>',readingTime:{minutes:.3,words:89},title:"",type:"article",s:"柯西中值定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./泰勒公式.html-C6zWP-D0.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学","公式"],excerpt:`<h2>简介</h2>
<p>泰勒公式用一个n次多项式来近似在某点具有<span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>阶导数的函数。其核心思想是局部的“以直代曲”：利用函数在展开点 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处的各阶导数值，来构建一个能够在该点附近高度“贴合”原函数的多项式。<br>
尽管展开后的多项式形式有时也相当复杂，但它成功地将复杂的函数性质转化为了更易于代数处理的多项式形式。</p>`,readingTime:{minutes:2.43,words:730},title:"",type:"article",s:"泰勒公式"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E6%B4%9B%E5%BF%85%E8%BE%BE%E6%B3%95%E5%88%99.html",{loader:()=>B(()=>import("./洛必达法则.html-BXyRjO3u.js"),[]),meta:{author:["Cyletix","Gemini"],tag:["数学","公式"],excerpt:`<h2>简介</h2>
<p>洛必达法则，是解决函数不定式极限问题的重要工具，由法国数学家<strong>洛必达</strong>（Guillaume de l'Hôpital）推广, 但实际上是由瑞士数学家约翰·伯努利所发现。它将复杂的极限问题转化为更简单的导数比值极限问题。该法则适用于 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow><annotation encoding="application/x-tex">\\frac{0}{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 或 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi mathvariant="normal">∞</mi><mi mathvariant="normal">∞</mi></mfrac></mrow><annotation encoding="application/x-tex">\\frac{\\infty}{\\infty}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 形式的极限。</p>`,readingTime:{minutes:4.56,words:1367},title:"",type:"article",s:"洛必达法则"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E7%BD%97%E5%B0%94%E5%AE%9A%E7%90%86.html",{loader:()=>B(()=>import("./罗尔定理.html-C8-pXYsr.js"),[]),meta:{author:["Cyletix"],tag:["数学"],excerpt:'<p>设函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 满足:</p>',readingTime:{minutes:.29,words:87},title:"",type:"article",s:"罗尔定理"}}],["/%E5%BE%AE%E7%A7%AF%E5%88%86/%E5%BE%AE%E5%88%86/%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%85%AC%E5%BC%8F.html",{loader:()=>B(()=>import("./麦克劳林公式.html-0KCqeDEL.js"),[]),meta:{author:["Cyletix"],tag:["数学","公式"],excerpt:'<p>麦克劳林公式是<a href="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2FPkmer-Math%2F%E5%BE%AE%E7%A7%AF%E5%88%86%2F%E5%BE%AE%E5%88%86%2F%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F.html" target="_blank">泰勒公式</a>在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x_0 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的特例。对于一个在 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 处具有所有阶导数的函数 <span v-pre class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span 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Uint16Array('ᵁ<Õıʊҝջאٵ۞ޢߖࠏ੊ઑඡ๭༉༦჊ረዡᐕᒝᓃᓟᔥ\0\0\0\0\0\0ᕫᛍᦍᰒᷝ὾⁠↰⊍⏀⏻⑂⠤⤒ⴈ⹈⿎〖㊺㘹㞬㣾㨨㩱㫠㬮ࠀEMabcfglmnoprstu\\bfms„‹•˜¦³¹ÈÏlig耻Æ䃆P耻&䀦cute耻Á䃁reve;䄂Āiyx}rc耻Â䃂;䐐r;쀀𝔄rave耻À䃀pha;䎑acr;䄀d;橓Āgp¡on;䄄f;쀀𝔸plyFunction;恡ing耻Å䃅Ācs¾Ãr;쀀𝒜ign;扔ilde耻Ã䃃ml耻Ä䃄ЀaceforsuåûþėĜĢħĪĀcrêòkslash;或Ŷöø;櫧ed;挆y;䐑ƀcrtąċĔause;戵noullis;愬a;䎒r;쀀𝔅pf;쀀𝔹eve;䋘còēmpeq;扎܀HOacdefhilorsuōőŖƀƞƢƵƷƺǜȕɳɸɾcy;䐧PY耻©䂩ƀcpyŝŢźute;䄆Ā;iŧŨ拒talDifferentialD;慅leys;愭ȀaeioƉƎƔƘron;䄌dil耻Ç䃇rc;䄈nint;戰ot;䄊ĀdnƧƭilla;䂸terDot;䂷òſi;䎧rcleȀDMPTLJNjǑǖot;抙inus;抖lus;投imes;抗oĀcsǢǸkwiseContourIntegral;戲eCurlyĀDQȃȏoubleQuote;思uote;怙ȀlnpuȞȨɇɕonĀ;eȥȦ户;橴ƀgitȯȶȺruent;扡nt;戯ourIntegral;戮ĀfrɌɎ;愂oduct;成nterClockwiseContourIntegral;戳oss;樯cr;쀀𝒞pĀ;Cʄʅ拓ap;才րDJSZacefiosʠʬʰʴʸˋ˗ˡ˦̳ҍĀ;oŹʥtrahd;椑cy;䐂cy;䐅cy;䐏ƀgrsʿ˄ˇger;怡r;憡hv;櫤Āayː˕ron;䄎;䐔lĀ;t˝˞戇a;䎔r;쀀𝔇Āaf˫̧Ācm˰̢riticalȀADGT̖̜̀̆cute;䂴oŴ̋̍;䋙bleAcute;䋝rave;䁠ilde;䋜ond;拄ferentialD;慆Ѱ̽\0\0\0͔͂\0Ѕf;쀀𝔻ƀ;DE͈͉͍䂨ot;惜qual;扐blèCDLRUVͣͲ΂ϏϢϸontourIntegraìȹoɴ͹\0\0ͻ»͉nArrow;懓Āeo·ΤftƀARTΐΖΡrrow;懐ightArrow;懔eåˊngĀLRΫτeftĀARγιrrow;柸ightArrow;柺ightArrow;柹ightĀATϘϞrrow;懒ee;抨pɁϩ\0\0ϯrrow;懑ownArrow;懕erticalBar;戥ǹABLRTaВЪаўѿͼrrowƀ;BUНОТ憓ar;椓pArrow;懵reve;䌑eft˒к\0ц\0ѐightVector;楐eeVector;楞ectorĀ;Bљњ憽ar;楖ightǔѧ\0ѱeeVector;楟ectorĀ;BѺѻ懁ar;楗eeĀ;A҆҇护rrow;憧ĀctҒҗr;쀀𝒟rok;䄐ࠀNTacdfglmopqstuxҽӀӄӋӞӢӧӮӵԡԯԶՒ՝ՠեG;䅊H耻Ð䃐cute耻É䃉ƀaiyӒӗӜron;䄚rc耻Ê䃊;䐭ot;䄖r;쀀𝔈rave耻È䃈ement;戈ĀapӺӾcr;䄒tyɓԆ\0\0ԒmallSquare;旻erySmallSquare;斫ĀgpԦԪon;䄘f;쀀𝔼silon;䎕uĀaiԼՉlĀ;TՂՃ橵ilde;扂librium;懌Āci՗՚r;愰m;橳a;䎗ml耻Ë䃋Āipժկsts;戃onentialE;慇ʀcfiosօֈ֍ֲ׌y;䐤r;쀀𝔉lledɓ֗\0\0֣mallSquare;旼erySmallSquare;斪Ͱֺ\0ֿ\0\0ׄf;쀀𝔽All;戀riertrf;愱cò׋؀JTabcdfgorstר׬ׯ׺؀ؒؖ؛؝أ٬ٲcy;䐃耻>䀾mmaĀ;d׷׸䎓;䏜reve;䄞ƀeiy؇،ؐdil;䄢rc;䄜;䐓ot;䄠r;쀀𝔊;拙pf;쀀𝔾eater̀EFGLSTصلَٖٛ٦qualĀ;Lؾؿ扥ess;招ullEqual;执reater;檢ess;扷lantEqual;橾ilde;扳cr;쀀𝒢;扫ЀAacfiosuڅڋږڛڞڪھۊRDcy;䐪Āctڐڔek;䋇;䁞irc;䄤r;愌lbertSpace;愋ǰگ\0ڲf;愍izontalLine;攀Āctۃۅòکrok;䄦mpńېۘownHumðįqual;扏܀EJOacdfgmnostuۺ۾܃܇܎ܚܞܡܨ݄ݸދޏޕcy;䐕lig;䄲cy;䐁cute耻Í䃍Āiyܓܘrc耻Î䃎;䐘ot;䄰r;愑rave耻Ì䃌ƀ;apܠܯܿĀcgܴܷr;䄪inaryI;慈lieóϝǴ݉\0ݢĀ;eݍݎ戬Āgrݓݘral;戫section;拂isibleĀCTݬݲomma;恣imes;恢ƀgptݿރވon;䄮f;쀀𝕀a;䎙cr;愐ilde;䄨ǫޚ\0ޞcy;䐆l耻Ï䃏ʀcfosuެ޷޼߂ߐĀiyޱ޵rc;䄴;䐙r;쀀𝔍pf;쀀𝕁ǣ߇\0ߌr;쀀𝒥rcy;䐈kcy;䐄΀HJacfosߤߨ߽߬߱ࠂࠈcy;䐥cy;䐌ppa;䎚Āey߶߻dil;䄶;䐚r;쀀𝔎pf;쀀𝕂cr;쀀𝒦րJTaceflmostࠥࠩࠬࡐࡣ঳সে্਷ੇcy;䐉耻<䀼ʀcmnpr࠷࠼ࡁࡄࡍute;䄹bda;䎛g;柪lacetrf;愒r;憞ƀaeyࡗ࡜ࡡron;䄽dil;䄻;䐛Āfsࡨ॰tԀACDFRTUVarࡾࢩࢱࣦ࣠ࣼयज़ΐ४Ānrࢃ࢏gleBracket;柨rowƀ;BR࢙࢚࢞憐ar;懤ightArrow;懆eiling;挈oǵࢷ\0ࣃbleBracket;柦nǔࣈ\0࣒eeVector;楡ectorĀ;Bࣛࣜ懃ar;楙loor;挊ightĀAV࣯ࣵrrow;憔ector;楎Āerँगeƀ;AVउऊऐ抣rrow;憤ector;楚iangleƀ;BEतथऩ抲ar;槏qual;抴pƀDTVषूौownVector;楑eeVector;楠ectorĀ;Bॖॗ憿ar;楘ectorĀ;B॥०憼ar;楒ightáΜs̀EFGLSTॾঋকঝঢভqualGreater;拚ullEqual;扦reater;扶ess;檡lantEqual;橽ilde;扲r;쀀𝔏Ā;eঽা拘ftarrow;懚idot;䄿ƀnpw৔ਖਛgȀLRlr৞৷ਂਐeftĀAR০৬rrow;柵ightArrow;柷ightArrow;柶eftĀarγਊightáοightáϊf;쀀𝕃erĀLRਢਬeftArrow;憙ightArrow;憘ƀchtਾੀੂòࡌ;憰rok;䅁;扪Ѐacefiosuਗ਼੝੠੷੼અઋ઎p;椅y;䐜Ādl੥੯iumSpace;恟lintrf;愳r;쀀𝔐nusPlus;戓pf;쀀𝕄cò੶;䎜ҀJacefostuણધભીଔଙඑ඗ඞcy;䐊cute;䅃ƀaey઴હાron;䅇dil;䅅;䐝ƀgswે૰଎ativeƀMTV૓૟૨ediumSpace;怋hiĀcn૦૘ë૙eryThiî૙tedĀGL૸ଆreaterGreateòٳessLesóੈLine;䀊r;쀀𝔑ȀBnptଢନଷ଺reak;恠BreakingSpace;䂠f;愕ڀ;CDEGHLNPRSTV୕ୖ୪୼஡௫ఄ౞಄ದ೘ൡඅ櫬Āou୛୤ngruent;扢pCap;扭oubleVerticalBar;戦ƀlqxஃஊ஛ement;戉ualĀ;Tஒஓ扠ilde;쀀≂̸ists;戄reater΀;EFGLSTஶஷ஽௉௓௘௥扯qual;扱ullEqual;쀀≧̸reater;쀀≫̸ess;批lantEqual;쀀⩾̸ilde;扵umpń௲௽ownHump;쀀≎̸qual;쀀≏̸eĀfsఊధtTriangleƀ;BEచఛడ拪ar;쀀⧏̸qual;括s̀;EGLSTవశ఼ౄోౘ扮qual;扰reater;扸ess;쀀≪̸lantEqual;쀀⩽̸ilde;扴estedĀGL౨౹reaterGreater;쀀⪢̸essLess;쀀⪡̸recedesƀ;ESಒಓಛ技qual;쀀⪯̸lantEqual;拠ĀeiಫಹverseElement;戌ghtTriangleƀ;BEೋೌ೒拫ar;쀀⧐̸qual;拭ĀquೝഌuareSuĀbp೨೹setĀ;E೰ೳ쀀⊏̸qual;拢ersetĀ;Eഃആ쀀⊐̸qual;拣ƀbcpഓതൎsetĀ;Eഛഞ쀀⊂⃒qual;抈ceedsȀ;ESTലള഻െ抁qual;쀀⪰̸lantEqual;拡ilde;쀀≿̸ersetĀ;E൘൛쀀⊃⃒qual;抉ildeȀ;EFT൮൯൵ൿ扁qual;扄ullEqual;扇ilde;扉erticalBar;戤cr;쀀𝒩ilde耻Ñ䃑;䎝܀Eacdfgmoprstuvලෂ෉෕ෛ෠෧෼ขภยา฿ไlig;䅒cute耻Ó䃓Āiy෎ීrc耻Ô䃔;䐞blac;䅐r;쀀𝔒rave耻Ò䃒ƀaei෮ෲ෶cr;䅌ga;䎩cron;䎟pf;쀀𝕆enCurlyĀDQฎบoubleQuote;怜uote;怘;橔Āclวฬr;쀀𝒪ash耻Ø䃘iŬื฼de耻Õ䃕es;樷ml耻Ö䃖erĀBP๋๠Āar๐๓r;怾acĀek๚๜;揞et;掴arenthesis;揜Ҁacfhilors๿ງຊຏຒດຝະ໼rtialD;戂y;䐟r;쀀𝔓i;䎦;䎠usMinus;䂱Āipຢອncareplanåڝf;愙Ȁ;eio຺ູ໠໤檻cedesȀ;EST່້໏໚扺qual;檯lantEqual;扼ilde;找me;怳Ādp໩໮uct;戏ortionĀ;aȥ໹l;戝Āci༁༆r;쀀𝒫;䎨ȀUfos༑༖༛༟OT耻"䀢r;쀀𝔔pf;愚cr;쀀𝒬؀BEacefhiorsu༾གྷཇའཱིྦྷྪྭ႖ႩႴႾarr;椐G耻®䂮ƀcnrཎནབute;䅔g;柫rĀ;tཛྷཝ憠l;椖ƀaeyཧཬཱron;䅘dil;䅖;䐠Ā;vླྀཹ愜erseĀEUྂྙĀlq྇ྎement;戋uilibrium;懋pEquilibrium;楯r»ཹo;䎡ghtЀACDFTUVa࿁࿫࿳ဢဨၛႇϘĀnr࿆࿒gleBracket;柩rowƀ;BL࿜࿝࿡憒ar;懥eftArrow;懄eiling;按oǵ࿹\0စbleBracket;柧nǔည\0နeeVector;楝ectorĀ;Bဝသ懂ar;楕loor;挋Āerိ၃eƀ;AVဵံြ抢rrow;憦ector;楛iangleƀ;BEၐၑၕ抳ar;槐qual;抵pƀDTVၣၮၸownVector;楏eeVector;楜ectorĀ;Bႂႃ憾ar;楔ectorĀ;B႑႒懀ar;楓Āpuႛ႞f;愝ndImplies;楰ightarrow;懛ĀchႹႼr;愛;憱leDelayed;槴ڀHOacfhimoqstuფჱჷჽᄙᄞᅑᅖᅡᅧᆵᆻᆿĀCcჩხHcy;䐩y;䐨FTcy;䐬cute;䅚ʀ;aeiyᄈᄉᄎᄓᄗ檼ron;䅠dil;䅞rc;䅜;䐡r;쀀𝔖ortȀDLRUᄪᄴᄾᅉownArrow»ОeftArrow»࢚ightArrow»࿝pArrow;憑gma;䎣allCircle;战pf;쀀𝕊ɲᅭ\0\0ᅰt;戚areȀ;ISUᅻᅼᆉᆯ斡ntersection;抓uĀbpᆏᆞsetĀ;Eᆗᆘ抏qual;抑ersetĀ;Eᆨᆩ抐qual;抒nion;抔cr;쀀𝒮ar;拆ȀbcmpᇈᇛሉላĀ;sᇍᇎ拐etĀ;Eᇍᇕqual;抆ĀchᇠህeedsȀ;ESTᇭᇮᇴᇿ扻qual;檰lantEqual;扽ilde;承Tháྌ;我ƀ;esሒሓሣ拑rsetĀ;Eሜም抃qual;抇et»ሓրHRSacfhiorsሾቄ቉ቕ቞ቱቶኟዂወዑORN耻Þ䃞ADE;愢ĀHc቎ቒcy;䐋y;䐦Ābuቚቜ;䀉;䎤ƀaeyብቪቯron;䅤dil;䅢;䐢r;쀀𝔗Āeiቻ኉Dzኀ\0ኇefore;戴a;䎘Ācn኎ኘkSpace;쀀  Space;怉ldeȀ;EFTካኬኲኼ戼qual;扃ullEqual;扅ilde;扈pf;쀀𝕋ipleDot;惛Āctዖዛr;쀀𝒯rok;䅦ૡዷጎጚጦ\0ጬጱ\0\0\0\0\0ጸጽ፷ᎅ\0᏿ᐄᐊᐐĀcrዻጁute耻Ú䃚rĀ;oጇገ憟cir;楉rǣጓ\0጖y;䐎ve;䅬Āiyጞጣrc耻Û䃛;䐣blac;䅰r;쀀𝔘rave耻Ù䃙acr;䅪Ādiፁ፩erĀBPፈ፝Āarፍፐr;䁟acĀekፗፙ;揟et;掵arenthesis;揝onĀ;P፰፱拃lus;抎Āgp፻፿on;䅲f;쀀𝕌ЀADETadps᎕ᎮᎸᏄϨᏒᏗᏳrrowƀ;BDᅐᎠᎤar;椒ownArrow;懅ownArrow;憕quilibrium;楮eeĀ;AᏋᏌ报rrow;憥ownáϳerĀLRᏞᏨeftArrow;憖ightArrow;憗iĀ;lᏹᏺ䏒on;䎥ing;䅮cr;쀀𝒰ilde;䅨ml耻Ü䃜ҀDbcdefosvᐧᐬᐰᐳᐾᒅᒊᒐᒖash;披ar;櫫y;䐒ashĀ;lᐻᐼ抩;櫦Āerᑃᑅ;拁ƀbtyᑌᑐᑺar;怖Ā;iᑏᑕcalȀBLSTᑡᑥᑪᑴar;戣ine;䁼eparator;杘ilde;所ThinSpace;怊r;쀀𝔙pf;쀀𝕍cr;쀀𝒱dash;抪ʀcefosᒧᒬᒱᒶᒼirc;䅴dge;拀r;쀀𝔚pf;쀀𝕎cr;쀀𝒲Ȁfiosᓋᓐᓒᓘr;쀀𝔛;䎞pf;쀀𝕏cr;쀀𝒳ҀAIUacfosuᓱᓵᓹᓽᔄᔏᔔᔚᔠcy;䐯cy;䐇cy;䐮cute耻Ý䃝Āiyᔉᔍrc;䅶;䐫r;쀀𝔜pf;쀀𝕐cr;쀀𝒴ml;䅸ЀHacdefosᔵᔹᔿᕋᕏᕝᕠᕤcy;䐖cute;䅹Āayᕄᕉron;䅽;䐗ot;䅻Dzᕔ\0ᕛoWidtè૙a;䎖r;愨pf;愤cr;쀀𝒵௡ᖃᖊᖐ\0ᖰᖶᖿ\0\0\0\0ᗆᗛᗫᙟ᙭\0ᚕ᚛ᚲᚹ\0ᚾcute耻á䃡reve;䄃̀;Ediuyᖜᖝᖡᖣᖨᖭ戾;쀀∾̳;房rc耻â䃢te肻´̆;䐰lig耻æ䃦Ā;r²ᖺ;쀀𝔞rave耻à䃠ĀepᗊᗖĀfpᗏᗔsym;愵èᗓha;䎱ĀapᗟcĀclᗤᗧr;䄁g;樿ɤᗰ\0\0ᘊʀ;adsvᗺᗻᗿᘁᘇ戧nd;橕;橜lope;橘;橚΀;elmrszᘘᘙᘛᘞᘿᙏᙙ戠;榤e»ᘙsdĀ;aᘥᘦ戡ѡᘰᘲᘴᘶᘸᘺᘼᘾ;榨;榩;榪;榫;榬;榭;榮;榯tĀ;vᙅᙆ戟bĀ;dᙌᙍ抾;榝Āptᙔᙗh;戢»¹arr;捼Āgpᙣᙧon;䄅f;쀀𝕒΀;Eaeiop዁ᙻᙽᚂᚄᚇᚊ;橰cir;橯;扊d;手s;䀧roxĀ;e዁ᚒñᚃing耻å䃥ƀctyᚡᚦᚨr;쀀𝒶;䀪mpĀ;e዁ᚯñʈilde耻ã䃣ml耻ä䃤Āciᛂᛈoninôɲnt;樑ࠀNabcdefiklnoprsu᛭ᛱᜰ᜼ᝃᝈ᝸᝽០៦ᠹᡐᜍ᤽᥈ᥰot;櫭Ācrᛶ᜞kȀcepsᜀᜅᜍᜓong;扌psilon;䏶rime;怵imĀ;e᜚᜛戽q;拍Ŷᜢᜦee;抽edĀ;gᜬᜭ挅e»ᜭrkĀ;t፜᜷brk;掶Āoyᜁᝁ;䐱quo;怞ʀcmprtᝓ᝛ᝡᝤᝨausĀ;eĊĉptyv;榰séᜌnoõēƀahwᝯ᝱ᝳ;䎲;愶een;扬r;쀀𝔟g΀costuvwឍឝឳេ៕៛៞ƀaiuបពរðݠrc;旯p»፱ƀdptឤឨឭot;樀lus;樁imes;樂ɱឹ\0\0ើcup;樆ar;昅riangleĀdu៍្own;施p;斳plus;樄eåᑄåᒭarow;植ƀako៭ᠦᠵĀcn៲ᠣkƀlst៺֫᠂ozenge;槫riangleȀ;dlr᠒᠓᠘᠝斴own;斾eft;旂ight;斸k;搣Ʊᠫ\0ᠳƲᠯ\0ᠱ;斒;斑4;斓ck;斈ĀeoᠾᡍĀ;qᡃᡆ쀀=⃥uiv;쀀≡⃥t;挐Ȁptwxᡙᡞᡧᡬf;쀀𝕓Ā;tᏋᡣom»Ꮜtie;拈؀DHUVbdhmptuvᢅᢖᢪᢻᣗᣛᣬ᣿ᤅᤊᤐᤡȀLRlrᢎᢐᢒᢔ;敗;敔;敖;敓ʀ;DUduᢡᢢᢤᢦᢨ敐;敦;敩;敤;敧ȀLRlrᢳᢵᢷᢹ;敝;敚;敜;教΀;HLRhlrᣊᣋᣍᣏᣑᣓᣕ救;敬;散;敠;敫;敢;敟ox;槉ȀLRlrᣤᣦᣨᣪ;敕;敒;攐;攌ʀ;DUduڽ᣷᣹᣻᣽;敥;敨;攬;攴inus;抟lus;択imes;抠ȀLRlrᤙᤛᤝ᤟;敛;敘;攘;攔΀;HLRhlrᤰᤱᤳᤵᤷ᤻᤹攂;敪;敡;敞;攼;攤;攜Āevģ᥂bar耻¦䂦Ȁceioᥑᥖᥚᥠr;쀀𝒷mi;恏mĀ;e᜚᜜lƀ;bhᥨᥩᥫ䁜;槅sub;柈Ŭᥴ᥾lĀ;e᥹᥺怢t»᥺pƀ;Eeįᦅᦇ;檮Ā;qۜۛೡᦧ\0᧨ᨑᨕᨲ\0ᨷᩐ\0\0᪴\0\0᫁\0\0ᬡᬮ᭍᭒\0᯽\0ᰌƀcpr᦭ᦲ᧝ute;䄇̀;abcdsᦿᧀᧄ᧊᧕᧙戩nd;橄rcup;橉Āau᧏᧒p;橋p;橇ot;橀;쀀∩︀Āeo᧢᧥t;恁îړȀaeiu᧰᧻ᨁᨅǰ᧵\0᧸s;橍on;䄍dil耻ç䃧rc;䄉psĀ;sᨌᨍ橌m;橐ot;䄋ƀdmnᨛᨠᨦil肻¸ƭptyv;榲t脀¢;eᨭᨮ䂢räƲr;쀀𝔠ƀceiᨽᩀᩍy;䑇ckĀ;mᩇᩈ朓ark»ᩈ;䏇r΀;Ecefms᩟᩠ᩢᩫ᪤᪪᪮旋;槃ƀ;elᩩᩪᩭ䋆q;扗eɡᩴ\0\0᪈rrowĀlr᩼᪁eft;憺ight;憻ʀRSacd᪒᪔᪖᪚᪟»ཇ;擈st;抛irc;抚ash;抝nint;樐id;櫯cir;槂ubsĀ;u᪻᪼晣it»᪼ˬ᫇᫔᫺\0ᬊonĀ;eᫍᫎ䀺Ā;qÇÆɭ᫙\0\0᫢aĀ;t᫞᫟䀬;䁀ƀ;fl᫨᫩᫫戁îᅠeĀmx᫱᫶ent»᫩eóɍǧ᫾\0ᬇĀ;dኻᬂot;橭nôɆƀfryᬐᬔᬗ;쀀𝕔oäɔ脀©;sŕᬝr;愗Āaoᬥᬩrr;憵ss;朗Ācuᬲᬷr;쀀𝒸Ābpᬼ᭄Ā;eᭁᭂ櫏;櫑Ā;eᭉᭊ櫐;櫒dot;拯΀delprvw᭠᭬᭷ᮂᮬᯔ᯹arrĀlr᭨᭪;椸;椵ɰ᭲\0\0᭵r;拞c;拟arrĀ;p᭿ᮀ憶;椽̀;bcdosᮏᮐᮖᮡᮥᮨ截rcap;橈Āauᮛᮞp;橆p;橊ot;抍r;橅;쀀∪︀Ȁalrv᮵ᮿᯞᯣrrĀ;mᮼᮽ憷;椼yƀevwᯇᯔᯘqɰᯎ\0\0ᯒreã᭳uã᭵ee;拎edge;拏en耻¤䂤earrowĀlrᯮ᯳eft»ᮀight»ᮽeäᯝĀciᰁᰇoninôǷnt;戱lcty;挭ঀAHabcdefhijlorstuwz᰸᰻᰿ᱝᱩᱵᲊᲞᲬᲷ᳻᳿ᴍᵻᶑᶫᶻ᷆᷍rò΁ar;楥Ȁglrs᱈ᱍ᱒᱔ger;怠eth;愸òᄳhĀ;vᱚᱛ怐»ऊūᱡᱧarow;椏aã̕Āayᱮᱳron;䄏;䐴ƀ;ao̲ᱼᲄĀgrʿᲁr;懊tseq;橷ƀglmᲑᲔᲘ耻°䂰ta;䎴ptyv;榱ĀirᲣᲨsht;楿;쀀𝔡arĀlrᲳᲵ»ࣜ»သʀaegsv᳂͸᳖᳜᳠mƀ;oș᳊᳔ndĀ;ș᳑uit;晦amma;䏝in;拲ƀ;io᳧᳨᳸䃷de脀÷;o᳧ᳰntimes;拇nø᳷cy;䑒cɯᴆ\0\0ᴊrn;挞op;挍ʀlptuwᴘᴝᴢᵉᵕlar;䀤f;쀀𝕕ʀ;emps̋ᴭᴷᴽᵂqĀ;d͒ᴳot;扑inus;戸lus;戔quare;抡blebarwedgåúnƀadhᄮᵝᵧownarrowóᲃarpoonĀlrᵲᵶefôᲴighôᲶŢᵿᶅkaro÷གɯᶊ\0\0ᶎrn;挟op;挌ƀcotᶘᶣᶦĀryᶝᶡ;쀀𝒹;䑕l;槶rok;䄑Ādrᶰᶴot;拱iĀ;fᶺ᠖斿Āah᷀᷃ròЩaòྦangle;榦Āci᷒ᷕy;䑟grarr;柿ऀDacdefglmnopqrstuxḁḉḙḸոḼṉṡṾấắẽỡἪἷὄ὎὚ĀDoḆᴴoôᲉĀcsḎḔute耻é䃩ter;橮ȀaioyḢḧḱḶron;䄛rĀ;cḭḮ扖耻ê䃪lon;払;䑍ot;䄗ĀDrṁṅot;扒;쀀𝔢ƀ;rsṐṑṗ檚ave耻è䃨Ā;dṜṝ檖ot;檘Ȁ;ilsṪṫṲṴ檙nters;揧;愓Ā;dṹṺ檕ot;檗ƀapsẅẉẗcr;䄓tyƀ;svẒẓẕ戅et»ẓpĀ1;ẝẤijạả;怄;怅怃ĀgsẪẬ;䅋p;怂ĀgpẴẸon;䄙f;쀀𝕖ƀalsỄỎỒrĀ;sỊị拕l;槣us;橱iƀ;lvỚớở䎵on»ớ;䏵ȀcsuvỪỳἋἣĀioữḱrc»Ḯɩỹ\0\0ỻíՈantĀglἂἆtr»ṝess»Ṻƀaeiἒ἖Ἒls;䀽st;扟vĀ;DȵἠD;橸parsl;槥ĀDaἯἳot;打rr;楱ƀcdiἾὁỸr;愯oô͒ĀahὉὋ;䎷耻ð䃰Āmrὓὗl耻ë䃫o;悬ƀcipὡὤὧl;䀡sôծĀeoὬὴctatioîՙnentialåչৡᾒ\0ᾞ\0ᾡᾧ\0\0ῆῌ\0ΐ\0ῦῪ \0 ⁚llingdotseñṄy;䑄male;晀ƀilrᾭᾳ῁lig;耀ffiɩᾹ\0\0᾽g;耀ffig;耀ffl;쀀𝔣lig;耀filig;쀀fjƀaltῙ῜ῡt;晭ig;耀flns;斱of;䆒ǰ΅\0ῳf;쀀𝕗ĀakֿῷĀ;vῼ´拔;櫙artint;樍Āao‌⁕Ācs‑⁒ႉ‸⁅⁈\0⁐β•‥‧‪‬\0‮耻½䂽;慓耻¼䂼;慕;慙;慛Ƴ‴\0‶;慔;慖ʴ‾⁁\0\0⁃耻¾䂾;慗;慜5;慘ƶ⁌\0⁎;慚;慝8;慞l;恄wn;挢cr;쀀𝒻ࢀEabcdefgijlnorstv₂₉₟₥₰₴⃰⃵⃺⃿℃ℒℸ̗ℾ⅒↞Ā;lٍ₇;檌ƀcmpₐₕ₝ute;䇵maĀ;dₜ᳚䎳;檆reve;䄟Āiy₪₮rc;䄝;䐳ot;䄡Ȁ;lqsؾق₽⃉ƀ;qsؾٌ⃄lanô٥Ȁ;cdl٥⃒⃥⃕c;檩otĀ;o⃜⃝檀Ā;l⃢⃣檂;檄Ā;e⃪⃭쀀⋛︀s;檔r;쀀𝔤Ā;gٳ؛mel;愷cy;䑓Ȁ;Eajٚℌℎℐ;檒;檥;檤ȀEaesℛℝ℩ℴ;扩pĀ;p℣ℤ檊rox»ℤĀ;q℮ℯ檈Ā;q℮ℛim;拧pf;쀀𝕘Āci⅃ⅆr;愊mƀ;el٫ⅎ⅐;檎;檐茀>;cdlqr׮ⅠⅪⅮⅳⅹĀciⅥⅧ;檧r;橺ot;拗Par;榕uest;橼ʀadelsↄⅪ←ٖ↛ǰ↉\0↎proø₞r;楸qĀlqؿ↖lesó₈ií٫Āen↣↭rtneqq;쀀≩︀Å↪ԀAabcefkosy⇄⇇⇱⇵⇺∘∝∯≨≽ròΠȀilmr⇐⇔⇗⇛rsðᒄf»․ilôکĀdr⇠⇤cy;䑊ƀ;cwࣴ⇫⇯ir;楈;憭ar;意irc;䄥ƀalr∁∎∓rtsĀ;u∉∊晥it»∊lip;怦con;抹r;쀀𝔥sĀew∣∩arow;椥arow;椦ʀamopr∺∾≃≞≣rr;懿tht;戻kĀlr≉≓eftarrow;憩ightarrow;憪f;쀀𝕙bar;怕ƀclt≯≴≸r;쀀𝒽asè⇴rok;䄧Ābp⊂⊇ull;恃hen»ᱛૡ⊣\0⊪\0⊸⋅⋎\0⋕⋳\0\0⋸⌢⍧⍢⍿\0⎆⎪⎴cute耻í䃭ƀ;iyݱ⊰⊵rc耻î䃮;䐸Ācx⊼⊿y;䐵cl耻¡䂡ĀfrΟ⋉;쀀𝔦rave耻ì䃬Ȁ;inoܾ⋝⋩⋮Āin⋢⋦nt;樌t;戭fin;槜ta;愩lig;䄳ƀaop⋾⌚⌝ƀcgt⌅⌈⌗r;䄫ƀelpܟ⌏⌓inåގarôܠh;䄱f;抷ed;䆵ʀ;cfotӴ⌬⌱⌽⍁are;愅inĀ;t⌸⌹戞ie;槝doô⌙ʀ;celpݗ⍌⍐⍛⍡al;抺Āgr⍕⍙eróᕣã⍍arhk;樗rod;樼Ȁcgpt⍯⍲⍶⍻y;䑑on;䄯f;쀀𝕚a;䎹uest耻¿䂿Āci⎊⎏r;쀀𝒾nʀ;EdsvӴ⎛⎝⎡ӳ;拹ot;拵Ā;v⎦⎧拴;拳Ā;iݷ⎮lde;䄩ǫ⎸\0⎼cy;䑖l耻ï䃯̀cfmosu⏌⏗⏜⏡⏧⏵Āiy⏑⏕rc;䄵;䐹r;쀀𝔧ath;䈷pf;쀀𝕛ǣ⏬\0⏱r;쀀𝒿rcy;䑘kcy;䑔Ѐacfghjos␋␖␢␧␭␱␵␻ppaĀ;v␓␔䎺;䏰Āey␛␠dil;䄷;䐺r;쀀𝔨reen;䄸cy;䑅cy;䑜pf;쀀𝕜cr;쀀𝓀஀ABEHabcdefghjlmnoprstuv⑰⒁⒆⒍⒑┎┽╚▀♎♞♥♹♽⚚⚲⛘❝❨➋⟀⠁⠒ƀart⑷⑺⑼rò৆òΕail;椛arr;椎Ā;gঔ⒋;檋ar;楢ॣ⒥\0⒪\0⒱\0\0\0\0\0⒵Ⓔ\0ⓆⓈⓍ\0⓹ute;䄺mptyv;榴raîࡌbda;䎻gƀ;dlࢎⓁⓃ;榑åࢎ;檅uo耻«䂫rЀ;bfhlpst࢙ⓞⓦⓩ⓫⓮⓱⓵Ā;f࢝ⓣs;椟s;椝ë≒p;憫l;椹im;楳l;憢ƀ;ae⓿─┄檫il;椙Ā;s┉┊檭;쀀⪭︀ƀabr┕┙┝rr;椌rk;杲Āak┢┬cĀek┨┪;䁻;䁛Āes┱┳;榋lĀdu┹┻;榏;榍Ȁaeuy╆╋╖╘ron;䄾Ādi═╔il;䄼ìࢰâ┩;䐻Ȁcqrs╣╦╭╽a;椶uoĀ;rนᝆĀdu╲╷har;楧shar;楋h;憲ʀ;fgqs▋▌উ◳◿扤tʀahlrt▘▤▷◂◨rrowĀ;t࢙□aé⓶arpoonĀdu▯▴own»њp»०eftarrows;懇ightƀahs◍◖◞rrowĀ;sࣴࢧarpoonó྘quigarro÷⇰hreetimes;拋ƀ;qs▋ও◺lanôবʀ;cdgsব☊☍☝☨c;檨otĀ;o☔☕橿Ā;r☚☛檁;檃Ā;e☢☥쀀⋚︀s;檓ʀadegs☳☹☽♉♋pproøⓆot;拖qĀgq♃♅ôউgtò⒌ôছiíলƀilr♕࣡♚sht;楼;쀀𝔩Ā;Eজ♣;檑š♩♶rĀdu▲♮Ā;l॥♳;楪lk;斄cy;䑙ʀ;achtੈ⚈⚋⚑⚖rò◁orneòᴈard;楫ri;旺Āio⚟⚤dot;䅀ustĀ;a⚬⚭掰che»⚭ȀEaes⚻⚽⛉⛔;扨pĀ;p⛃⛄檉rox»⛄Ā;q⛎⛏檇Ā;q⛎⚻im;拦Ѐa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di㩀㩑Ābg㩅㩉ar;機eĀ;qᗺ㩏;扙erp;愘r;쀀𝔴pf;쀀𝕨Ā;eᑹ㩦atèᑹcr;쀀𝓌ૣណ㪇\0㪋\0㪐㪛\0\0㪝㪨㪫㪯\0\0㫃㫎\0㫘ៜ៟tré៑r;쀀𝔵ĀAa㪔㪗ròσrò৶;䎾ĀAa㪡㪤ròθrò৫að✓is;拻ƀdptឤ㪵㪾Āfl㪺ឩ;쀀𝕩imåឲĀAa㫇㫊ròώròਁĀcq㫒ីr;쀀𝓍Āpt៖㫜ré។Ѐacefiosu㫰㫽㬈㬌㬑㬕㬛㬡cĀuy㫶㫻te耻ý䃽;䑏Āiy㬂㬆rc;䅷;䑋n耻¥䂥r;쀀𝔶cy;䑗pf;쀀𝕪cr;쀀𝓎Ācm㬦㬩y;䑎l耻ÿ䃿Ԁacdefhiosw㭂㭈㭔㭘㭤㭩㭭㭴㭺㮀cute;䅺Āay㭍㭒ron;䅾;䐷ot;䅼Āet㭝㭡træᕟa;䎶r;쀀𝔷cy;䐶grarr;懝pf;쀀𝕫cr;쀀𝓏Ājn㮅㮇;怍j;怌'.split("").map(s=>s.charCodeAt(0))),QV=new 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p=[],r=[],m=[],o=[],c=s.md.block.ruler.getRules("blockquote"),h=s.parentType;s.parentType="blockquote";let u=!1,y;for(y=a;y<t;y++){const A=s.sCount[y]<s.blkIndent;if(e=s.bMarks[y]+s.tShift[y],l=s.eMarks[y],e>=l)break;if(s.src.charCodeAt(e++)===62&&!A){let I=s.sCount[y]+1,$,U;s.src.charCodeAt(e)===32?(e++,I++,U=!1,$=!0):s.src.charCodeAt(e)===9?($=!0,(s.bsCount[y]+I)%4===3?(e++,I++,U=!1):U=!0):$=!1;let L=I;for(p.push(s.bMarks[y]),s.bMarks[y]=e;e<l;){const P=s.src.charCodeAt(e);if(At(P))P===9?L+=4-(L+s.bsCount[y]+(U?1:0))%4:L++;else break;e++}u=e>=l,r.push(s.bsCount[y]),s.bsCount[y]=s.sCount[y]+1+($?1:0),m.push(s.sCount[y]),s.sCount[y]=L-I,o.push(s.tShift[y]),s.tShift[y]=e-s.bMarks[y];continue}if(u)break;let C=!1;for(let I=0,$=c.length;I<$;I++)if(c[I](s,y,t,!0)){C=!0;break}if(C){s.lineMax=y,s.blkIndent!==0&&(p.push(s.bMarks[y]),r.push(s.bsCount[y]),o.push(s.tShift[y]),m.push(s.sCount[y]),s.sCount[y]-=s.blkIndent);break}p.push(s.bMarks[y]),r.push(s.bsCount[y]),o.push(s.tShift[y]),m.push(s.sCount[y]),s.sCount[y]=-1}const x=s.blkIndent;s.blkIndent=0;const b=s.push("blockquote_open","blockquote",1);b.markup=">";const _=[a,0];b.map=_,s.md.block.tokenize(s,a,y);const T=s.push("blockquote_close","blockquote",-1);T.markup=">",s.lineMax=i,s.parentType=h,_[1]=s.line;for(let A=0;A<o.length;A++)s.bMarks[A+a]=p[A],s.tShift[A+a]=o[A],s.sCount[A+a]=m[A],s.bsCount[A+a]=r[A];return s.blkIndent=x,!0}function JH(s,a,t,n){const e=s.eMarks[a];if(s.sCount[a]-s.blkIndent>=4)return!1;let l=s.bMarks[a]+s.tShift[a];const i=s.src.charCodeAt(l++);if(i!==42&&i!==45&&i!==95)return!1;let p=1;for(;l<e;){const m=s.src.charCodeAt(l++);if(m!==i&&!At(m))return!1;m===i&&p++}if(p<3)return!1;if(n)return!0;s.line=a+1;const r=s.push("hr","hr",0);return r.map=[a,s.line],r.markup=Array(p+1).join(String.fromCharCode(i)),!0}function W6(s,a){const t=s.eMarks[a];let n=s.bMarks[a]+s.tShift[a];const e=s.src.charCodeAt(n++);if(e!==42&&e!==45&&e!==43)return-1;if(n<t){const l=s.src.charCodeAt(n);if(!At(l))return-1}return n}function Y6(s,a){const t=s.bMarks[a]+s.tShift[a],n=s.eMarks[a];let e=t;if(e+1>=n)return-1;let l=s.src.charCodeAt(e++);if(l<48||l>57)return-1;for(;;){if(e>=n)return-1;if(l=s.src.charCodeAt(e++),l>=48&&l<=57){if(e-t>=10)return-1;continue}if(l===41||l===46)break;return-1}return e<n&&(l=s.src.charCodeAt(e),!At(l))?-1:e}function sq(s,a){const t=s.level+2;for(let n=a+2,e=s.tokens.length-2;n<e;n++)s.tokens[n].level===t&&s.tokens[n].type==="paragraph_open"&&(s.tokens[n+2].hidden=!0,s.tokens[n].hidden=!0,n+=2)}function aq(s,a,t,n){let e,l,i,p,r=a,m=!0;if(s.sCount[r]-s.blkIndent>=4||s.listIndent>=0&&s.sCount[r]-s.listIndent>=4&&s.sCount[r]<s.blkIndent)return!1;let o=!1;n&&s.parentType==="paragraph"&&s.sCount[r]>=s.blkIndent&&(o=!0);let c,h,u;if((u=Y6(s,r))>=0){if(c=!0,i=s.bMarks[r]+s.tShift[r],h=Number(s.src.slice(i,u-1)),o&&h!==1)return!1}else if((u=W6(s,r))>=0)c=!1;else return!1;if(o&&s.skipSpaces(u)>=s.eMarks[r])return!1;if(n)return!0;const y=s.src.charCodeAt(u-1),x=s.tokens.length;c?(p=s.push("ordered_list_open","ol",1),h!==1&&(p.attrs=[["start",h]])):p=s.push("bullet_list_open","ul",1);const b=[r,0];p.map=b,p.markup=String.fromCharCode(y);let _=!1;const T=s.md.block.ruler.getRules("list"),A=s.parentType;for(s.parentType="list";r<t;){l=u,e=s.eMarks[r];const C=s.sCount[r]+u-(s.bMarks[r]+s.tShift[r]);let I=C;for(;l<e;){const ss=s.src.charCodeAt(l);if(ss===9)I+=4-(I+s.bsCount[r])%4;else if(ss===32)I++;else break;l++}const $=l;let U;$>=e?U=1:U=I-C,U>4&&(U=1);const L=C+U;p=s.push("list_item_open","li",1),p.markup=String.fromCharCode(y);const P=[r,0];p.map=P,c&&(p.info=s.src.slice(i,u-1));const G=s.tight,X=s.tShift[r],S=s.sCount[r],is=s.listIndent;if(s.listIndent=s.blkIndent,s.blkIndent=L,s.tight=!0,s.tShift[r]=$-s.bMarks[r],s.sCount[r]=I,$>=e&&s.isEmpty(r+1)?s.line=Math.min(s.line+2,t):s.md.block.tokenize(s,r,t,!0),(!s.tight||_)&&(m=!1),_=s.line-r>1&&s.isEmpty(s.line-1),s.blkIndent=s.listIndent,s.listIndent=is,s.tShift[r]=X,s.sCount[r]=S,s.tight=G,p=s.push("list_item_close","li",-1),p.markup=String.fromCharCode(y),r=s.line,P[1]=r,r>=t||s.sCount[r]<s.blkIndent||s.sCount[r]-s.blkIndent>=4)break;let R=!1;for(let ss=0,Y=T.length;ss<Y;ss++)if(T[ss](s,r,t,!0)){R=!0;break}if(R)break;if(c){if(u=Y6(s,r),u<0)break;i=s.bMarks[r]+s.tShift[r]}else if(u=W6(s,r),u<0)break;if(y!==s.src.charCodeAt(u-1))break}return c?p=s.push("ordered_list_close","ol",-1):p=s.push("bullet_list_close","ul",-1),p.markup=String.fromCharCode(y),b[1]=r,s.line=r,s.parentType=A,m&&sq(s,x),!0}function tq(s,a,t,n){let e=s.bMarks[a]+s.tShift[a],l=s.eMarks[a],i=a+1;if(s.sCount[a]-s.blkIndent>=4||s.src.charCodeAt(e)!==91)return!1;function p(T){const A=s.lineMax;if(T>=A||s.isEmpty(T))return null;let C=!1;if(s.sCount[T]-s.blkIndent>3&&(C=!0),s.sCount[T]<0&&(C=!0),!C){const U=s.md.block.ruler.getRules("reference"),L=s.parentType;s.parentType="reference";let P=!1;for(let G=0,X=U.length;G<X;G++)if(U[G](s,T,A,!0)){P=!0;break}if(s.parentType=L,P)return null}const I=s.bMarks[T]+s.tShift[T],$=s.eMarks[T];return s.src.slice(I,$+1)}let r=s.src.slice(e,l+1);l=r.length;let m=-1;for(e=1;e<l;e++){const T=r.charCodeAt(e);if(T===91)return!1;if(T===93){m=e;break}else if(T===10){const A=p(i);A!==null&&(r+=A,l=r.length,i++)}else if(T===92&&(e++,e<l&&r.charCodeAt(e)===10)){const A=p(i);A!==null&&(r+=A,l=r.length,i++)}}if(m<0||r.charCodeAt(m+1)!==58)return!1;for(e=m+2;e<l;e++){const T=r.charCodeAt(e);if(T===10){const A=p(i);A!==null&&(r+=A,l=r.length,i++)}else if(!At(T))break}const o=s.md.helpers.parseLinkDestination(r,e,l);if(!o.ok)return!1;const c=s.md.normalizeLink(o.str);if(!s.md.validateLink(c))return!1;e=o.pos;const h=e,u=i,y=e;for(;e<l;e++){const T=r.charCodeAt(e);if(T===10){const A=p(i);A!==null&&(r+=A,l=r.length,i++)}else if(!At(T))break}let x=s.md.helpers.parseLinkTitle(r,e,l);for(;x.can_continue;){const T=p(i);if(T===null)break;r+=T,e=l,l=r.length,i++,x=s.md.helpers.parseLinkTitle(r,e,l,x)}let b;for(e<l&&y!==e&&x.ok?(b=x.str,e=x.pos):(b="",e=h,i=u);e<l;){const T=r.charCodeAt(e);if(!At(T))break;e++}if(e<l&&r.charCodeAt(e)!==10&&b)for(b="",e=h,i=u;e<l;){const T=r.charCodeAt(e);if(!At(T))break;e++}if(e<l&&r.charCodeAt(e)!==10)return!1;const _=zd(r.slice(1,m));return _?(n||(typeof s.env.references>"u"&&(s.env.references={}),typeof s.env.references[_]>"u"&&(s.env.references[_]={title:b,href:c}),s.line=i),!0):!1}const nq=["address","article","aside","base","basefont","blockquote","body","caption","center","col","colgroup","dd","details","dialog","dir","div","dl","dt","fieldset","figcaption","figure","footer","form","frame","frameset","h1","h2","h3","h4","h5","h6","head","header","hr","html","iframe","legend","li","link","main","menu","menuitem","nav","noframes","ol","optgroup","option","p","param","search","section","summary","table","tbody","td","tfoot","th","thead","title","tr","track","ul"],eq="[a-zA-Z_:][a-zA-Z0-9:._-]*",lq="[^\"'=<>`\\x00-\\x20]+",iq="'[^']*'",pq='"[^"]*"',rq="(?:"+lq+"|"+iq+"|"+pq+")",mq="(?:\\s+"+eq+"(?:\\s*=\\s*"+rq+")?)",Y_="<[A-Za-z][A-Za-z0-9\\-]*"+mq+"*\\s*\\/?>",X_="<\\/[A-Za-z][A-Za-z0-9\\-]*\\s*>",oq="<!---?>|<!--(?:[^-]|-[^-]|--[^>])*-->",cq="<[?][\\s\\S]*?[?]>",hq="<![A-Za-z][^>]*>",uq="<!\\[CDATA\\[[\\s\\S]*?\\]\\]>",dq=new RegExp("^(?:"+Y_+"|"+X_+"|"+oq+"|"+cq+"|"+hq+"|"+uq+")"),gq=new RegExp("^(?:"+Y_+"|"+X_+")"),Or=[[/^<(script|pre|style|textarea)(?=(\s|>|$))/i,/<\/(script|pre|style|textarea)>/i,!0],[/^<!--/,/-->/,!0],[/^<\?/,/\?>/,!0],[/^<![A-Z]/,/>/,!0],[/^<!\[CDATA\[/,/\]\]>/,!0],[new RegExp("^</?("+nq.join("|")+")(?=(\\s|/?>|$))","i"),/^$/,!0],[new RegExp(gq.source+"\\s*$"),/^$/,!1]];function fq(s,a,t,n){let e=s.bMarks[a]+s.tShift[a],l=s.eMarks[a];if(s.sCount[a]-s.blkIndent>=4||!s.md.options.html||s.src.charCodeAt(e)!==60)return!1;let i=s.src.slice(e,l),p=0;for(;p<Or.length&&!Or[p][0].test(i);p++);if(p===Or.length)return!1;if(n)return Or[p][2];let r=a+1;if(!Or[p][1].test(i)){for(;r<t&&!(s.sCount[r]<s.blkIndent);r++)if(e=s.bMarks[r]+s.tShift[r],l=s.eMarks[r],i=s.src.slice(e,l),Or[p][1].test(i)){i.length!==0&&r++;break}}s.line=r;const m=s.push("html_block","",0);return m.map=[a,r],m.content=s.getLines(a,r,s.blkIndent,!0),!0}function yq(s,a,t,n){let e=s.bMarks[a]+s.tShift[a],l=s.eMarks[a];if(s.sCount[a]-s.blkIndent>=4)return!1;let i=s.src.charCodeAt(e);if(i!==35||e>=l)return!1;let p=1;for(i=s.src.charCodeAt(++e);i===35&&e<l&&p<=6;)p++,i=s.src.charCodeAt(++e);if(p>6||e<l&&!At(i))return!1;if(n)return!0;l=s.skipSpacesBack(l,e);const r=s.skipCharsBack(l,35,e);r>e&&At(s.src.charCodeAt(r-1))&&(l=r),s.line=a+1;const m=s.push("heading_open","h"+String(p),1);m.markup="########".slice(0,p),m.map=[a,s.line];const o=s.push("inline","",0);o.content=s.src.slice(e,l).trim(),o.map=[a,s.line],o.children=[];const c=s.push("heading_close","h"+String(p),-1);return c.markup="########".slice(0,p),!0}function vq(s,a,t){const n=s.md.block.ruler.getRules("paragraph");if(s.sCount[a]-s.blkIndent>=4)return!1;const e=s.parentType;s.parentType="paragraph";let l=0,i,p=a+1;for(;p<t&&!s.isEmpty(p);p++){if(s.sCount[p]-s.blkIndent>3)continue;if(s.sCount[p]>=s.blkIndent){let u=s.bMarks[p]+s.tShift[p];const y=s.eMarks[p];if(u<y&&(i=s.src.charCodeAt(u),(i===45||i===61)&&(u=s.skipChars(u,i),u=s.skipSpaces(u),u>=y))){l=i===61?1:2;break}}if(s.sCount[p]<0)continue;let h=!1;for(let u=0,y=n.length;u<y;u++)if(n[u](s,p,t,!0)){h=!0;break}if(h)break}if(!l)return!1;const r=s.getLines(a,p,s.blkIndent,!1).trim();s.line=p+1;const m=s.push("heading_open","h"+String(l),1);m.markup=String.fromCharCode(i),m.map=[a,s.line];const o=s.push("inline","",0);o.content=r,o.map=[a,s.line-1],o.children=[];const c=s.push("heading_close","h"+String(l),-1);return c.markup=String.fromCharCode(i),s.parentType=e,!0}function Eq(s,a,t){const n=s.md.block.ruler.getRules("paragraph"),e=s.parentType;let l=a+1;for(s.parentType="paragraph";l<t&&!s.isEmpty(l);l++){if(s.sCount[l]-s.blkIndent>3||s.sCount[l]<0)continue;let m=!1;for(let o=0,c=n.length;o<c;o++)if(n[o](s,l,t,!0)){m=!0;break}if(m)break}const i=s.getLines(a,l,s.blkIndent,!1).trim();s.line=l;const p=s.push("paragraph_open","p",1);p.map=[a,s.line];const r=s.push("inline","",0);return r.content=i,r.map=[a,s.line],r.children=[],s.push("paragraph_close","p",-1),s.parentType=e,!0}const r0=[["table",XH,["paragraph","reference"]],["code",KH],["fence",ZH,["paragraph","reference","blockquote","list"]],["blockquote",QH,["paragraph","reference","blockquote","list"]],["hr",JH,["paragraph","reference","blockquote","list"]],["list",aq,["paragraph","reference","blockquote"]],["reference",tq],["html_block",fq,["paragraph","reference","blockquote"]],["heading",yq,["paragraph","reference","blockquote"]],["lheading",vq],["paragraph",Eq]];function Dd(){this.ruler=new ye;for(let s=0;s<r0.length;s++)this.ruler.push(r0[s][0],r0[s][1],{alt:(r0[s][2]||[]).slice()})}Dd.prototype.tokenize=function(s,a,t){const n=this.ruler.getRules(""),e=n.length,l=s.md.options.maxNesting;let i=a,p=!1;for(;i<t&&(s.line=i=s.skipEmptyLines(i),!(i>=t||s.sCount[i]<s.blkIndent));){if(s.level>=l){s.line=t;break}const r=s.line;let m=!1;for(let o=0;o<e;o++)if(m=n[o](s,i,t,!1),m){if(r>=s.line)throw new Error("block rule didn't increment state.line");break}if(!m)throw new Error("none of the block rules matched");s.tight=!p,s.isEmpty(s.line-1)&&(p=!0),i=s.line,i<t&&s.isEmpty(i)&&(p=!0,i++,s.line=i)}};Dd.prototype.parse=function(s,a,t,n){if(!s)return;const e=new this.State(s,a,t,n);this.tokenize(e,e.line,e.lineMax)};Dd.prototype.State=Xl;function dh(s,a,t,n){this.src=s,this.env=t,this.md=a,this.tokens=n,this.tokens_meta=Array(n.length),this.pos=0,this.posMax=this.src.length,this.level=0,this.pending="",this.pendingLevel=0,this.cache={},this.delimiters=[],this._prev_delimiters=[],this.backticks={},this.backticksScanned=!1,this.linkLevel=0}dh.prototype.pushPending=function(){const s=new vl("text","",0);return s.content=this.pending,s.level=this.pendingLevel,this.tokens.push(s),this.pending="",s};dh.prototype.push=function(s,a,t){this.pending&&this.pushPending();const n=new vl(s,a,t);let e=null;return t<0&&(this.level--,this.delimiters=this._prev_delimiters.pop()),n.level=this.level,t>0&&(this.level++,this._prev_delimiters.push(this.delimiters),this.delimiters=[],e={delimiters:this.delimiters}),this.pendingLevel=this.level,this.tokens.push(n),this.tokens_meta.push(e),n};dh.prototype.scanDelims=function(s,a){const t=this.posMax,n=this.src.charCodeAt(s),e=s>0?this.src.charCodeAt(s-1):32;let l=s;for(;l<t&&this.src.charCodeAt(l)===n;)l++;const i=l-s,p=l<t?this.src.charCodeAt(l):32,r=_c(e)||Ac(String.fromCharCode(e)),m=_c(p)||Ac(String.fromCharCode(p)),o=bc(e),c=bc(p),h=!c&&(!m||o||r),u=!o&&(!r||c||m);return{can_open:h&&(a||!u||r),can_close:u&&(a||!h||m),length:i}};dh.prototype.Token=vl;function xq(s){switch(s){case 10:case 33:case 35:case 36:case 37:case 38:case 42:case 43:case 45:case 58:case 60:case 61:case 62:case 64:case 91:case 92:case 93:case 94:case 95:case 96:case 123:case 125:case 126:return!0;default:return!1}}function wq(s,a){let t=s.pos;for(;t<s.posMax&&!xq(s.src.charCodeAt(t));)t++;return t===s.pos?!1:(a||(s.pending+=s.src.slice(s.pos,t)),s.pos=t,!0)}const bq=/(?:^|[^a-z0-9.+-])([a-z][a-z0-9.+-]*)$/i;function Aq(s,a){if(!s.md.options.linkify||s.linkLevel>0)return!1;const t=s.pos,n=s.posMax;if(t+3>n||s.src.charCodeAt(t)!==58||s.src.charCodeAt(t+1)!==47||s.src.charCodeAt(t+2)!==47)return!1;const e=s.pending.match(bq);if(!e)return!1;const l=e[1],i=s.md.linkify.matchAtStart(s.src.slice(t-l.length));if(!i)return!1;let p=i.url;if(p.length<=l.length)return!1;let r=p.length;for(;r>0&&p.charCodeAt(r-1)===42;)r--;r!==p.length&&(p=p.slice(0,r));const m=s.md.normalizeLink(p);if(!s.md.validateLink(m))return!1;if(!a){s.pending=s.pending.slice(0,-l.length);const o=s.push("link_open","a",1);o.attrs=[["href",m]],o.markup="linkify",o.info="auto";const c=s.push("text","",0);c.content=s.md.normalizeLinkText(p);const h=s.push("link_close","a",-1);h.markup="linkify",h.info="auto"}return s.pos+=p.length-l.length,!0}function _q(s,a){let t=s.pos;if(s.src.charCodeAt(t)!==10)return!1;const n=s.pending.length-1,e=s.posMax;if(!a)if(n>=0&&s.pending.charCodeAt(n)===32)if(n>=1&&s.pending.charCodeAt(n-1)===32){let l=n-1;for(;l>=1&&s.pending.charCodeAt(l-1)===32;)l--;s.pending=s.pending.slice(0,l),s.push("hardbreak","br",0)}else s.pending=s.pending.slice(0,-1),s.push("softbreak","br",0);else s.push("softbreak","br",0);for(t++;t<e&&At(s.src.charCodeAt(t));)t++;return s.pos=t,!0}const u8=[];for(let s=0;s<256;s++)u8.push(0);"\\!\"#$%&'()*+,./:;<=>?@[]^_`{|}~-".split("").forEach(function(s){u8[s.charCodeAt(0)]=1});function kq(s,a){let t=s.pos;const n=s.posMax;if(s.src.charCodeAt(t)!==92||(t++,t>=n))return!1;let e=s.src.charCodeAt(t);if(e===10){for(a||s.push("hardbreak","br",0),t++;t<n&&(e=s.src.charCodeAt(t),!!At(e));)t++;return s.pos=t,!0}let l=s.src[t];if(e>=55296&&e<=56319&&t+1<n){const p=s.src.charCodeAt(t+1);p>=56320&&p<=57343&&(l+=s.src[t+1],t++)}const i="\\"+l;if(!a){const p=s.push("text_special","",0);e<256&&u8[e]!==0?p.content=l:p.content=i,p.markup=i,p.info="escape"}return s.pos=t+1,!0}function Bq(s,a){let t=s.pos;if(s.src.charCodeAt(t)!==96)return!1;const e=t;t++;const l=s.posMax;for(;t<l&&s.src.charCodeAt(t)===96;)t++;const i=s.src.slice(e,t),p=i.length;if(s.backticksScanned&&(s.backticks[p]||0)<=e)return a||(s.pending+=i),s.pos+=p,!0;let r=t,m;for(;(m=s.src.indexOf("`",r))!==-1;){for(r=m+1;r<l&&s.src.charCodeAt(r)===96;)r++;const o=r-m;if(o===p){if(!a){const c=s.push("code_inline","code",0);c.markup=i,c.content=s.src.slice(t,m).replace(/\n/g," ").replace(/^ (.+) $/,"$1")}return s.pos=r,!0}s.backticks[o]=m}return s.backticksScanned=!0,a||(s.pending+=i),s.pos+=p,!0}function Mq(s,a){const t=s.pos,n=s.src.charCodeAt(t);if(a||n!==126)return!1;const e=s.scanDelims(s.pos,!0);let l=e.length;const i=String.fromCharCode(n);if(l<2)return!1;let p;l%2&&(p=s.push("text","",0),p.content=i,l--);for(let r=0;r<l;r+=2)p=s.push("text","",0),p.content=i+i,s.delimiters.push({marker:n,length:0,token:s.tokens.length-1,end:-1,open:e.can_open,close:e.can_close});return s.pos+=e.length,!0}function X6(s,a){let t;const n=[],e=a.length;for(let l=0;l<e;l++){const i=a[l];if(i.marker!==126||i.end===-1)continue;const p=a[i.end];t=s.tokens[i.token],t.type="s_open",t.tag="s",t.nesting=1,t.markup="~~",t.content="",t=s.tokens[p.token],t.type="s_close",t.tag="s",t.nesting=-1,t.markup="~~",t.content="",s.tokens[p.token-1].type==="text"&&s.tokens[p.token-1].content==="~"&&n.push(p.token-1)}for(;n.length;){const l=n.pop();let i=l+1;for(;i<s.tokens.length&&s.tokens[i].type==="s_close";)i++;i--,l!==i&&(t=s.tokens[i],s.tokens[i]=s.tokens[l],s.tokens[l]=t)}}function Tq(s){const a=s.tokens_meta,t=s.tokens_meta.length;X6(s,s.delimiters);for(let n=0;n<t;n++)a[n]&&a[n].delimiters&&X6(s,a[n].delimiters)}const K_={tokenize:Mq,postProcess:Tq};function Cq(s,a){const t=s.pos,n=s.src.charCodeAt(t);if(a||n!==95&&n!==42)return!1;const e=s.scanDelims(s.pos,n===42);for(let l=0;l<e.length;l++){const i=s.push("text","",0);i.content=String.fromCharCode(n),s.delimiters.push({marker:n,length:e.length,token:s.tokens.length-1,end:-1,open:e.can_open,close:e.can_close})}return s.pos+=e.length,!0}function K6(s,a){const t=a.length;for(let n=t-1;n>=0;n--){const e=a[n];if(e.marker!==95&&e.marker!==42||e.end===-1)continue;const l=a[e.end],i=n>0&&a[n-1].end===e.end+1&&a[n-1].marker===e.marker&&a[n-1].token===e.token-1&&a[e.end+1].token===l.token+1,p=String.fromCharCode(e.marker),r=s.tokens[e.token];r.type=i?"strong_open":"em_open",r.tag=i?"strong":"em",r.nesting=1,r.markup=i?p+p:p,r.content="";const m=s.tokens[l.token];m.type=i?"strong_close":"em_close",m.tag=i?"strong":"em",m.nesting=-1,m.markup=i?p+p:p,m.content="",i&&(s.tokens[a[n-1].token].content="",s.tokens[a[e.end+1].token].content="",n--)}}function zq(s){const a=s.tokens_meta,t=s.tokens_meta.length;K6(s,s.delimiters);for(let n=0;n<t;n++)a[n]&&a[n].delimiters&&K6(s,a[n].delimiters)}const Z_={tokenize:Cq,postProcess:zq};function Dq(s,a){let t,n,e,l,i="",p="",r=s.pos,m=!0;if(s.src.charCodeAt(s.pos)!==91)return!1;const o=s.pos,c=s.posMax,h=s.pos+1,u=s.md.helpers.parseLinkLabel(s,s.pos,!0);if(u<0)return!1;let y=u+1;if(y<c&&s.src.charCodeAt(y)===40){for(m=!1,y++;y<c&&(t=s.src.charCodeAt(y),!(!At(t)&&t!==10));y++);if(y>=c)return!1;if(r=y,e=s.md.helpers.parseLinkDestination(s.src,y,s.posMax),e.ok){for(i=s.md.normalizeLink(e.str),s.md.validateLink(i)?y=e.pos:i="",r=y;y<c&&(t=s.src.charCodeAt(y),!(!At(t)&&t!==10));y++);if(e=s.md.helpers.parseLinkTitle(s.src,y,s.posMax),y<c&&r!==y&&e.ok)for(p=e.str,y=e.pos;y<c&&(t=s.src.charCodeAt(y),!(!At(t)&&t!==10));y++);}(y>=c||s.src.charCodeAt(y)!==41)&&(m=!0),y++}if(m){if(typeof s.env.references>"u")return!1;if(y<c&&s.src.charCodeAt(y)===91?(r=y+1,y=s.md.helpers.parseLinkLabel(s,y),y>=0?n=s.src.slice(r,y++):y=u+1):y=u+1,n||(n=s.src.slice(h,u)),l=s.env.references[zd(n)],!l)return s.pos=o,!1;i=l.href,p=l.title}if(!a){s.pos=h,s.posMax=u;const x=s.push("link_open","a",1),b=[["href",i]];x.attrs=b,p&&b.push(["title",p]),s.linkLevel++,s.md.inline.tokenize(s),s.linkLevel--,s.push("link_close","a",-1)}return s.pos=y,s.posMax=c,!0}function Lq(s,a){let t,n,e,l,i,p,r,m,o="";const c=s.pos,h=s.posMax;if(s.src.charCodeAt(s.pos)!==33||s.src.charCodeAt(s.pos+1)!==91)return!1;const u=s.pos+2,y=s.md.helpers.parseLinkLabel(s,s.pos+1,!1);if(y<0)return!1;if(l=y+1,l<h&&s.src.charCodeAt(l)===40){for(l++;l<h&&(t=s.src.charCodeAt(l),!(!At(t)&&t!==10));l++);if(l>=h)return!1;for(m=l,p=s.md.helpers.parseLinkDestination(s.src,l,s.posMax),p.ok&&(o=s.md.normalizeLink(p.str),s.md.validateLink(o)?l=p.pos:o=""),m=l;l<h&&(t=s.src.charCodeAt(l),!(!At(t)&&t!==10));l++);if(p=s.md.helpers.parseLinkTitle(s.src,l,s.posMax),l<h&&m!==l&&p.ok)for(r=p.str,l=p.pos;l<h&&(t=s.src.charCodeAt(l),!(!At(t)&&t!==10));l++);else r="";if(l>=h||s.src.charCodeAt(l)!==41)return s.pos=c,!1;l++}else{if(typeof s.env.references>"u")return!1;if(l<h&&s.src.charCodeAt(l)===91?(m=l+1,l=s.md.helpers.parseLinkLabel(s,l),l>=0?e=s.src.slice(m,l++):l=y+1):l=y+1,e||(e=s.src.slice(u,y)),i=s.env.references[zd(e)],!i)return s.pos=c,!1;o=i.href,r=i.title}if(!a){n=s.src.slice(u,y);const x=[];s.md.inline.parse(n,s.md,s.env,x);const b=s.push("image","img",0),_=[["src",o],["alt",""]];b.attrs=_,b.children=x,b.content=n,r&&_.push(["title",r])}return s.pos=l,s.posMax=h,!0}const Sq=/^([a-zA-Z0-9.!#$%&'*+/=?^_`{|}~-]+@[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?(?:\.[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?)*)$/,Fq=/^([a-zA-Z][a-zA-Z0-9+.-]{1,31}):([^<>\x00-\x20]*)$/;function Oq(s,a){let t=s.pos;if(s.src.charCodeAt(t)!==60)return!1;const n=s.pos,e=s.posMax;for(;;){if(++t>=e)return!1;const i=s.src.charCodeAt(t);if(i===60)return!1;if(i===62)break}const l=s.src.slice(n+1,t);if(Fq.test(l)){const i=s.md.normalizeLink(l);if(!s.md.validateLink(i))return!1;if(!a){const p=s.push("link_open","a",1);p.attrs=[["href",i]],p.markup="autolink",p.info="auto";const r=s.push("text","",0);r.content=s.md.normalizeLinkText(l);const m=s.push("link_close","a",-1);m.markup="autolink",m.info="auto"}return s.pos+=l.length+2,!0}if(Sq.test(l)){const i=s.md.normalizeLink("mailto:"+l);if(!s.md.validateLink(i))return!1;if(!a){const p=s.push("link_open","a",1);p.attrs=[["href",i]],p.markup="autolink",p.info="auto";const r=s.push("text","",0);r.content=s.md.normalizeLinkText(l);const m=s.push("link_close","a",-1);m.markup="autolink",m.info="auto"}return s.pos+=l.length+2,!0}return!1}function Iq(s){return/^<a[>\s]/i.test(s)}function Nq(s){return/^<\/a\s*>/i.test(s)}function Pq(s){const a=s|32;return a>=97&&a<=122}function Rq(s,a){if(!s.md.options.html)return!1;const t=s.posMax,n=s.pos;if(s.src.charCodeAt(n)!==60||n+2>=t)return!1;const e=s.src.charCodeAt(n+1);if(e!==33&&e!==63&&e!==47&&!Pq(e))return!1;const l=s.src.slice(n).match(dq);if(!l)return!1;if(!a){const i=s.push("html_inline","",0);i.content=l[0],Iq(i.content)&&s.linkLevel++,Nq(i.content)&&s.linkLevel--}return s.pos+=l[0].length,!0}const $q=/^&#((?:x[a-f0-9]{1,6}|[0-9]{1,7}));/i,Vq=/^&([a-z][a-z0-9]{1,31});/i;function Hq(s,a){const t=s.pos,n=s.posMax;if(s.src.charCodeAt(t)!==38||t+1>=n)return!1;if(s.src.charCodeAt(t+1)===35){const l=s.src.slice(t).match($q);if(l){if(!a){const i=l[1][0].toLowerCase()==="x"?parseInt(l[1].slice(1),16):parseInt(l[1],10),p=s.push("text_special","",0);p.content=c8(i)?wu(i):wu(65533),p.markup=l[0],p.info="entity"}return s.pos+=l[0].length,!0}}else{const l=s.src.slice(t).match(Vq);if(l){const i=q_(l[0]);if(i!==l[0]){if(!a){const p=s.push("text_special","",0);p.content=i,p.markup=l[0],p.info="entity"}return s.pos+=l[0].length,!0}}}return!1}function Z6(s){const a={},t=s.length;if(!t)return;let n=0,e=-2;const l=[];for(let i=0;i<t;i++){const p=s[i];if(l.push(0),(s[n].marker!==p.marker||e!==p.token-1)&&(n=i),e=p.token,p.length=p.length||0,!p.close)continue;a.hasOwnProperty(p.marker)||(a[p.marker]=[-1,-1,-1,-1,-1,-1]);const r=a[p.marker][(p.open?3:0)+p.length%3];let m=n-l[n]-1,o=m;for(;m>r;m-=l[m]+1){const c=s[m];if(c.marker===p.marker&&c.open&&c.end<0){let h=!1;if((c.close||p.open)&&(c.length+p.length)%3===0&&(c.length%3!==0||p.length%3!==0)&&(h=!0),!h){const u=m>0&&!s[m-1].open?l[m-1]+1:0;l[i]=i-m+u,l[m]=u,p.open=!1,c.end=i,c.close=!1,o=-1,e=-2;break}}}o!==-1&&(a[p.marker][(p.open?3:0)+(p.length||0)%3]=o)}}function qq(s){const a=s.tokens_meta,t=s.tokens_meta.length;Z6(s.delimiters);for(let n=0;n<t;n++)a[n]&&a[n].delimiters&&Z6(a[n].delimiters)}function Uq(s){let a,t,n=0;const e=s.tokens,l=s.tokens.length;for(a=t=0;a<l;a++)e[a].nesting<0&&n--,e[a].level=n,e[a].nesting>0&&n++,e[a].type==="text"&&a+1<l&&e[a+1].type==="text"?e[a+1].content=e[a].content+e[a+1].content:(a!==t&&(e[t]=e[a]),t++);a!==t&&(e.length=t)}const F1=[["text",wq],["linkify",Aq],["newline",_q],["escape",kq],["backticks",Bq],["strikethrough",K_.tokenize],["emphasis",Z_.tokenize],["link",Dq],["image",Lq],["autolink",Oq],["html_inline",Rq],["entity",Hq]],O1=[["balance_pairs",qq],["strikethrough",K_.postProcess],["emphasis",Z_.postProcess],["fragments_join",Uq]];function gh(){this.ruler=new ye;for(let s=0;s<F1.length;s++)this.ruler.push(F1[s][0],F1[s][1]);this.ruler2=new ye;for(let s=0;s<O1.length;s++)this.ruler2.push(O1[s][0],O1[s][1])}gh.prototype.skipToken=function(s){const a=s.pos,t=this.ruler.getRules(""),n=t.length,e=s.md.options.maxNesting,l=s.cache;if(typeof l[a]<"u"){s.pos=l[a];return}let i=!1;if(s.level<e){for(let p=0;p<n;p++)if(s.level++,i=t[p](s,!0),s.level--,i){if(a>=s.pos)throw new Error("inline rule didn't increment state.pos");break}}else s.pos=s.posMax;i||s.pos++,l[a]=s.pos};gh.prototype.tokenize=function(s){const a=this.ruler.getRules(""),t=a.length,n=s.posMax,e=s.md.options.maxNesting;for(;s.pos<n;){const l=s.pos;let i=!1;if(s.level<e){for(let p=0;p<t;p++)if(i=a[p](s,!1),i){if(l>=s.pos)throw new Error("inline rule didn't increment state.pos");break}}if(i){if(s.pos>=n)break;continue}s.pending+=s.src[s.pos++]}s.pending&&s.pushPending()};gh.prototype.parse=function(s,a,t,n){const e=new this.State(s,a,t,n);this.tokenize(e);const l=this.ruler2.getRules(""),i=l.length;for(let p=0;p<i;p++)l[p](e)};gh.prototype.State=dh;function jq(s){const a={};s=s||{},a.src_Any=P_.source,a.src_Cc=R_.source,a.src_Z=V_.source,a.src_P=m8.source,a.src_ZPCc=[a.src_Z,a.src_P,a.src_Cc].join("|"),a.src_ZCc=[a.src_Z,a.src_Cc].join("|");const t="[><｜]";return 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a.match=="string")?a.id:null,this.detail=a.detail??"",this.process_alias=a.process_alias??"",this.process_param=a.process_param??null,this.process_return=a.process_return??null,this.process=a.process,this.is_disable=!1,this.register_from="内置"}constructor_user(a){this.id=a.id,this.name=a.name,this.match=/^\//.test(a.match)?RegExp(a.match):a.match,this.default=null,this.detail="",this.process_alias=a.process_alias,this.process_param=null,this.process_return=null,this.process=()=>{},this.is_disable=!1,this.register_from="用户"}destructor(){const a=Ot.getInstance().list_abConvert.indexOf(this);a>-1&&Ot.getInstance().list_abConvert.splice(a,1)}}function AU(s,a,t){s.trimEnd().endsWith("|")||(s=s+"|"),s.trimStart().startsWith("|")||(s="|"+s),a=="mdit"?s="|:::_140lne"+s.trimStart():a=="list"||It.reg_list_noprefix.test(t.trimStart())?s="|list_140lne"+s:a=="heading"||It.reg_heading_noprefix.test(t.trimStart())?s="|heading_140lne"+s:a=="code"||It.reg_code_noprefix.test(t.trimStart())?s="|code_140lne"+s:a=="quote"||It.reg_quote_noprefix.test(t.trimStart())?s="|quote_140lne"+s:(a=="table"||It.reg_table_noprefix.test(t.trimStart()))&&(s="|table_140lne"+s);for(const n of mk())s=s.replace(n.regex,n.replacement);for(const n of _U)s=s.replace(n.regex,(e,...l)=>n.replacement.replace(/\$(\d+)/g,(i,p)=>l[p-1]??""));for(const n of OU)s=s.replace(n.regex,n.replacement);return s}const _U=[{regex:/\|(note|warning|caution|attention|error|info|danger|tip|hint|example|abstract|summary|tldr|quote|cite|todo|success|check|done|important|question|help|faq|failure|fail|missing|bug)([+-]?)(\s.*)?\|/,replacement:"|add([!$1]$2$3)|addQuote|"},{regex:/\|(callout|alert) ([^+-\s]+)([+-]?)\s?(.*)\|/,replacement:"|add([!$2]$3 $4)|addQuote|"}],kU=[{regex:/\|:::_140lne\|(2?tabs?|标签页?)\|/,replacement:"|mditTabs|"},{regex:"|:::_140lne|demo|",replacement:"|mditDemo|"},{regex:"|:::_140lne|abDemo|",replacement:"|mditABDemo|"},{regex:/\|:::_140lne\|(2?col|分栏)\|/,replacement:"|mditCol|"},{regex:/\|:::_140lne\|(2?card|卡片)\|/,replacement:"|mditCard|"},{regex:/\|:::_140lne\|(2?chat|聊天)\|/,replacement:"|mditChat|code(chat)|"}],BU=[{regex:"|title2list|",replacement:"|title2listdata|listdata2strict|listdata2list|"},{regex:/\|heading_140lne\|2?(timeline|时间线)\|/,replacement:"|title2timeline|"},{regex:/\|heading_140lne\|2?(tabs?|标签页?)\||\|title2tabs?\|/,replacement:"|title2c2listdata|c2listdata2tab|"},{regex:/\|heading_140lne\|2?(col|分栏)\||\|title2col\|/,replacement:"|title2c2listdata|c2listdata2items|addClass(ab-col)|"},{regex:/\|heading_140lne\|2?(card|卡片)\||\|title2card\|/,replacement:"|title2c2listdata|c2listdata2items|addClass(ab-card)|addClass(ab-lay-vfall)|"},{regex:/\|heading_140lne\|2?(nodes?|节点)\||\|(title2node|title2abMindmap)\|/,replacement:"|title2listdata|listdata2strict|listdata2nodes|"},{regex:/\|heading_140lne\|2?(mermaid|flow|流程图)\|/,replacement:"|title2list|list2mermaid|"},{regex:/\|heading_140lne\|2?(mehrmaid|mdmermaid)\|/,replacement:"|title2list|list2mehrmaidText|code(mehrmaid)|"},{regex:/\|heading_140lne\|2?(puml)?(plantuml|mindmap|脑图|思维导图)\|/,replacement:"|title2list|list2pumlMindmap|"},{regex:/\|heading_140lne\|2?(markmap|mdMindmap|md脑图|md思维导图)\|/,replacement:"|title2list|list2markmap|"},{regex:/\|heading_140lne\|2?(wbs|(工作)?分解(图|结构))\|/,replacement:"|title2list|list2pumlWBS|"},{regex:/\|heading_140lne\|2?(table|multiWayTable|multiCrossTable|表格?|多叉表格?|跨行表格?)\|/,replacement:"|title2list|list2table|"},{regex:/\|heading_140lne\|2?(lt|listTable|treeTable|listGrid|treeGrid|列表格|树形表格?)\|/,replacement:"|title2list|list2lt|"},{regex:/\|heading_140lne\|2?(list|列表)\|/,replacement:"|title2list|list2lt|addClass(ab-listtable-likelist)|"},{regex:/\|heading_140lne\|2?(dir|dirTree|目录树?|目录结构)\|/,replacement:"|title2list|list2dt|"},{regex:/\|heading_140lne\|(fakeList|仿列表)\|/,replacement:"|title2list|list2table|addClass(ab-table-fc)|addClass(ab-table-likelist)|"}],MU=[{regex:"|listXinline|",replacement:"|list2listdata|listdata2strict|listdata2list|"},{regex:"|list2task|",replacement:"|list2listdata|listdata2task|listdata2list|"},{regex:"|task2",replacement:"|list2listdata|listdata2task|listdata2list|list2"},{regex:/\|list_140lne\|2?(timeline|时间线)\|/,replacement:"|list2timeline|"},{regex:/\|list_140lne\|2?(tabs?|标签页?)\||\|list2tabs?\|/,replacement:"|list2c2listdata|c2listdata2tab|"},{regex:/\|list_140lne\|2?(col|分栏)\||\|list2col\|/,replacement:"|list2c2listdata|c2listdata2items|addClass(ab-col)|"},{regex:/\|list_140lne\|2?(card|卡片)\||\|list2card\|/,replacement:"|list2c2listdata|c2listdata2items|addClass(ab-card)|addClass(ab-lay-vfall)|"},{regex:/\|list_140lne\|2?(nodes?|节点)\||\|(list2node|list2abMindmap)\|/,replacement:"|list2listdata|listdata2strict|listdata2nodes|"},{regex:/\|list_140lne\|2?(mermaid|flow|流程图)\|/,replacement:"|list2mermaid|"},{regex:/\|list_140lne\|2?(mehrmaid|mdmermaid)\|/,replacement:"|list2mehrmaidText|code(mehrmaid)|"},{regex:/\|list_140lne\|2?(puml)?(plantuml|mindmap|脑图|思维导图)\|/,replacement:"|list2pumlMindmap|"},{regex:/\|list_140lne\|2?(markmap|mdMindmap|md脑图|md思维导图)\|/,replacement:"|list2markmap|"},{regex:/\|list_140lne\|2?(wbs|(工作)?分解(图|结构))\|/,replacement:"|list2pumlWBS|"},{regex:/\|list_140lne\|2?(table|multiWayTable|multiCrossTable|表格?|多叉表格?|跨行表格?)\|/,replacement:"|list2table|"},{regex:/\|list_140lne\|2?(lt|listTable|treeTable|listGrid|treeGrid|列表格|树形表格?)\|/,replacement:"|list2lt|"},{regex:/\|list_140lne\|2?(list|列表)\|/,replacement:"|list2lt|addClass(ab-listtable-likelist)|"},{regex:/\|list_140lne\|2?(dir|dirTree|目录树?|目录结构)\|/,replacement:"|list2dt|"},{regex:/\|list_140lne\|(fakeList|仿列表)\|/,replacement:"|list2table|addClass(ab-table-fc)|addClass(ab-table-likelist)|"}],TU=[{regex:"|code_140lne|X|",replacement:"|xCode|"},{regex:"|code_140lne|x|",replacement:"|xCode|"},{regex:"|code2list|",replacement:"|xCode|region2indent|addList|"},{regex:"|echarts|",replacement:"|xCode|code(echarts)|"},{regex:"|list2code|",replacement:"|xList|code(js)|"}],CU=[],zU=[],DU=[{regex:"|echarts|",replacement:"|xCode|code(echarts)|"},{regex:"|黑幕|",replacement:"|addClass(ab-deco-heimu)|"},{regex:"|折叠|",replacement:"|fold|"},{regex:"|滚动|",replacement:"|scroll|"},{regex:"|超出折叠|",replacement:"|overfold|"},{regex:"|转置|",replacement:"|transpose|"},{regex:"|T|",replacement:"|transpose|"},{regex:"|红字|",replacement:"|addClass(ab-custom-text-red)|"},{regex:"|橙字|",replacement:"|addClass(ab-custom-text-orange)|"},{regex:"|黄字|",replacement:"|addClass(ab-custom-text-yellow)|"},{regex:"|绿字|",replacement:"|addClass(ab-custom-text-green)|"},{regex:"|青字|",replacement:"|addClass(ab-custom-text-cyan)|"},{regex:"|蓝字|",replacement:"|addClass(ab-custom-text-blue)|"},{regex:"|紫字|",replacement:"|addClass(ab-custom-text-purple)|"},{regex:"|白字|",replacement:"|addClass(ab-custom-text-white)|"},{regex:"|黑字|",replacement:"|addClass(ab-custom-text-black)|"},{regex:"|红底|",replacement:"|addClass(ab-custom-bg-red)|"},{regex:"|橙底|",replacement:"|addClass(ab-custom-bg-orange)|"},{regex:"|黄底|",replacement:"|addClass(ab-custom-bg-yellow)|"},{regex:"|绿底|",replacement:"|addClass(ab-custom-bg-green)|"},{regex:"|青底|",replacement:"|addClass(ab-custom-bg-cyan)|"},{regex:"|蓝底|",replacement:"|addClass(ab-custom-bg-blue)|"},{regex:"|紫底|",replacement:"|addClass(ab-custom-bg-purple)|"},{regex:"|白底|",replacement:"|addClass(ab-custom-bg-white)|"},{regex:"|黑底|",replacement:"|addClass(ab-custom-bg-black)|"},{regex:"|靠上|",replacement:"|addClass(ab-custom-dire-top)|"},{regex:"|靠下|",replacement:"|addClass(ab-custom-dire-down)|"},{regex:"|靠左|",replacement:"|addClass(ab-custom-dire-left)|"},{regex:"|靠右|",replacement:"|addClass(ab-custom-dire-right)|"},{regex:"|居中|",replacement:"|addClass(ab-custom-dire-center)|"},{regex:"|水平居中|",replacement:"|addClass(ab-custom-dire-hcenter)|"},{regex:"|垂直居中|",replacement:"|addClass(ab-custom-dire-vcenter)|"},{regex:"|两端对齐|",replacement:"|addClass(ab-custom-dire-justify)|"},{regex:"|大字|",replacement:"|addClass(ab-custom-font-large)|"},{regex:"|超大字|",replacement:"|addClass(ab-custom-font-largex)|"},{regex:"|超超大字|",replacement:"|addClass(ab-custom-font-largexx)|"},{regex:"|小字|",replacement:"|addClass(ab-custom-font-small)|"},{regex:"|超小字|",replacement:"|addClass(ab-custom-font-smallx)|"},{regex:"|超超小字|",replacement:"|addClass(ab-custom-font-smallxx)|"},{regex:"|加粗|",replacement:"|addClass(ab-custom-font-bold)|"}],LU=[],SU=[],FU=[...kU,...BU,...MU,...TU,...CU,...zU,...DU];function*mk(){yield*LU,yield*SU,yield*FU}const OU=[{regex:"|:::_140lne",replacement:""},{regex:"|heading_140lne",replacement:""},{regex:"|list_140lne",replacement:""},{regex:"|code_140lne",replacement:""},{regex:"|qutoe_140lne",replacement:""},{regex:"|table_140lne",replacement:""}];class Ot{constructor(){this.list_abConvert=[],this.m_renderMarkdownFn=(a,t)=>{t.classList.add("markdown-rendered"),console.error("Please use renderMarkdownFn redefine render function")},console.log(typeof obsidian>"u"&&typeof app>"u"?"[environment]: markdown-it, without obsidian":"[environment]: obsidian")}static getInstance(){return Ot.m_instance||(Ot.m_instance=new Ot),Ot.m_instance}getConvertOptions(){return this.list_abConvert.filter(a=>a.default).map(a=>({id:a.default,name:a.name}))}redefine_renderMarkdown(a){this.m_renderMarkdownFn=a}static autoABConvert(a,t,n,e="",l){let i=n,p="string",r=Cs.text,o={prev_result:i,prev_type:p,prev_type2:r,prev_processor:void 0};{t=AU(t,e,i);let c=t.split("|");i=this.autoABConvert_runConvert(a,c,o,l),this.autoABConvert_last(a,t,e,o,l)}}static autoABConvert_runConvert(a,t,n,e){for(let l of t)for(let i of Ot.getInstance().list_abConvert){if(typeof i.match=="string"){if(i.match!=l)continue}else if(!i.match.test(l))continue;if(i.process_alias){let p=i.process_alias;(()=>{if(i.process_alias.indexOf("%")<0||typeof i.match=="string")return;const r=l.match(i.match);if(!r)return;const m=r.length;if(m!=1)for(let o=1;o<m;o++)r[o]&&(p=p.replace(RegExp(`%${o}`),r[o]))})(),n.prev_result=this.autoABConvert_runConvert(a,p.split("|"),n,e)}else if(i.process){if(i.process_param!=n.prev_type2)if(i.process_param==Cs.el&&n.prev_type2==Cs.text){const p=document.createElement("div");a.appendChild(p),Ot.getInstance().m_renderMarkdownFn(n.prev_result,p),n.prev_result=a,n.prev_type=typeof n.prev_result,n.prev_type2=Cs.el,n.prev_processor="md"}else if(i.process_param==Cs.text&&(n.prev_type2==Cs.list_stream||n.prev_type2==Cs.c2list_stream))n.prev_result=JSON.stringify(n.prev_result,null,2),n.prev_type=typeof n.prev_result,n.prev_type2=Cs.text,n.prev_processor="stream to text";else if(i.process_param==Cs.text&&n.prev_type2==Cs.json)n.prev_type2=Cs.text,n.prev_processor="json to text";else{console.warn(`处理器输入类型错误, id:${i.id}, virtualParam:${i.process_param}, realParam:${n.prev_type2}`);break}n.prev_result=i.process(a,l,n.prev_result,e),n.prev_type=typeof n.prev_result,n.prev_type2=i.process_return,n.prev_processor=i.process}else console.warn("处理器必须实现process或process_alias方法")}return n}static autoABConvert_last(a,t,n,e,l){if(e.prev_type=="string"&&e.prev_type2==Cs.text){const i=document.createElement("div");a.appendChild(i),Ot.getInstance().m_renderMarkdownFn(e.prev_result,i),e.prev_result=a,e.prev_type="object",e.prev_type2=Cs.el,e.process="md"}else if(e.prev_type=="string"&&e.prev_type2==Cs.json){const i="```json\n"+e.prev_result+"\n```\n",p=document.createElement("div");a.appendChild(p),Ot.getInstance().m_renderMarkdownFn(i,p),e.prev_result=a,e.prev_type="object",e.prev_type2=Cs.el,e.process="show_json"}else if(e.prev_type=="object"&&(e.prev_type2==Cs.list_stream||e.prev_type2==Cs.c2list_stream||e.prev_type2==Cs.json)){const i="```json\n"+JSON.stringify(e.prev_result,null,2)+"\n```\n",p=document.createElement("div");a.appendChild(p),Ot.getInstance().m_renderMarkdownFn(i,p),e.prev_result=a,e.prev_type="object",e.prev_type2=Cs.el,e.process="show_listStream"}else{if(e.prev_type=="object"&&e.prev_type2==Cs.el)return e;console.warn("other type in tail, can not tail processor:",e.prev_type,e.prev_type2,e.prev_result)}return e}}fa.factory({id:"addQuote",name:"增加引用块",match:"addQuote",detail:"在文本的每行前面加上 `> `",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>t.split(`
`).map(n=>"> "+n).join(`
`)});fa.factory({id:"addCode",name:"增加代码块",match:/^(addCode|code)(\((.*)\))?$/,default:"code()",detail:"在文本的前后均加上一行代码块围栏。不加`()`表示用原文本的第一行作为代码类型，括号类型为空表示代码类型为空",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=a.match(/^(addCode|code)(\((.*)\))?$/);return n?(n[2]&&(t=n[3]+`
`+t),"``````"+t+"\n``````"):t}});const IU=fa.factory({id:"xQuote",name:"去除引用块",match:/^(xQuote|Xquote)$/,detail:"在文本的每行前面删除 `> `",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>t.split(`
`).map(n=>n.replace(/^>\s/,"")).join(`
`)}),NU=fa.factory({id:"xCode",name:"去除代码块",match:/^(xCode|Xcode)(\((true|false|)\))?$/,default:"xCode(true)",detail:"参数为是否移除代码类型, Xcode默认为false, Xcode默认为true。记法: code|Xcode 或 code()|Xcode()内容不变",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=a.match(/^(xCode|Xcode)(\((true|false|)\))?$/);if(!n)return t;let e;n[2]==""?e=!1:e=n[3]!="false";let l=t.split(`
`),i="",p=-1,r=-1;for(let m=0;m<l.length;m++)if(i==""){const o=l[m].match(It.reg_code);o&&(i=o[3],p=m)}else if(l[m].indexOf(i)>=0){r=m;break}return p>=0&&r>0&&(e?l[p]=l[p].replace(/^```(.*)$|^~~~(.*)$/,""):l[p]=l[p].replace(/^```|^~~~/,""),l[r]=l[r].replace(/^```|^~~~/,""),t=l.join(`
`)),t}});fa.factory({id:"x",name:"去除代码或引用块",match:/^(x|X)$/,process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n="";for(let e of t.split(`
`))if(It.reg_code.test(e)){n="code";break}else if(It.reg_quote.test(e)){n="quote";break}return n=="code"?NU.process(s,a,t):n=="quote"?IU.process(s,a,t):t}});fa.factory({id:"slice",name:"切片",match:/^slice\((\s*\d+\s*)(,\s*-?\d+\s*)?\)$/,detail:"和js的slice方法是一样的。例如 `[slice(1, -1)]`",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=a.match(/^slice\((\s*\d+\s*)(,\s*-?\d+\s*)?\)$/);if(!n)return t;const e=Number(n[1].trim());if(isNaN(e))return t;if(!n[2])return t.split(`
`).slice(e).join(`
`);const l=Number(n[2].replace(",","").trim());return isNaN(l)?t.split(`
`).slice(e).join(`
`):t.split(`
`).slice(e,l).join(`
`)}});fa.factory({id:"add",name:"增添内容",match:/^add\((.*?)(,\s*-?\d+\s*)?\)$/,detail:"增添. 参数2为行序, 默认0, 行尾-1。会插行增添",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=a.match(/^add\((.*?)(,\s*-?\d+\s*)?\)$/);if(!n||!n[1])return t;const e=n[1].trim();if(!e)return t;let l;n[2]?(l=Number(n[2].replace(",","").trim()),isNaN(l)&&(l=0)):l=0;const i=t.split(`
`);return l>=0&&l<i.length?i[l]=e+`
`+i[l]:l<0&&l*-1<=i.length&&(l=i.length+l,i[l]=e+`
`+i[l]),i.join(`
`)}});fa.factory({id:"listroot",name:"增加列表根",match:/^listroot\((.*)\)$/,default:"listroot(root)",detail:"每行前面加两空格，并在首行插入 `- ` 开头的根列表项",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=a.match(/^listroot\((.*)\)$/);if(!n)return t;const e=n[1].trim();return t=t.split(`
`).map(l=>"  "+l).join(`
`),t="- "+e+`
`+t,t}});fa.factory({id:"addList",name:"缩进转列表",detail:"缩进转列表",default:"缩进转列表。对于空行的处理有两种策略：一是空行表示下一个列表，单换行表示同一列表项。二是忽略空行。暂时仅为策略二",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=t.trimEnd().split(`
`);let e="";for(const l of n){if(l.trim()=="")continue;const i=l.match(/^(\s*)(.*)/);i&&(e+=`
`+i[1]+"- "+i[2])}return e}});fa.factory({id:"xList",name:"列表转缩进",match:/^(xList|Xlist)$/,detail:"列表转缩进。对于多行内容的列表项，默认换行项删除前置空格并使用 `; ` 拼接，拼接符暂不支持自定义",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=t.trimEnd().split(`
`);let e="";for(const l of n){const i=l.match(It.reg_list_noprefix);i?e+=`
`+i[1]+i[4]:e!=""&&(e+="; "+l.trimStart())}return e.slice(1)}});fa.factory({id:"region2indent",name:"代码注释转缩进",detail:"代码块注释转缩进 (识别 `//` 和 `#` 的region注释对)，通常配合code2list使用。默认补充两缩进",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=t.trimEnd().split(`
`);let e="",l=0;const i=/^([ \t]*)(#|\/\/)\s*#?(region|endregion)(.*)/;for(let p=0;p<n.length;p++){const r=n[p],m=r.match(i);if(m)m[3]=="region"?(e+=`
`+"  ".repeat(l)+m[4].trimStart(),l++):l--;else{e+=`
`+"  ".repeat(l)+r;continue}}return e.slice(1)}});fa.factory({id:"mdit2code",name:"mdit转代码块",detail:"mdit转代码块 (允许嵌套)。注意 `:*n` 会转化为 `~*(n+3)`, `@aaa bbb` 会转换为 `# bbb` (h1标题)",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=t.trimEnd().split(`
`);let e="";for(let l=0;l<n.length;l++){const i=n[l],p=i.match(It.reg_mdit_head),r=i.trim().match(/^@(\S*?)\s(.*?)$/);if(r)e+=`
# `+r[2];else if(p){const m="~".repeat(p[3].length+3);p[4].trim()!==""?e+=`
`+m+`anyblock
[`+p[4].trimStart()+"]":e+=`
`+m;continue}else{e+=`
`+i;continue}}return e.slice(1)}});class Ga{static list2data(a,t=!0){let n=[];function e(r,m){if(n.length==0&&m==0)return 0;for(;n.length&&n[n.length-1].level>=r;)n.pop();if(n.length==0&&m==0)return 0;let o;return n.length==0?o=0:o=n[n.length-1].inline_comp,m>0&&n.push({level:r,inline_comp:m+o}),o}let l=[],i=null;const p=a.split(`
`);for(let r of p){const m=r.match(It.reg_list_noprefix);if(m){i=m;let o=m[4].split(It.inline_split),c=m[1].length,h=e(c,o.length-1);for(let u=0;u<o.length;u++)l.push({content:o[u],level:c+u+h})}else{let o=l.pop();o&&(i&&/^\s*$/.test(r.slice(0,i[1].length+2))?r=r.slice(i[1].length+2):r=r.trim(),l.push({content:o.content+`
`+r,level:o.level}))}}return l}static list2listnode(a){let t=Ga.list2data(a,!1);t=Ga.data2strict(t);let n=[],e=[],l;for(let i=0;i<t.length;i++){const p=t[i];if(l={content:p.content,children:[]},e[p.level]=l,p.level>=1&&e.hasOwnProperty(p.level-1))e[p.level-1].children.push(l);else if(p.level==0)n.push(l);else return console.error(`list数据不合规，没有正规化. level:${p.level}, prev_nodes:${e}`),n}return n}static list2json(a){let t=Ga.list2data(a,!1);t=Ga.data2strict(t);let n={},e=[];for(let p=0;p<t.length;p++){const r=t[p],m=r.content,o={};if(e[r.level]=o,r.level>=1&&e.hasOwnProperty(r.level-1)){let c=e[r.level-1];if(typeof c!="object"||Array.isArray(c))return console.error("list数据不合规，父节点的value值不是{}类型"),n;c[m]=o}else if(r.level==0)n[m]=o;else return console.error(`list数据不合规，没有正规化. level:${r.level}, prev_nodes:${e}`),n}let l=n;return i(l),l;function i(p,r,m){if(Array.isArray(p))return;const o=Object.keys(p);let c=0;for(let h=0;h<o.length;h++){const u=o[h];if(!p.hasOwnProperty(u))continue;const y=p[u];if(typeof y=="object"&&!Array.isArray(y))if(Object.keys(y).length===0){let x=u.indexOf(": ");x>0?(delete p[u],h--,p[u.slice(0,x)]=u.slice(x+1)):(p[u]="",c++)}else i(y,p,u)}if(r&&m){let h=[];if(c==o.length){for(let u=0;u<o.length;u++){const y=o[u];p.hasOwnProperty(y)&&h.push(y)}r[m]=h}}}}static title2data(a){let t=[];const n=a.split(`
`);let e="",l="";for(let p of n){if(l==""){const o=p.match(It.reg_code);if(o&&o[3]){l=o[1]+o[3],t[t.length-1].content=t[t.length-1].content+`
`+p;continue}}else{p.indexOf(l)==0&&(l=""),t[t.length-1].content=t[t.length-1].content+`
`+p;continue}const r=p.match(It.reg_heading_noprefix),m=p.match(It.reg_list_noprefix);if(r&&!r[1])i(),t.push({content:r[4],level:r[3].length-1-10}),e="heading";else if(m)i(),t.push({content:m[4],level:m[1].length+1}),e="list";else if(/^\S/.test(p)&&e=="list")t[t.length-1].content=t[t.length-1].content+`
`+p;else if(e=="para")t[t.length-1].content=t[t.length-1].content+`
`+p;else{if(/^\s*$/.test(p))continue;t.push({content:p,level:0}),e="para"}}return i(),t;function i(){(e=="para"||e=="list")&&(t[t.length-1].content=t[t.length-1].content.replace(/\s*$/,""))}}static old_ulist2data(a){let t=[],n=-1,e=-1;const l=a.split(`
`);for(let p of l){const r=p.match(It.reg_list_noprefix);if(r){let m=r[1].length,o;if(n<0)n=m,o=1;else if(n>=m)o=1;else if(e<0)e=m,o=2;else if(e>=m)o=2;else{let c=t.pop();c&&t.push({content:c.content+`
`+p.trim(),level:c.level});continue}t.push({content:r[4],level:o})}else{let m=t.pop();m&&t.push({content:m.content+`
`+p.trim(),level:m.level})}}let i=0;for(let p of t)p.level==2?(p.level+=i,i++):i=0;return t}static data2strict(a){let t=[-999],n=[];for(let e of a){let l=0;for(let i=0;i<t.length;i++)if(!(t[i]<e.level))if(t[i]==e.level){t=t.slice(0,i+1),l=i;break}else{t=t.slice(0,i),t.push(e.level),l=i;break}l==0&&(t.push(e.level),l=t.length-1),n.push({content:e.content,level:l-1})}return n}static data2taskData(a){for(let t of a)t.content=t.content.split(`
`).map(n=>n.match(/\[.\] /)?"- "+n:n).join(`
`);return a}static data_2L_2_mL1B(a){let t=[],n=0;for(let e of a)e.level!=0?(t.push({content:e.content,level:e.level+n}),n++):(t.push({content:e.content,level:e.level}),n=0);return t}static data2list(a){let t=[];for(let e of a){const l="  ".repeat(e.level);let i=e.content.split(`
`);for(let p=0;p<i.length;p++)p==0?t.push(l+"- "+i[p]):t.push(l+"  "+i[p])}return t.join(`
`)}static data2nodes(a,t){const n=document.createElement("div");t.appendChild(n),n.classList.add("ab-nodes");const e=document.createElement("div");n.appendChild(e),e.classList.add("ab-nodes-children");let l=[];for(let i of a){const p=document.createElement("div");p.classList.add("ab-nodes-node"),p.setAttribute("has_children","false");const r=document.createElement("div");p.appendChild(r),r.classList.add("ab-nodes-content"),Ot.getInstance().m_renderMarkdownFn(i.content,r);const m=document.createElement("div");p.appendChild(m),m.classList.add("ab-nodes-children");const o=document.createElement("div");m.appendChild(o),o.classList.add("ab-nodes-bracket");const c=document.createElement("div");if(m.appendChild(c),c.classList.add("ab-nodes-bracket2"),l[i.level]={node:p,content:r,children:m},i.level==0)e.appendChild(p);else if(i.level>=1&&l.hasOwnProperty(i.level-1))l[i.level-1].children.appendChild(p),l[i.level-1].node.setAttribute("has_children","true");else return console.error("节点错误"),t}return t}}const PU=fa.factory({id:"list2listdata",name:"列表到listdata",process_param:Cs.text,process_return:Cs.list_stream,detail:"列表到listdata",process:(s,a,t)=>Ga.list2data(t)}),RU=fa.factory({id:"title2listdata",name:"标题到listdata",process_param:Cs.text,process_return:Cs.list_stream,detail:"标题到listdata",process:(s,a,t)=>Ga.title2data(t)});fa.factory({id:"listdata2list",name:"listdata到列表",process_param:Cs.list_stream,process_return:Cs.text,detail:"listdata到列表",process:(s,a,t)=>Ga.data2list(t)});fa.factory({id:"listdata2nodes",name:"listdata到节点图",process_param:Cs.list_stream,process_return:Cs.el,detail:"listdata到节点图",process:(s,a,t)=>Ga.data2nodes(t,s)});fa.factory({id:"listdata2strict",name:"listdata严格化",process_param:Cs.list_stream,process_return:Cs.list_stream,detail:"将列表数据转化为更规范的列表数据。统一缩进符(2空格 4空格 tab混用)为level 1、禁止跳等级(h1直接就到h3)",process:(s,a,t)=>Ga.data2strict(t)});fa.factory({id:"listdata2task",name:"listdata支持任务列表",process_param:Cs.list_stream,process_return:Cs.list_stream,detail:"当列表中存在任务列表项时，令此列表项支持任务项",process:(s,a,t)=>Ga.data2taskData(t)});fa.factory({id:"list2listnode",name:"列表到listnode (beta)",process_param:Cs.text,process_return:Cs.json,detail:"列表到listnode",process:(s,a,t)=>{const n=Ga.list2listnode(t);return JSON.stringify(n,null,2)}});fa.factory({id:"list2json",name:"列表到json (beta)",process_param:Cs.text,process_return:Cs.json,detail:"列表到json",process:(s,a,t)=>{const n=Ga.list2json(t);return JSON.stringify(n,null,2)}});class je{static data_mL_2_2L1B(a){const t=[];let l=!1;for(const i of a){if(0>=i.level){t.push({content:i.content.trim(),level:0}),l=!1;continue}if(l){const p=t.pop();if(p){let r=i.content.trim();i.level>1&&(r="- "+r);for(let m=0;m<i.level-1;m++)r=" "+r;r=p.content+`
`+r,t.push({content:r,level:1})}}else{t.push({content:i.content.trim(),level:1}),l=!0;continue}}return t}static data_mL_2_2L(a){const t=[];for(const l of a){if(0>=l.level){t.push({content:l.content.trim(),level:0});continue}if(1>=l.level){t.push({content:l.content.trim(),level:1});continue}else{const i=t.pop();if(i){let p=l.content.trim();l.level>1&&(p="- "+p);for(let r=0;r<l.level-1;r++)p=" "+p;p=i.content+`
`+p,t.push({content:p,level:1})}}}return t}static list2c2data(a,t=!0){const n=[],e=a.trimStart().split(`
`),l=e[0].match(It.reg_list_noprefix);if(!l||l[1])return console.error("不是列表内容:",e[0]),n;const i=l[1].length;let p="",r="";for(let o of e){const c=o.match(It.reg_list_noprefix);if(c&&!c[1]&&c[1].length<=i){m();let h=c[4];if(t){const u=c[4].split(It.inline_split);if(u.length>1){const y=h.indexOf(u[1]);p+=h.slice(y)+`
`,r="  ",h=u[0]}}n.push({content:h,level:0})}else p.trim()==""&&(c&&c[1]?r=c[1]:r="  "),o.startsWith("	")?o=o.substring(1):o.startsWith(r)&&(o=o.substring(r.length)),p+=o+`
`}return m(),n;function m(){p.trim()!=""&&(n.push({content:p,level:1}),p="")}}static title2c2data(a){const t=[],n=a.trimStart().split(`
`),e=n[0].match(It.reg_heading_noprefix);if(!e||e[1])return console.error("不是标题内容:",n[0]),t;const l=e[3].length-1;let i="",p="";for(const m of n){if(p==""){const c=m.match(It.reg_code);if(c&&c[3]){p=c[1]+c[3],i+=m+`
`;continue}}else{m.indexOf(p)==0&&(p=""),i+=m+`
`;continue}const o=m.match(It.reg_heading_noprefix);o&&!o[1]&&o[3].length-1<=l?(r(),t.push({content:o[4],level:0})):i+=m+`
`}return r(),t;function r(){i.trim()!=""&&(t.push({content:i,level:1}),i="")}}static c2data2tab(a,t,n){const e=document.createElement("div");t.appendChild(e),e.classList.add("ab-tab-root"),n&&e.setAttribute("modeT","true");const l=document.createElement("div");e.appendChild(l),l.classList.add("ab-tab-nav");const i=document.createElement("div");e.appendChild(i),i.classList.add("ab-tab-content");let p=null;for(let o=0;o<a.length;o++){const c=a[o];if(c.level==0){const h=document.createElement("button");l.appendChild(h),h.classList.add("ab-tab-nav-item"),h.setAttribute("is_activate",o==0?"true":"false"),h.setAttribute("data_index",o.toString()),h.textContent=c.content.slice(0,20),p=document.createElement("div"),i.appendChild(p),p.classList.add("ab-tab-content-item"),p.setAttribute("style",o==0?"display:block":"display:none"),p.setAttribute("is_activate",o==0?"true":"false")}else if(p)p.setAttribute("data_index",o.toString()),Ot.getInstance().m_renderMarkdownFn(c.content,p),p=null;else continue}const r=e.querySelectorAll(":scope>.ab-tab-nav>.ab-tab-nav-item"),m=e.querySelectorAll(":scope>.ab-tab-content>.ab-tab-content-item");r.length!=m.length&&console.warn("ab-tab-nav-item和ab-tab-content-item的数量不一致");for(let o=0;o<r.length;o++)Dm.env.startsWith("obsidian")?r[o].onclick=()=>{for(let c=0;c<m.length;c++)r[c].setAttribute("is_activate","false"),m[c].setAttribute("is_activate","false"),m[c].setAttribute("style","display:none");r[o].setAttribute("is_activate","true"),m[o].setAttribute("is_activate","true"),m[o].setAttribute("style","display:block")}:r[o].setAttribute("onclick",`
          const i = ${o}
          const tab_current = this
          const tab_nav = this.parentNode
          const tab_root = tab_nav.parentNode
          const tab_content = tab_root.querySelector(":scope>.ab-tab-content")
          const tab_nav_items = tab_nav.querySelectorAll(":scope>.ab-tab-nav-item")
          const tab_content_items = tab_content.querySelectorAll(":scope>.ab-tab-content-item")
          for (let j=0; j<tab_content_items.length; j++){
            tab_nav_items[j].setAttribute("is_activate", "false")
            tab_content_items[j].setAttribute("is_activate", "false")
            tab_content_items[j].setAttribute("style", "display:none")
          }
          tab_current.setAttribute("is_activate", "true")
          tab_content_items[i].setAttribute("is_activate", "true")
          tab_content_items[i].setAttribute("style", "display:block")
        `);return t}static c2data2items(a,t){const n=document.createElement("div");t.appendChild(n),n.classList.add("ab-items");let e=null;for(const l of a)if(l.level==0){e=document.createElement("div"),n.appendChild(e),e.classList.add("ab-items-item");const i=document.createElement("div");e.appendChild(i),i.classList.add("ab-items-title"),Ot.getInstance().m_renderMarkdownFn(l.content,i)}else if(e){const i=document.createElement("div");e.appendChild(i),i.classList.add("ab-items-content"),Ot.getInstance().m_renderMarkdownFn(l.content,i)}else continue;return t}}fa.factory({id:"list2c2listdata",name:"列表转c2listdata",match:"list2c2listdata",default:"list2c2listdata",process_param:Cs.text,process_return:Cs.c2list_stream,process:(s,a,t)=>je.list2c2data(t)});fa.factory({id:"title2c2listdata",name:"标题转c2listdata",match:"title2c2listdata",default:"title2c2listdata",process_param:Cs.text,process_return:Cs.c2list_stream,process:(s,a,t)=>je.title2c2data(t)});fa.factory({id:"c2listdata2tab",name:"c2listdata转标签",match:"c2listdata2tab",default:"c2listdata2tab",process_param:Cs.c2list_stream,process_return:Cs.el,process:(s,a,t)=>je.c2data2tab(t,s,!1)});fa.factory({id:"c2listdata2items",name:"c2listdata转容器结构",match:"c2listdata2items",default:"c2listdata2items",process_param:Cs.c2list_stream,process_return:Cs.el,process:(s,a,t)=>je.c2data2items(t,s)});fa.factory({id:"c2listdata2easytimeline",name:"适配到easy_timeline",match:"c2listdata2easytimeline",detail:"适配到easy_timeline格式，需要安装easy timeline插件",process_param:Cs.c2list_stream,process_return:Cs.text,process:(s,a,t)=>{let n="",e="";for(const l of t)l.level==0?(e!=""&&(n+=e+`

`),e=l.content+". "):(e==""&&(e=" . "),e+=l.content);return e!=""&&(n+=e),"````timeline\n"+n+"\n````"}});class Pl{static list2ut(a,t,n=!1){let e=Ga.list2data(a);return e=Ga.data2strict(e),e=je.data_mL_2_2L(e),e=Ga.data_2L_2_mL1B(e),Pl.data2table(e,t,n)}static list2timeline(a,t,n=!1){let e=je.list2c2data(a);t=Pl.data2table(e,t,n);const l=t.querySelector("table");return l&&l.classList.add("ab-table-timeline","ab-table-fc"),t}static title2timeline(a,t,n=!1){let e=je.title2c2data(a);t=Pl.data2table(e,t,n);const l=t.querySelector("table");return l&&l.classList.add("ab-table-timeline","ab-table-fc"),t}static data2table(a,t,n){let e=[],l=-1,i=999;for(let p=0;p<a.length;p++){let r=a[p],m=1,o=a[p].level;for(let c=p+1;c<a.length;c++)if(a[c].level>o)o=a[c].level;else if(a[c].level>a[p].level)o=a[c].level,m++;else break;r.level<=i&&l++,i=r.level,e.push({content:r.content,level:r.level,tableRowSpan:m,tableRow:l})}{const p=document.createElement("table");t.appendChild(p),p.classList.add("ab-table","ab-branch-table"),n&&p.setAttribute("modeT","true");let r;e[0].content.indexOf("< ")==0&&(r=document.createElement("thead"),p.appendChild(r),e[0].content=e[0].content.replace(/^\<\s/,""));const m=document.createElement("tbody");p.appendChild(m);for(let o=0;o<l+1;o++){let c,h;o==0&&r?(h=document.createElement("tr"),r.appendChild(h),c=!0):(c=!1,h=document.createElement("tr"),m.appendChild(h));for(let u of e){if(u.tableRow!=o)continue;let y=document.createElement(c?"th":"td");h.appendChild(y),y.setAttribute("rowspan",u.tableRowSpan.toString()),y.setAttribute("col_index",u.level.toString()),Ot.getInstance().m_renderMarkdownFn(u.content,y)}}}return t}}fa.factory({id:"title2table",name:"标题到表格",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=RU.process(s,a,t);return s=Pl.data2table(n,s,!1)}});fa.factory({id:"list2table",name:"列表转表格",match:/list2(md)?table(T)?/,default:"list2table",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/list2(md)?table(T)?/);if(!n)return s;const e=PU.process(s,a,t);return s=Pl.data2table(e,s,n[2]=="T")}});fa.factory({id:"list2c2t",name:"列表转二列表格",match:"list2c2t",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{let n=je.list2c2data(t);return Pl.data2table(n,s,!1),s}});fa.factory({id:"list2ut",name:"列表转二维表格",match:/list2(md)?ut(T)?/,default:"list2ut",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/list2(md)?ut(T)?/);return n&&Pl.list2ut(t,s,n[2]=="T"),s}});fa.factory({id:"list2timeline",name:"列表转时间线",match:/list2(md)?timeline(T)?/,default:"list2mdtimeline",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/list2(md)?timeline(T)?/);return n&&Pl.list2timeline(t,s,n[2]=="T"),s}});fa.factory({id:"title2timeline",name:"标题转时间线",match:/title2(md)?timeline(T)?/,default:"title2mdtimeline",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/title2(md)?timeline(T)?/);return n&&Pl.title2timeline(t,s,n[2]=="T"),s}});class Yp{static list2lt(a,t,n=!1){let e=Yp.list2dtdata(a);return e=Ga.data2strict(e).map((l,i)=>({content:e[i].content,level:l.level,tableRow:e[i].tableRow,tableColumn:e[i].tableColumn,tableRowSpan:e[i].tableRowSpan,type:e[i].type})),Yp.dtdata2dt(e,t,n)}static list2dt(a,t,n=!1){let e=Yp.list2dtdata(a);return e=Ga.data2strict(e).map((l,i)=>({content:e[i].content,level:l.level,tableRow:e[i].tableRow,tableColumn:e[i].tableColumn,tableRowSpan:e[i].tableRowSpan,type:e[i].type})),Yp.dtdata2dt(e,t,n,!0)}static list2dtdata(a){const t=[],n=a.split(`
`);let e=-1;for(const i of n){const p=i.match(It.reg_list_noprefix);if(p){e++;const r=p[4],m=p[1].length;t.push({content:r.trimStart(),level:m,tableRow:e,tableColumn:0,type:"",tableRowSpan:1})}else{const r=t.pop();r&&t.push({content:r.content+`
`+i.trim(),level:r.level,tableRow:r.tableRow,tableColumn:r.tableColumn,type:r.type,tableRowSpan:r.tableRowSpan})}}const l=[];for(const i of t){const p=i.content.split(It.inline_split);for(let r=0;r<p.length;r++){let m="";if(r==0)if(p[r].trimEnd().endsWith("/"))m="folder";else{const o=p[r].split(".");o.length===0||o[o.length-1]===""?m="":m=o[o.length-1]}l.push({content:p[r].trimEnd(),level:i.level,tableRow:e,tableColumn:r,type:m,tableRowSpan:i.tableRowSpan})}}return l}static dtdata2dt(a,t,n,e=!1){const l=document.createElement("div");t.appendChild(l),l.classList.add("ab-list-table-parent");let i,p,r;{i=document.createElement("table"),l.appendChild(i),i.classList.add("ab-table","ab-list-table"),e&&i.classList.add("ab-table-folder"),n&&i.setAttribute("modeT","true"),a[0].content.indexOf("< ")==0&&(p=document.createElement("thead"),i.appendChild(p),a[0].content=a[0].content.replace(/^\<\s/,"")),r=document.createElement("tbody"),i.appendChild(r);let m,o=!1,c=null;for(let h=0;h<a.length;h++){const u=a[h];u.tableColumn==0&&(h==0&&p?(o=!0,m=document.createElement("tr"),p.appendChild(m)):(o=!1,m=document.createElement("tr"),r.appendChild(m),m.classList.add("ab-foldable-tr"),m.setAttribute("tr_level",u.level.toString()),m.setAttribute("is_fold","false"),m.setAttribute("able_fold","false"),m.setAttribute("type",u.type)),c&&!isNaN(Number(c.getAttribute("tr_level")))&&Number(c.getAttribute("tr_level"))<u.level&&c.setAttribute("able_fold","true"),c=m);const y=document.createElement(o?"th":"td");if(m.appendChild(y),y.setAttribute("rowspan",u.tableRowSpan.toString()),u.tableColumn==0&&e){const b=document.createElement("div");y.appendChild(b),b.classList.add("ab-list-table-svg"),o||(u.type=="folder"?(b.innerHTML='<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--!Font Awesome Free 6.6.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free Copyright 2024 Fonticons, Inc.--><path d="M64 480H448c35.3 0 64-28.7 64-64V160c0-35.3-28.7-64-64-64H288c-10.1 0-19.6-4.7-25.6-12.8L243.2 57.6C231.1 41.5 212.1 32 192 32H64C28.7 32 0 60.7 0 96V416c0 35.3 28.7 64 64 64z"/></svg>',u.content=u.content.trimEnd().slice(0,-1)):b.innerHTML='<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 384 512"><!--!Font Awesome Free 6.6.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free Copyright 2024 Fonticons, Inc.--><path d="M320 464c8.8 0 16-7.2 16-16l0-288-80 0c-17.7 0-32-14.3-32-32l0-80L64 48c-8.8 0-16 7.2-16 16l0 384c0 8.8 7.2 16 16 16l256 0zM0 64C0 28.7 28.7 0 64 0L229.5 0c17 0 33.3 6.7 45.3 18.7l90.5 90.5c12 12 18.7 28.3 18.7 45.3L384 448c0 35.3-28.7 64-64 64L64 512c-35.3 0-64-28.7-64-64L0 64z"/></svg>')}const x=document.createElement("div");y.appendChild(x),x.classList.add("ab-list-table-witharrow"),u.tableColumn==0&&e?x.innerHTML=u.content:Ot.getInstance().m_renderMarkdownFn(u.content,x)}}{const m=r.querySelectorAll("tr");for(let o=0;o<m.length;o++){const c=m[o],h=c.querySelector(":scope>td:first-child")??c;Dm.env.startsWith("obsidian")?h.onclick=()=>{const u=Number(c.getAttribute("tr_level"));if(isNaN(u))return;const y=c.getAttribute("is_fold");if(!y)return;let x=!1;for(let b=o+1;b<m.length;b++){const _=m[b],T=Number(_.getAttribute("tr_level"));if(isNaN(T)||T<=u)break;y=="true"?_.style.display="":_.style.display="none",x=!0}x&&c.setAttribute("is_fold",y=="true"?"false":"true")}:h.setAttribute("onclick",`            const tr = (this.tagName == "TD") ? this.parentNode : this
            const l_tr = tr.parentNode.querySelectorAll("tr")
            const i = ${o}
            
            const tr_level = Number(tr.getAttribute("tr_level"))
            if (isNaN(tr_level)) return
            const tr_isfold = tr.getAttribute("is_fold")
            if (!tr_isfold) return
            let flag_do_fold = false  // 防止折叠最小层
            for (let j=i+1; j<l_tr.length; j++){
              const tr2 = l_tr[j]
              const tr_level2 = Number(tr2.getAttribute("tr_level"))
              if (isNaN(tr_level2)) break
              if (tr_level2 <= tr_level) break // 影响所有后代级
              // if (tr_level2 != tr_level+1) break // 影响下一级的 (话说这里可能有列表规范性问题?)
              (tr_isfold == "true") ? tr2.style.display = "" : tr2.style.display = "none"
              flag_do_fold = true
            }
            if (flag_do_fold) tr.setAttribute("is_fold", tr_isfold=="true"?"false":"true")
          `)}}{const m=document.createElement("button");l.appendChild(m),m.classList.add("ab-table-fold");const o='<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-fold-vertical-icon lucide-fold-vertical"><path d="M12 22v-6"/><path d="M12 8V2"/><path d="M4 12H2"/><path d="M10 12H8"/><path d="M16 12h-2"/><path d="M22 12h-2"/><path d="m15 19-3-3-3 3"/><path d="m15 5-3 3-3-3"/></svg>',c='<svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-unfold-vertical-icon lucide-unfold-vertical"><path d="M12 22v-6"/><path d="M12 8V2"/><path d="M4 12H2"/><path d="M10 12H8"/><path d="M16 12h-2"/><path d="M22 12h-2"/><path d="m15 19-3 3-3-3"/><path d="m15 5-3-3-3 3"/></svg>';m.setAttribute("is_fold","false"),m.innerHTML=o,Dm.env.startsWith("obsidian")?m.onclick=()=>{m.getAttribute("is_fold")=="true"?(m.setAttribute("is_fold","false"),m.innerHTML=o,i.querySelectorAll("tr[is_fold='true']>td:first-child").forEach(y=>{y.click()})):(m.setAttribute("is_fold","true"),m.innerHTML=c,i.querySelectorAll("tr[is_fold='false']>td:first-child").forEach(y=>{y.click()}))}:m.setAttribute("onclick",`          const btn = this;
          const svgStr_fold = \`${o}\`;
          const svgStr_unfold = \`${c}\`;
          const table = btn.parentNode?.querySelector("table");
          if (!table) return;
          
          const is_all_fold = btn.getAttribute("is_fold")
          if (is_all_fold=="true") {
            btn.setAttribute("is_fold", "false"); btn.innerHTML = svgStr_fold;
            const btn_subs = table.querySelectorAll("tr[is_fold='true']>td:first-child");
              btn_subs.forEach((btn_sub) => { btn_sub.click() }) // 注意setAttr版本无断言
          }
          else {
            btn.setAttribute("is_fold", "true"); btn.innerHTML = svgStr_unfold;
            const btn_subs = table.querySelectorAll("tr[is_fold='false']>td:first-child");
              btn_subs.forEach((btn_sub) => { btn_sub.click() }) // 注意setAttr版本无断言
          }`)}return t}}fa.factory({id:"list2lt",name:"列表转列表表格",match:/list2(md)?lt(T)?/,default:"list2lt",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/list2(md)?lt(T)?/);return n&&Yp.list2lt(t,s,n[2]=="T"),s}});fa.factory({id:"list2dt",name:"列表转树状目录",match:/list2(md)?dt(T)?/,default:"list2dt",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/list2(md)?dt(T)?/);return n&&Yp.list2dt(t,s,n[2]=="T"),s}});function $U(s){const a=[],t=[];for(let n=0;n<s.length;n++){const e=s[n];let l;if(e.content.trimEnd().endsWith("/"))l="folder";else{const r=e.content.split(".");r.length===0||r[r.length-1]===""?l="":l=r[r.length-1]}let i=!0;for(let r=n+1;r<s.length;r++)if(s[r].level<e.level){i=!0;break}else if(s[r].level==e.level){i=!1;break}else continue;a[e.level]=!i;let p="";if(e.level>1)for(let r=1;r<e.level;r++)a.hasOwnProperty(r)?a[r]?p+="|  ":p+="   ":p+="[e]";t.push({content:e.content,level:e.level,type:l,is_last:i,pre_as_text:p})}return t}fa.factory({id:"list2astreeH",name:"列表到sacii目录树",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=Ga.list2data(t);n=Ga.data2strict(n);const e=$U(n);let l="";for(const i of e)i.level==0?l+=i.content+`
`:l+=i.pre_as_text+(i.is_last?"└─ ":"├─ ")+i.content+`
`;return l=l.trimEnd(),l}});fa.factory({id:"md",name:"md",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=document.createElement("div");return s.appendChild(n),Ot.getInstance().m_renderMarkdownFn(t,n),s}});fa.factory({id:"text",name:"纯文本",detail:"其实一般会更推荐用code()代替，那个更精确",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>(s.innerHTML=`<p>${t.replace(/ /g,"&nbsp;").split(`
`).join("<br/>")}</p>`,s)});fa.factory({id:"fold",name:"折叠",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{if(t.children.length!=1)return t;const n=t.children[0];n.remove(),n.setAttribute("is_hide","true"),n.classList.add("ab-deco-fold-content"),n.style.display="none";const e=document.createElement("div");t.appendChild(e),e.classList.add("ab-deco-fold");const l=document.createElement("div");e.appendChild(l),l.classList.add("ab-deco-fold-button"),l.textContent="展开";const i=()=>{const m=n.getAttribute("is_hide");m&&m=="false"?(n.setAttribute("is_hide","true"),n.style.display="none",l.textContent="展开"):m&&m=="true"&&(n.setAttribute("is_hide","false"),n.style.display="",l.textContent="折叠")};Dm.env.startsWith("obsidian")?l.onclick=i:l.setAttribute("onclick",`
        const sub_button = this;
        const sub_el = this.nextElementSibling;

        const is_hide = sub_el.getAttribute("is_hide")
        if (is_hide && is_hide=="false") {
          sub_el.setAttribute("is_hide", "true"); 
          sub_el.style.display = "none"
          sub_button.textContent = "展开"
        }
        else if(is_hide && is_hide=="true") {
          sub_el.setAttribute("is_hide", "false");
          sub_el.style.display = ""
          sub_button.textContent = "折叠"
        }
      `),e.appendChild(l),e.appendChild(n);const p=n.classList.contains("ab-list-table-parent"),r=n.querySelector(".ab-table-fold");if(p&&r&&Dm.env.startsWith("obsidian")){i(),l.textContent="折叠/展开";const m=()=>{const o=new MouseEvent("click",{view:window,bubbles:!0,cancelable:!0});r.dispatchEvent(o)};m(),l.onclick=m}return t}});fa.factory({id:"scroll",name:"滚动",match:/^scroll(X)?(\((\d+)\))?$/,default:"scroll(460)",detail:"默认是纵向滚动。可以指定溢出滚动的范围，可以使用scrollX进行横向滚动",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^scroll(X)?(\((\d+)\))?$/);if(!n)return t;let e=0;if(n[2]&&n[3]&&(e=Number(n[3]),isNaN(e))||t.children.length!=1)return t;const l=t.children[0];l.remove();const i=document.createElement("div");return t.appendChild(i),i.classList.add("ab-deco-scroll"),i.appendChild(l),n[1]?(i.classList.add("ab-deco-scroll-x"),i.setAttribute("style",`max-height: ${e!==0?e+"px":"100%"}`)):(i.classList.add("ab-deco-scroll-y"),i.setAttribute("style",`max-height: ${e!==0?e+"px":"460px"}`)),t}});fa.factory({id:"overfold",name:"超出折叠",match:/^overfold(\((\d+)\))?$/,default:"overfold(380)",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^overfold(\((\d+)\))?$/);if(!n)return t;let e;if(!n[1])e=460;else if(!n[2]||(e=Number(n[2]),isNaN(e)))return t;if(t.children.length!=1)return t;const l=t.children[0];l.remove();const i=document.createElement("div");t.appendChild(i),i.classList.add("ab-deco-overfold");const p=document.createElement("div");return i.appendChild(p),p.classList.add("ab-deco-overfold-button"),p.textContent="展开",l.classList.add("ab-deco-overfold-content"),i.appendChild(l),i.appendChild(p),i.setAttribute("style",`max-height: ${e}px`),i.setAttribute("is-fold","true"),p.onclick=()=>{const r=i.getAttribute("is-fold");r&&(r=="true"?(i.setAttribute("style",""),i.setAttribute("is-fold","false"),p.textContent="折叠"):(i.setAttribute("style",`max-height: ${e}px`),i.setAttribute("is-fold","true"),p.textContent="展开"))},t}});fa.factory({id:"width",name:"宽度控制",match:/^width\(((?:\d*\.?\d+(?:%|px|rem)?,\s*)*\d*\.?\d+(?:%|px|rem)?)\)$/,detail:"用于控制表格或分栏的每列的宽度",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^width\(((?:\d*\.?\d+(?:%|px|rem)?,\s*)*\d*\.?\d+(?:%|px|rem)?)\)$/);if(!n||t.children.length!=1)return t;const e=n[1].split(",").map(i=>/^\d*\.?\d+$/.test(i.trim())?`${i.trim()}%`:i.trim());if(t.children[0].classList.contains("ab-col")){const i=t.children[0].children;if(i.length==0)return t;for(let p=0;p<Math.min(i.length,e.length);p++){const r=i[p];e[p].endsWith("%")?r.style.flex=`0 1 ${e[p]}`:(r.style.width=e[p],r.style.flex="0 0 auto")}return t}const l=t.querySelector("table");return l!==null&&(l.style.tableLayout="fixed",l.style.width=e.some(i=>i.endsWith("%"))?"100%":"fit-content",l.querySelectorAll("tr").forEach(i=>{for(let p=0;p<Math.min(i.children.length,e.length);p++){const r=i.children[p];r.style.width=r.style.minWidth=r.style.maxWidth=e[p]}})),t}});fa.factory({id:"addClass",name:"增加class",detail:"给当前块增加一个类名。支持正常使用空格来添加多个class, 不需要加dot符, 就像在class=''里写的那样",match:/^addClass\((.*)\)$/,process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^addClass\((.*)\)$/);if(!n||!n[1]||t.children.length!=1)return t;const e=t.children[0],l=n[1].split(" ");for(const i of l)e.classList.add(i);return t}});fa.factory({id:"addStyle",name:"增加style",detail:"给当前块增加一个样式, 注意最外的括号往内要留一个空格, 避免rotate这种用括号时冲突。添加多个则正常使用分号",match:/^addStyle\(\s(.*)\s\)$/,process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^addStyle\(\s(.*)\s\)$/);if(!n||!n[1]||t.children.length!=1)return t;const e=t.children[0];return e.style.cssText+=String(n[1]),t}});fa.factory({id:"addDiv",name:"增加div和class",detail:"给当前块增加一个父类，需要给这个父类一个类名",match:/^addDiv\((.*)\)$/,process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^addDiv\((.*)\)$/);if(!n||!n[1]||t.children.length!=1)return t;const e=t.children[0];e.remove();const l=document.createElement("div");t.appendChild(l);const i=n[1].split(" ");for(const p of i)l.classList.add(p);return l.appendChild(e),t}});fa.factory({id:"title",name:"标题",match:/^#(.*)/,detail:"若直接处理代码或表格块，则会有特殊风格",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=a.match(/^#(.*)/);if(!n||!n[1])return t;const e=n[1],l=document.createElement("div");for(;t.firstChild;){const c=t.firstChild;t.removeChild(c),l.appendChild(c)}const i=document.createElement("div");t.appendChild(i),i.classList.add("ab-deco-title");const p=document.createElement("div");i.appendChild(p),p.classList.add("ab-deco-title-title");const r=document.createElement("p");p.appendChild(r),r.textContent=e,i.appendChild(l),l.classList.add("ab-deco-title-content");let m=l.childNodes[0];if(!m)return t;m instanceof HTMLDivElement&&l.childNodes.length==1&&l.childNodes[0].childNodes[0]&&(m=l.childNodes[0].childNodes[0]);let o="none";if(m instanceof HTMLQuoteElement){o="quote",i.classList.add("callout"),p.classList.add("callout-title"),l.classList.add("callout-content");const c=m.parentNode;if(!c)return t;for(;m.firstChild;)c.insertBefore(m.firstChild,m);c.removeChild(m)}else m instanceof HTMLTableElement?o="table":m instanceof HTMLUListElement?o="ul":m instanceof HTMLPreElement&&(o="pre");return p.setAttribute("title-type",o),t}});fa.factory({id:"transposition",name:"表格转置",match:"transposition",detail:"将表格进行转置，就像矩阵转置那样。该版本不支持有跨行跨列单元格。若复杂表格，请换用trs版本",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=t.querySelector("table");if(!n)return t;const e=n.rows,l=e.length,i=e[0].cells.length,p=document.createElement("table");n.classList.add("ab-transposition","ab-table"),n.classList.forEach(m=>{p.classList.add(m)});const r=document.createElement("tbody");p.appendChild(r);for(let m=0;m<i;m++){const o=r.insertRow();for(let c=0;c<l;c++){const h=e[c].cells[m];if(!h)continue;const u=o.insertCell();u.innerHTML=h.innerHTML}}return n.innerHTML=p.innerHTML,t}});fa.factory({id:"transpose",name:"表格转置",match:"trs",detail:"将表格进行转置，就像矩阵转置那样。该版本支持有跨行跨列单元格",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=t.querySelector("table");if(!n)return t;let{tableMap:e,origi_rowCount:l,origi_colCount:i}=g8(n);const p=new Array(i).fill(null).map(()=>new Array(l).fill(null));for(let o=0;o<l;o++)for(let c=0;c<i;c++){const h=e[o][c];if(h)if(h=="<")p[c][o]="^";else if(h=="^")p[c][o]="<";else{let u=h.html;u.innerHTML=="<"||u.innerHTML=="&lt;"?u.innerHTML="^":u.innerHTML=="^"&&(u.innerHTML="<"),p[c][o]={html:h.html,rowSpan:h.colSpan||1,colSpan:h.rowSpan||1,rowIndex:h.colIndex,colIndex:h.rowIndex}}else continue}const r=i;i=l,l=r;const m=f8(p,l,i);return n.classList.forEach(o=>{m.classList.add(o)}),m.classList.add("ab-transposition","ab-table"),n.innerHTML=m.innerHTML,t}});fa.factory({id:"exTable",name:"表格扩展",match:"exTable",detail:"将表格应用sheet-table语法 (使用 `</^` 标注合并单元格)",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=t.querySelector("table");if(!n)return t;let{tableMap:e,origi_rowCount:l,origi_colCount:i}=g8(n,!0);const r=f8(e,l,i);return n.classList.forEach(m=>{r.classList.add(m)}),r.classList.add("ab-extable","ab-table"),n.innerHTML=r.innerHTML,t}});fa.factory({id:"strictTable",name:"正规化表格",match:"strictTable",detail:"补全表格的尾丢失项，list2table|trs时，可以有效避免bug",process_param:Cs.el,process_return:Cs.el,process:(s,a,t)=>{const n=t.querySelector("table");if(!n)return t;let{tableMap:e,origi_rowCount:l,origi_colCount:i}=g8(n);for(let r=0;r<l;r++)for(let m=0;m<i;m++)e[r][m]||(e[r][m]={html:document.createElement("td"),rowSpan:1,colSpan:1,rowIndex:r,colIndex:m});const p=f8(e,l,i);return n.classList.forEach(r=>{p.classList.add(r)}),p.classList.add("ab-extable","ab-table"),n.innerHTML=p.innerHTML,t}});function g8(s,a=!1){const t=s.rows;let n=t.length,e=0;{let i=[];for(let p=0;p<n;p++)for(const r of t[p].cells){const m=r.colSpan||1,o=r.rowSpan||1;for(let c=p;c<p+o;c++)i[c]?i[c]+=m:i[c]=m}n=i.length,e=Math.max(...i)}const l=new Array(n).fill(null).map(()=>new Array(e).fill(null));for(let i=0;i<n;i++)for(let p=0;p<t[i].cells.length;p++){const r=t[i].cells[p],m=i;let o=p;for(;;){if(o>=l[m].length)throw console.error(`表格解析错误: colIndex超出范围: [${m}][${o}] overflow tableMap[${n-1}][${e-1}]`,l),new Error("表格解析错误: colIndex超出范围");if(l[m][o])o++;else break}if(r.rowSpan>1)for(let c=1;c<r.rowSpan&&!(m+c>=l.length);c++)l[m+c][o]="^";if(r.colSpan>1)for(let c=1;c<r.rowSpan&&!(o+c>=l[m].length);c++)l[m][o+c]="<";if(a)if(r.rowSpan==1&&r.colSpan==1&&r.textContent=="^"){l[m][o]="^";for(let c=m-1;c>=0;c--){const h=l[c][o];if(!h||h=="<")break;if(h!="^"){if(h.html.textContent=="<")break;if(h.html.textContent!="^"||c==0){h.rowSpan+=1;break}}}}else if(r.rowSpan==1&&r.colSpan==1&&r.textContent=="<"){l[m][o]="<";for(let c=o-1;c>=0;c--){const h=l[m][c];if(!h||h=="^")break;if(h!="<"){if(h.html.textContent=="^")break;if(h.html.textContent!="<"||c==0){h.colSpan+=1;break}}}}else l[m][o]={html:r,rowSpan:r.rowSpan,colSpan:r.colSpan,rowIndex:m,colIndex:o};else l[m][o]={html:r,rowSpan:r.rowSpan,colSpan:r.colSpan,rowIndex:m,colIndex:o}}return{tableMap:l,origi_rowCount:n,origi_colCount:e}}function f8(s,a,t){const n=document.createElement("table"),e=document.createElement("tbody");n.appendChild(e);for(let l=0;l<a;l++){const i=e.insertRow();for(let p=0;p<t;p++){const r=s[l][p];if(!r||r=="<"||r=="^")continue;const m=i.insertCell();m.innerHTML=r.html.innerHTML,m.rowSpan=r.rowSpan,m.colSpan=r.colSpan,m.setAttribute("rowIndex",String(r.rowIndex)),m.setAttribute("colIndex",String(r.colIndex))}}return n}fa.factory({id:"faq",name:"FAQ",match:"FAQ",detail:"渲染常见问题/对话。每个项需以 `/^([a-zA-Z])(: |：)(.*)/` 开头",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=document.createElement("div");s.appendChild(n),n.classList.add("ab-faq");const e=t.split(`
`);let l="",i="";for(let p of e){const r=p.match(/^([\S]+)(:|：)(.*)/);if(r){if(l){const m=document.createElement("div");n.appendChild(m),m.classList.add("ab-faq-line",`ab-faq-${l}`);const o=document.createElement("div");m.appendChild(o),o.classList.add("ab-faq-bubble",`ab-faq-${l}`);const c=document.createElement("div");o.appendChild(c),c.classList.add("ab-faq-content"),Ot.getInstance().m_renderMarkdownFn(i,c)}l=r[1],i=r[3]}else{l&&(i=i+`
`+p);continue}}if(l){const p=document.createElement("div");n.appendChild(p),p.classList.add("ab-faq-line",`ab-faq-${l}`);const r=document.createElement("div");p.appendChild(r),r.classList.add("ab-faq-bubble",`ab-faq-${l}`);const m=document.createElement("div");r.appendChild(m),m.classList.add("ab-faq-content"),Ot.getInstance().m_renderMarkdownFn(i,m)}return s}});fa.factory({id:"info_converter",name:"INFO",detail:"查看当前软件版本下的注册处理器表",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=document.createElement("div");s.appendChild(n),n.classList.add("ab-setting","md-table-fig1");const e=document.createElement("table");n.appendChild(e),e.classList.add("ab-setting","md-table-fig2");{const i=document.createElement("thead");e.appendChild(i);const p=document.createElement("tr");i.appendChild(p);let r;r=document.createElement("th"),p.appendChild(r),r.textContent=`处理器名
Processor name`,r=document.createElement("th"),p.appendChild(r),r.textContent=`下拉框默认项
The default drop-down box`,r=document.createElement("th"),p.appendChild(r),r.textContent=`用途描述
Purpose description`,r=document.createElement("th"),p.appendChild(r),r.textContent=`输入类型
Input type`,r=document.createElement("th"),p.appendChild(r),r.textContent=`输出类型
Output type`,r=document.createElement("th"),p.appendChild(r),r.textContent=`正则
RegExp`,r=document.createElement("th"),p.appendChild(r),r.textContent=`是否启用
Is enable`,r=document.createElement("th"),p.appendChild(r),r.textContent=`定义来源
Source`,r=document.createElement("th"),p.appendChild(r),r.textContent=`别名替换
Alias substitution`}const l=document.createElement("tbody");e.appendChild(l);for(let i of Ot.getInstance().list_abConvert){const p=document.createElement("tr");l.appendChild(p);let r;r=document.createElement("td"),p.appendChild(r),r.textContent=i.name,r=document.createElement("td"),p.appendChild(r),r.textContent=String(i.default),r=document.createElement("td"),p.appendChild(r),r.textContent=i.detail,r.setAttribute("style","max-width:240px;overflow-x:auto;white-space:nowrap;"),r=document.createElement("td"),p.appendChild(r),r.textContent=String(i.process_param),r=document.createElement("td"),p.appendChild(r),r.textContent=String(i.process_return),r=document.createElement("td"),p.appendChild(r),r.textContent=String(i.match),r=document.createElement("td"),p.appendChild(r),r.textContent=i.is_disable?"No":"Yes",r=document.createElement("td"),p.appendChild(r),r.textContent=i.register_from,r=document.createElement("td"),p.appendChild(r),r.textContent=i.process_alias}return s}});fa.factory({id:"info_alias",name:"INFO_Alias",match:"info_alias",detail:"查看当前软件版本下的注册别名表",process_param:Cs.text,process_return:Cs.json,process:(s,a,t)=>JSON.stringify(Array.from(mk(),n=>({regex:n.regex.toString(),replacement:n.replacement})),null,2)});function y8(s,a){const t=s.split(`
`);let n="";const e=[];for(let i=0;i<t.length;i++){const p=t[i],r=p.match(a);if(r){l(),e.push({content:r[1].trim(),level:0});continue}else n+=p+`
`}return l(),e;function l(){n.trim()!=""&&(e.push({content:n,level:1}),n="")}}const ok=fa.factory({id:"mditTabs",name:"mdit标签页",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=y8(t,/^@tab(.*)$/);return je.c2data2tab(n,s,!1),s}});fa.factory({id:"mditDemo",name:"mdit展示对比",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=`@tab show
${t}
@tab mdSource
~~~~~md
${t}
~~~~~`;return ok.process(s,a,n),s}});fa.factory({id:"mditABDemo",name:"AnyBlock专用展示对比",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{{const n=`@tab show
${t}
@tab withoutPlugin
(noPlugin)${t.trimStart()}
@tab mdSource
~~~~~md
${t}
~~~~~`;return ok.process(s,a,n),s}}});fa.factory({id:"mditCol",name:"mdit分栏",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=y8(t,/^@col(.*)$/);return je.c2data2items(n,s),s.querySelector("div")?.classList.add("ab-col"),s}});fa.factory({id:"mditCard",name:"mdit卡片",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=y8(t,/^@card(.*)$/);return je.c2data2items(n,s),s.querySelector("div")?.classList.add("ab-card"),s.querySelector("div")?.classList.add("ab-lay-vfall"),s}});fa.factory({id:"mditChat",name:"mdit对话",detail:"显示渲染对话，需要配合 obsidian-view-chat-qq 插件使用",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{const n=t.split(`
`);let e="";for(let l=0;l<n.length;l++){const i=n[l];if(/^@chat(.*)$/.test(i)){const p=i.match(/^@chat(.*)$/);if(p&&p[1]){e+=`
`+p[1]+`:
`;continue}}e+=i+`
`}return e}});class _r extends Error{}class VU extends _r{constructor(a){super(`Invalid DateTime: ${a.toMessage()}`)}}class HU extends _r{constructor(a){super(`Invalid Interval: ${a.toMessage()}`)}}class qU extends _r{constructor(a){super(`Invalid Duration: ${a.toMessage()}`)}}class Zr extends _r{}class ck extends _r{constructor(a){super(`Invalid unit ${a}`)}}class Rn extends _r{}class ji extends _r{constructor(){super("Zone is an abstract class")}}const da="numeric",ul="short",Be="long",Au={year:da,month:da,day:da},hk={year:da,month:ul,day:da},UU={year:da,month:ul,day:da,weekday:ul},uk={year:da,month:Be,day:da},dk={year:da,month:Be,day:da,weekday:Be},gk={hour:da,minute:da},fk={hour:da,minute:da,second:da},yk={hour:da,minute:da,second:da,timeZoneName:ul},vk={hour:da,minute:da,second:da,timeZoneName:Be},Ek={hour:da,minute:da,hourCycle:"h23"},xk={hour:da,minute:da,second:da,hourCycle:"h23"},wk={hour:da,minute:da,second:da,hourCycle:"h23",timeZoneName:ul},bk={hour:da,minute:da,second:da,hourCycle:"h23",timeZoneName:Be},Ak={year:da,month:da,day:da,hour:da,minute:da},_k={year:da,month:da,day:da,hour:da,minute:da,second:da},kk={year:da,month:ul,day:da,hour:da,minute:da},Bk={year:da,month:ul,day:da,hour:da,minute:da,second:da},jU={year:da,month:ul,day:da,weekday:ul,hour:da,minute:da},Mk={year:da,month:Be,day:da,hour:da,minute:da,timeZoneName:ul},Tk={year:da,month:Be,day:da,hour:da,minute:da,second:da,timeZoneName:ul},Ck={year:da,month:Be,day:da,weekday:Be,hour:da,minute:da,timeZoneName:Be},zk={year:da,month:Be,day:da,weekday:Be,hour:da,minute:da,second:da,timeZoneName:Be};class fh{get type(){throw new ji}get name(){throw new ji}get ianaName(){return this.name}get isUniversal(){throw new ji}offsetName(a,t){throw new ji}formatOffset(a,t){throw new ji}offset(a){throw new ji}equals(a){throw new ji}get isValid(){throw new ji}}let P1=null;class Sd extends fh{static get instance(){return P1===null&&(P1=new Sd),P1}get type(){return"system"}get name(){return new Intl.DateTimeFormat().resolvedOptions().timeZone}get isUniversal(){return!1}offsetName(a,{format:t,locale:n}){return Vk(a,t,n)}formatOffset(a,t){return Jo(this.offset(a),t)}offset(a){return-new Date(a).getTimezoneOffset()}equals(a){return a.type==="system"}get isValid(){return!0}}const av=new Map;function GU(s){let a=av.get(s);return a===void 0&&(a=new Intl.DateTimeFormat("en-US",{hour12:!1,timeZone:s,year:"numeric",month:"2-digit",day:"2-digit",hour:"2-digit",minute:"2-digit",second:"2-digit",era:"short"}),av.set(s,a)),a}const WU={year:0,month:1,day:2,era:3,hour:4,minute:5,second:6};function YU(s,a){const t=s.format(a).replace(/\u200E/g,""),n=/(\d+)\/(\d+)\/(\d+) (AD|BC),? (\d+):(\d+):(\d+)/.exec(t),[,e,l,i,p,r,m,o]=n;return[i,e,l,p,r,m,o]}function XU(s,a){const t=s.formatToParts(a),n=[];for(let e=0;e<t.length;e++){const{type:l,value:i}=t[e],p=WU[l];l==="era"?n[p]=i:Sa(p)||(n[p]=parseInt(i,10))}return n}const R1=new Map;class Ti extends fh{static create(a){let t=R1.get(a);return t===void 0&&R1.set(a,t=new Ti(a)),t}static resetCache(){R1.clear(),av.clear()}static isValidSpecifier(a){return this.isValidZone(a)}static isValidZone(a){if(!a)return!1;try{return new Intl.DateTimeFormat("en-US",{timeZone:a}).format(),!0}catch{return!1}}constructor(a){super(),this.zoneName=a,this.valid=Ti.isValidZone(a)}get type(){return"iana"}get name(){return this.zoneName}get isUniversal(){return!1}offsetName(a,{format:t,locale:n}){return Vk(a,t,n,this.name)}formatOffset(a,t){return Jo(this.offset(a),t)}offset(a){if(!this.valid)return NaN;const t=new Date(a);if(isNaN(t))return NaN;const n=GU(this.name);let[e,l,i,p,r,m,o]=n.formatToParts?XU(n,t):YU(n,t);p==="BC"&&(e=-Math.abs(e)+1);const h=Od({year:e,month:l,day:i,hour:r===24?0:r,minute:m,second:o,millisecond:0});let u=+t;const y=u%1e3;return u-=y>=0?y:1e3+y,(h-u)/(60*1e3)}equals(a){return a.type==="iana"&&a.name===this.name}get isValid(){return this.valid}}let a7={};function KU(s,a={}){const t=JSON.stringify([s,a]);let n=a7[t];return n||(n=new Intl.ListFormat(s,a),a7[t]=n),n}const tv=new Map;function nv(s,a={}){const t=JSON.stringify([s,a]);let n=tv.get(t);return n===void 0&&(n=new Intl.DateTimeFormat(s,a),tv.set(t,n)),n}const ev=new Map;function ZU(s,a={}){const t=JSON.stringify([s,a]);let n=ev.get(t);return n===void 0&&(n=new Intl.NumberFormat(s,a),ev.set(t,n)),n}const lv=new Map;function QU(s,a={}){const{base:t,...n}=a,e=JSON.stringify([s,n]);let l=lv.get(e);return l===void 0&&(l=new Intl.RelativeTimeFormat(s,a),lv.set(e,l)),l}let Do=null;function JU(){return Do||(Do=new Intl.DateTimeFormat().resolvedOptions().locale,Do)}const iv=new Map;function Dk(s){let a=iv.get(s);return a===void 0&&(a=new Intl.DateTimeFormat(s).resolvedOptions(),iv.set(s,a)),a}const pv=new Map;function sj(s){let a=pv.get(s);if(!a){const t=new Intl.Locale(s);a="getWeekInfo"in t?t.getWeekInfo():t.weekInfo,"minimalDays"in a||(a={...Lk,...a}),pv.set(s,a)}return a}function aj(s){const a=s.indexOf("-x-");a!==-1&&(s=s.substring(0,a));const t=s.indexOf("-u-");if(t===-1)return[s];{let n,e;try{n=nv(s).resolvedOptions(),e=s}catch{const r=s.substring(0,t);n=nv(r).resolvedOptions(),e=r}const{numberingSystem:l,calendar:i}=n;return[e,l,i]}}function tj(s,a,t){return(t||a)&&(s.includes("-u-")||(s+="-u"),t&&(s+=`-ca-${t}`),a&&(s+=`-nu-${a}`)),s}function nj(s){const a=[];for(let t=1;t<=12;t++){const n=La.utc(2009,t,1);a.push(s(n))}return a}function ej(s){const a=[];for(let t=1;t<=7;t++){const n=La.utc(2016,11,13+t);a.push(s(n))}return a}function m0(s,a,t,n){const e=s.listingMode();return e==="error"?null:e==="en"?t(a):n(a)}function lj(s){return s.numberingSystem&&s.numberingSystem!=="latn"?!1:s.numberingSystem==="latn"||!s.locale||s.locale.startsWith("en")||Dk(s.locale).numberingSystem==="latn"}class ij{constructor(a,t,n){this.padTo=n.padTo||0,this.floor=n.floor||!1;const{padTo:e,floor:l,...i}=n;if(!t||Object.keys(i).length>0){const p={useGrouping:!1,...n};n.padTo>0&&(p.minimumIntegerDigits=n.padTo),this.inf=ZU(a,p)}}format(a){if(this.inf){const t=this.floor?Math.floor(a):a;return this.inf.format(t)}else{const t=this.floor?Math.floor(a):b8(a,3);return tn(t,this.padTo)}}}class pj{constructor(a,t,n){this.opts=n,this.originalZone=void 0;let e;if(this.opts.timeZone)this.dt=a;else if(a.zone.type==="fixed"){const i=-1*(a.offset/60),p=i>=0?`Etc/GMT+${i}`:`Etc/GMT${i}`;a.offset!==0&&Ti.create(p).valid?(e=p,this.dt=a):(e="UTC",this.dt=a.offset===0?a:a.setZone("UTC").plus({minutes:a.offset}),this.originalZone=a.zone)}else a.zone.type==="system"?this.dt=a:a.zone.type==="iana"?(this.dt=a,e=a.zone.name):(e="UTC",this.dt=a.setZone("UTC").plus({minutes:a.offset}),this.originalZone=a.zone);const l={...this.opts};l.timeZone=l.timeZone||e,this.dtf=nv(t,l)}format(){return this.originalZone?this.formatToParts().map(({value:a})=>a).join(""):this.dtf.format(this.dt.toJSDate())}formatToParts(){const a=this.dtf.formatToParts(this.dt.toJSDate());return this.originalZone?a.map(t=>{if(t.type==="timeZoneName"){const n=this.originalZone.offsetName(this.dt.ts,{locale:this.dt.locale,format:this.opts.timeZoneName});return{...t,value:n}}else return t}):a}resolvedOptions(){return this.dtf.resolvedOptions()}}class rj{constructor(a,t,n){this.opts={style:"long",...n},!t&&Rk()&&(this.rtf=QU(a,n))}format(a,t){return this.rtf?this.rtf.format(a,t):zj(t,a,this.opts.numeric,this.opts.style!=="long")}formatToParts(a,t){return this.rtf?this.rtf.formatToParts(a,t):[]}}const Lk={firstDay:1,minimalDays:4,weekend:[6,7]};class dt{static 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e={years:["year","yr."],quarters:["quarter","qtr."],months:["month","mo."],weeks:["week","wk."],days:["day","day","days"],hours:["hour","hr."],minutes:["minute","min."],seconds:["second","sec."]},l=["hours","minutes","seconds"].indexOf(s)===-1;if(t==="auto"&&l){const c=s==="days";switch(a){case 1:return c?"tomorrow":`next ${e[s][0]}`;case-1:return c?"yesterday":`last ${e[s][0]}`;case 0:return c?"today":`this ${e[s][0]}`}}const i=Object.is(a,-0)||a<0,p=Math.abs(a),r=p===1,m=e[s],o=n?r?m[1]:m[2]||m[1]:r?e[s][0]:s;return i?`${p} ${o} ago`:`in ${p} ${o}`}function f7(s,a){let t="";for(const n of s)n.literal?t+=n.val:t+=a(n.val);return t}const Dj={D:Au,DD:hk,DDD:uk,DDDD:dk,t:gk,tt:fk,ttt:yk,tttt:vk,T:Ek,TT:xk,TTT:wk,TTTT:bk,f:Ak,ff:kk,fff:Mk,ffff:Ck,F:_k,FF:Bk,FFF:Tk,FFFF:zk};class Vn{static create(a,t={}){return new Vn(a,t)}static parseFormat(a){let t=null,n="",e=!1;const l=[];for(let i=0;i<a.length;i++){const p=a.charAt(i);p==="'"?((n.length>0||e)&&l.push({literal:e||/^\s+$/.test(n),val:n===""?"'":n}),t=null,n="",e=!e):e||p===t?n+=p:(n.length>0&&l.push({literal:/^\s+$/.test(n),val:n}),n=p,t=p)}return n.length>0&&l.push({literal:e||/^\s+$/.test(n),val:n}),l}static macroTokenToFormatOpts(a){return Dj[a]}constructor(a,t){this.opts=t,this.loc=a,this.systemLoc=null}formatWithSystemDefault(a,t){return this.systemLoc===null&&(this.systemLoc=this.loc.redefaultToSystem()),this.systemLoc.dtFormatter(a,{...this.opts,...t}).format()}dtFormatter(a,t={}){return this.loc.dtFormatter(a,{...this.opts,...t})}formatDateTime(a,t){return this.dtFormatter(a,t).format()}formatDateTimeParts(a,t){return this.dtFormatter(a,t).formatToParts()}formatInterval(a,t){return this.dtFormatter(a.start,t).dtf.formatRange(a.start.toJSDate(),a.end.toJSDate())}resolvedOptions(a,t){return this.dtFormatter(a,t).resolvedOptions()}num(a,t=0,n=void 0){if(this.opts.forceSimple)return tn(a,t);const e={...this.opts};return t>0&&(e.padTo=t),n&&(e.signDisplay=n),this.loc.numberFormatter(e).format(a)}formatDateTimeFromString(a,t){const n=this.loc.listingMode()==="en",e=this.loc.outputCalendar&&this.loc.outputCalendar!=="gregory",l=(u,y)=>this.loc.extract(a,u,y),i=u=>a.isOffsetFixed&&a.offset===0&&u.allowZ?"Z":a.isValid?a.zone.formatOffset(a.ts,u.format):"",p=()=>n?Bj(a):l({hour:"numeric",hourCycle:"h12"},"dayperiod"),r=(u,y)=>n?Tj(a,u):l(y?{month:u}:{month:u,day:"numeric"},"month"),m=(u,y)=>n?Mj(a,u):l(y?{weekday:u}:{weekday:u,month:"long",day:"numeric"},"weekday"),o=u=>{const y=Vn.macroTokenToFormatOpts(u);return y?this.formatWithSystemDefault(a,y):u},c=u=>n?Cj(a,u):l({era:u},"era"),h=u=>{switch(u){case"S":return this.num(a.millisecond);case"u":case"SSS":return this.num(a.millisecond,3);case"s":return this.num(a.second);case"ss":return this.num(a.second,2);case"uu":return this.num(Math.floor(a.millisecond/10),2);case"uuu":return this.num(Math.floor(a.millisecond/100));case"m":return this.num(a.minute);case"mm":return this.num(a.minute,2);case"h":return this.num(a.hour%12===0?12:a.hour%12);case"hh":return this.num(a.hour%12===0?12:a.hour%12,2);case"H":return this.num(a.hour);case"HH":return this.num(a.hour,2);case"Z":return i({format:"narrow",allowZ:this.opts.allowZ});case"ZZ":return i({format:"short",allowZ:this.opts.allowZ});case"ZZZ":return i({format:"techie",allowZ:this.opts.allowZ});case"ZZZZ":return a.zone.offsetName(a.ts,{format:"short",locale:this.loc.locale});case"ZZZZZ":return a.zone.offsetName(a.ts,{format:"long",locale:this.loc.locale});case"z":return a.zoneName;case"a":return p();case"d":return e?l({day:"numeric"},"day"):this.num(a.day);case"dd":return e?l({day:"2-digit"},"day"):this.num(a.day,2);case"c":return this.num(a.weekday);case"ccc":return m("short",!0);case"cccc":return m("long",!0);case"ccccc":return m("narrow",!0);case"E":return this.num(a.weekday);case"EEE":return m("short",!1);case"EEEE":return m("long",!1);case"EEEEE":return m("narrow",!1);case"L":return e?l({month:"numeric",day:"numeric"},"month"):this.num(a.month);case"LL":return e?l({month:"2-digit",day:"numeric"},"month"):this.num(a.month,2);case"LLL":return r("short",!0);case"LLLL":return r("long",!0);case"LLLLL":return r("narrow",!0);case"M":return e?l({month:"numeric"},"month"):this.num(a.month);case"MM":return e?l({month:"2-digit"},"month"):this.num(a.month,2);case"MMM":return r("short",!1);case"MMMM":return r("long",!1);case"MMMMM":return r("narrow",!1);case"y":return e?l({year:"numeric"},"year"):this.num(a.year);case"yy":return e?l({year:"2-digit"},"year"):this.num(a.year.toString().slice(-2),2);case"yyyy":return e?l({year:"numeric"},"year"):this.num(a.year,4);case"yyyyyy":return e?l({year:"numeric"},"year"):this.num(a.year,6);case"G":return c("short");case"GG":return c("long");case"GGGGG":return c("narrow");case"kk":return this.num(a.weekYear.toString().slice(-2),2);case"kkkk":return this.num(a.weekYear,4);case"W":return this.num(a.weekNumber);case"WW":return this.num(a.weekNumber,2);case"n":return this.num(a.localWeekNumber);case"nn":return this.num(a.localWeekNumber,2);case"ii":return this.num(a.localWeekYear.toString().slice(-2),2);case"iiii":return this.num(a.localWeekYear,4);case"o":return this.num(a.ordinal);case"ooo":return this.num(a.ordinal,3);case"q":return this.num(a.quarter);case"qq":return this.num(a.quarter,2);case"X":return this.num(Math.floor(a.ts/1e3));case"x":return this.num(a.ts);default:return o(u)}};return f7(Vn.parseFormat(t),h)}formatDurationFromString(a,t){const n=this.opts.signMode==="negativeLargestOnly"?-1:1,e=o=>{switch(o[0]){case"S":return"milliseconds";case"s":return"seconds";case"m":return"minutes";case"h":return"hours";case"d":return"days";case"w":return"weeks";case"M":return"months";case"y":return"years";default:return null}},l=(o,c)=>h=>{const u=e(h);if(u){const y=c.isNegativeDuration&&u!==c.largestUnit?n:1;let x;return this.opts.signMode==="negativeLargestOnly"&&u!==c.largestUnit?x="never":this.opts.signMode==="all"?x="always":x="auto",this.num(o.get(u)*y,h.length,x)}else return h},i=Vn.parseFormat(t),p=i.reduce((o,{literal:c,val:h})=>c?o:o.concat(h),[]),r=a.shiftTo(...p.map(e).filter(o=>o)),m={isNegativeDuration:r<0,largestUnit:Object.keys(r.values)[0]};return f7(i,l(r,m))}}const Kk=/[A-Za-z_+-]{1,256}(?::?\/[A-Za-z0-9_+-]{1,256}(?:\/[A-Za-z0-9_+-]{1,256})?)?/;function Qm(...s){const a=s.reduce((t,n)=>t+n.source,"");return RegExp(`^${a}$`)}function Jm(...s){return a=>s.reduce(([t,n,e],l)=>{const[i,p,r]=l(a,e);return[{...t,...i},p||n,r]},[{},null,1]).slice(0,2)}function so(s,...a){if(s==null)return[null,null];for(const[t,n]of a){const e=t.exec(s);if(e)return n(e)}return[null,null]}function Zk(...s){return(a,t)=>{const n={};let e;for(e=0;e<s.length;e++)n[s[e]]=Ji(a[t+e]);return[n,null,t+e]}}const Qk=/(?:([Zz])|([+-]\d\d)(?::?(\d\d))?)/,Lj=`(?:${Qk.source}?(?:\\[(${Kk.source})\\])?)?`,A8=/(\d\d)(?::?(\d\d)(?::?(\d\d)(?:[.,](\d{1,30}))?)?)?/,Jk=RegExp(`${A8.source}${Lj}`),_8=RegExp(`(?:[Tt]${Jk.source})?`),Sj=/([+-]\d{6}|\d{4})(?:-?(\d\d)(?:-?(\d\d))?)?/,Fj=/(\d{4})-?W(\d\d)(?:-?(\d))?/,Oj=/(\d{4})-?(\d{3})/,Ij=Zk("weekYear","weekNumber","weekDay"),Nj=Zk("year","ordinal"),Pj=/(\d{4})-(\d\d)-(\d\d)/,sB=RegExp(`${A8.source} ?(?:${Qk.source}|(${Kk.source}))?`),Rj=RegExp(`(?: ${sB.source})?`);function hm(s,a,t){const n=s[a];return Sa(n)?t:Ji(n)}function $j(s,a){return[{year:hm(s,a),month:hm(s,a+1,1),day:hm(s,a+2,1)},null,a+3]}function ao(s,a){return[{hours:hm(s,a,0),minutes:hm(s,a+1,0),seconds:hm(s,a+2,0),milliseconds:w8(s[a+3])},null,a+4]}function vh(s,a){const t=!s[a]&&!s[a+1],n=Id(s[a+1],s[a+2]),e=t?null:Jn.instance(n);return[{},e,a+3]}function Eh(s,a){const t=s[a]?Ti.create(s[a]):null;return[{},t,a+1]}const Vj=RegExp(`^T?${A8.source}$`),Hj=/^-?P(?:(?:(-?\d{1,20}(?:\.\d{1,20})?)Y)?(?:(-?\d{1,20}(?:\.\d{1,20})?)M)?(?:(-?\d{1,20}(?:\.\d{1,20})?)W)?(?:(-?\d{1,20}(?:\.\d{1,20})?)D)?(?:T(?:(-?\d{1,20}(?:\.\d{1,20})?)H)?(?:(-?\d{1,20}(?:\.\d{1,20})?)M)?(?:(-?\d{1,20})(?:[.,](-?\d{1,20}))?S)?)?)$/;function qj(s){const[a,t,n,e,l,i,p,r,m]=s,o=a[0]==="-",c=r&&r[0]==="-",h=(u,y=!1)=>u!==void 0&&(y||u&&o)?-u:u;return[{years:h(Ip(t)),months:h(Ip(n)),weeks:h(Ip(e)),days:h(Ip(l)),hours:h(Ip(i)),minutes:h(Ip(p)),seconds:h(Ip(r),r==="-0"),milliseconds:h(w8(m),c)}]}const Uj={GMT:0,EDT:-240,EST:-300,CDT:-300,CST:-360,MDT:-360,MST:-420,PDT:-420,PST:-480};function k8(s,a,t,n,e,l,i){const p={year:a.length===2?ov(Ji(a)):Ji(a),month:qk.indexOf(t)+1,day:Ji(n),hour:Ji(e),minute:Ji(l)};return i&&(p.second=Ji(i)),s&&(p.weekday=s.length>3?jk.indexOf(s)+1:Gk.indexOf(s)+1),p}const jj=/^(?:(Mon|Tue|Wed|Thu|Fri|Sat|Sun),\s)?(\d{1,2})\s(Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec)\s(\d{2,4})\s(\d\d):(\d\d)(?::(\d\d))?\s(?:(UT|GMT|[ECMP][SD]T)|([Zz])|(?:([+-]\d\d)(\d\d)))$/;function Gj(s){const[,a,t,n,e,l,i,p,r,m,o,c]=s,h=k8(a,e,n,t,l,i,p);let u;return r?u=Uj[r]:m?u=0:u=Id(o,c),[h,new Jn(u)]}function Wj(s){return s.replace(/\([^()]*\)|[\n\t]/g," ").replace(/(\s\s+)/g," ").trim()}const Yj=/^(Mon|Tue|Wed|Thu|Fri|Sat|Sun), (\d\d) (Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec) (\d{4}) (\d\d):(\d\d):(\d\d) GMT$/,Xj=/^(Monday|Tuesday|Wednesday|Thursday|Friday|Saturday|Sunday), (\d\d)-(Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec)-(\d\d) (\d\d):(\d\d):(\d\d) GMT$/,Kj=/^(Mon|Tue|Wed|Thu|Fri|Sat|Sun) (Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec) ( \d|\d\d) (\d\d):(\d\d):(\d\d) (\d{4})$/;function y7(s){const[,a,t,n,e,l,i,p]=s;return[k8(a,e,n,t,l,i,p),Jn.utcInstance]}function Zj(s){const[,a,t,n,e,l,i,p]=s;return[k8(a,p,t,n,e,l,i),Jn.utcInstance]}const Qj=Qm(Sj,_8),Jj=Qm(Fj,_8),sG=Qm(Oj,_8),aG=Qm(Jk),aB=Jm($j,ao,vh,Eh),tG=Jm(Ij,ao,vh,Eh),nG=Jm(Nj,ao,vh,Eh),eG=Jm(ao,vh,Eh);function lG(s){return so(s,[Qj,aB],[Jj,tG],[sG,nG],[aG,eG])}function iG(s){return so(Wj(s),[jj,Gj])}function pG(s){return so(s,[Yj,y7],[Xj,y7],[Kj,Zj])}function rG(s){return so(s,[Hj,qj])}const mG=Jm(ao);function oG(s){return so(s,[Vj,mG])}const cG=Qm(Pj,Rj),hG=Qm(sB),uG=Jm(ao,vh,Eh);function dG(s){return so(s,[cG,aB],[hG,uG])}const v7="Invalid Duration",tB={weeks:{days:7,hours:168,minutes:10080,seconds:10080*60,milliseconds:10080*60*1e3},days:{hours:24,minutes:1440,seconds:1440*60,milliseconds:1440*60*1e3},hours:{minutes:60,seconds:3600,milliseconds:3600*1e3},minutes:{seconds:60,milliseconds:60*1e3},seconds:{milliseconds:1e3}},gG={years:{quarters:4,months:12,weeks:52,days:365,hours:365*24,minutes:365*24*60,seconds:365*24*60*60,milliseconds:365*24*60*60*1e3},quarters:{months:3,weeks:13,days:91,hours:2184,minutes:2184*60,seconds:2184*60*60,milliseconds:2184*60*60*1e3},months:{weeks:4,days:30,hours:720,minutes:720*60,seconds:720*60*60,milliseconds:720*60*60*1e3},...tB},Ne=146097/400,Ir=146097/4800,fG={years:{quarters:4,months:12,weeks:Ne/7,days:Ne,hours:Ne*24,minutes:Ne*24*60,seconds:Ne*24*60*60,milliseconds:Ne*24*60*60*1e3},quarters:{months:3,weeks:Ne/28,days:Ne/4,hours:Ne*24/4,minutes:Ne*24*60/4,seconds:Ne*24*60*60/4,milliseconds:Ne*24*60*60*1e3/4},months:{weeks:Ir/7,days:Ir,hours:Ir*24,minutes:Ir*24*60,seconds:Ir*24*60*60,milliseconds:Ir*24*60*60*1e3},...tB},Xp=["years","quarters","months","weeks","days","hours","minutes","seconds","milliseconds"],yG=Xp.slice(0).reverse();function ei(s,a,t=!1){const n={values:t?a.values:{...s.values,...a.values||{}},loc:s.loc.clone(a.loc),conversionAccuracy:a.conversionAccuracy||s.conversionAccuracy,matrix:a.matrix||s.matrix};return new st(n)}function nB(s,a){let t=a.milliseconds??0;for(const n of yG.slice(1))a[n]&&(t+=a[n]*s[n].milliseconds);return t}function E7(s,a){const t=nB(s,a)<0?-1:1;Xp.reduceRight((n,e)=>{if(Sa(a[e]))return n;if(n){const l=a[n]*t,i=s[e][n],p=Math.floor(l/i);a[e]+=p*t,a[n]-=p*i*t}return e},null),Xp.reduce((n,e)=>{if(Sa(a[e]))return n;if(n){const l=a[n]%1;a[n]-=l,a[e]+=l*s[n][e]}return e},null)}function x7(s){const a={};for(const[t,n]of Object.entries(s))n!==0&&(a[t]=n);return a}class st{constructor(a){const t=a.conversionAccuracy==="longterm"||!1;let n=t?fG:gG;a.matrix&&(n=a.matrix),this.values=a.values,this.loc=a.loc||dt.create(),this.conversionAccuracy=t?"longterm":"casual",this.invalid=a.invalid||null,this.matrix=n,this.isLuxonDuration=!0}static fromMillis(a,t){return st.fromObject({milliseconds:a},t)}static fromObject(a,t={}){if(a==null||typeof a!="object")throw new Rn(`Duration.fromObject: argument expected to be an object, got ${a===null?"null":typeof a}`);return new st({values:Bu(a,st.normalizeUnit),loc:dt.fromObject(t),conversionAccuracy:t.conversionAccuracy,matrix:t.matrix})}static fromDurationLike(a){if(dp(a))return st.fromMillis(a);if(st.isDuration(a))return a;if(typeof a=="object")return st.fromObject(a);throw new Rn(`Unknown duration argument ${a} of type ${typeof a}`)}static fromISO(a,t){const[n]=rG(a);return n?st.fromObject(n,t):st.invalid("unparsable",`the input "${a}" can't be parsed as ISO 8601`)}static fromISOTime(a,t){const[n]=oG(a);return n?st.fromObject(n,t):st.invalid("unparsable",`the input "${a}" can't be parsed as ISO 8601`)}static invalid(a,t=null){if(!a)throw new Rn("need to specify a reason the Duration is invalid");const n=a instanceof pl?a:new pl(a,t);if(jt.throwOnInvalid)throw new qU(n);return new st({invalid:n})}static normalizeUnit(a){const t={year:"years",years:"years",quarter:"quarters",quarters:"quarters",month:"months",months:"months",week:"weeks",weeks:"weeks",day:"days",days:"days",hour:"hours",hours:"hours",minute:"minutes",minutes:"minutes",second:"seconds",seconds:"seconds",millisecond:"milliseconds",milliseconds:"milliseconds"}[a&&a.toLowerCase()];if(!t)throw new ck(a);return t}static isDuration(a){return a&&a.isLuxonDuration||!1}get locale(){return this.isValid?this.loc.locale:null}get numberingSystem(){return this.isValid?this.loc.numberingSystem:null}toFormat(a,t={}){const n={...t,floor:t.round!==!1&&t.floor!==!1};return this.isValid?Vn.create(this.loc,n).formatDurationFromString(this,a):v7}toHuman(a={}){if(!this.isValid)return v7;const t=a.showZeros!==!1,n=Xp.map(e=>{const l=this.values[e];return Sa(l)||l===0&&!t?null:this.loc.numberFormatter({style:"unit",unitDisplay:"long",...a,unit:e.slice(0,-1)}).format(l)}).filter(e=>e);return this.loc.listFormatter({type:"conjunction",style:a.listStyle||"narrow",...a}).format(n)}toObject(){return this.isValid?{...this.values}:{}}toISO(){if(!this.isValid)return null;let a="P";return this.years!==0&&(a+=this.years+"Y"),(this.months!==0||this.quarters!==0)&&(a+=this.months+this.quarters*3+"M"),this.weeks!==0&&(a+=this.weeks+"W"),this.days!==0&&(a+=this.days+"D"),(this.hours!==0||this.minutes!==0||this.seconds!==0||this.milliseconds!==0)&&(a+="T"),this.hours!==0&&(a+=this.hours+"H"),this.minutes!==0&&(a+=this.minutes+"M"),(this.seconds!==0||this.milliseconds!==0)&&(a+=b8(this.seconds+this.milliseconds/1e3,3)+"S"),a==="P"&&(a+="T0S"),a}toISOTime(a={}){if(!this.isValid)return null;const t=this.toMillis();return t<0||t>=864e5?null:(a={suppressMilliseconds:!1,suppressSeconds:!1,includePrefix:!1,format:"extended",...a,includeOffset:!1},La.fromMillis(t,{zone:"UTC"}).toISOTime(a))}toJSON(){return this.toISO()}toString(){return this.toISO()}[Symbol.for("nodejs.util.inspect.custom")](){return this.isValid?`Duration { values: ${JSON.stringify(this.values)} }`:`Duration { Invalid, reason: ${this.invalidReason} }`}toMillis(){return this.isValid?nB(this.matrix,this.values):NaN}valueOf(){return this.toMillis()}plus(a){if(!this.isValid)return this;const t=st.fromDurationLike(a),n={};for(const e of Xp)(Lm(t.values,e)||Lm(this.values,e))&&(n[e]=t.get(e)+this.get(e));return ei(this,{values:n},!0)}minus(a){if(!this.isValid)return this;const t=st.fromDurationLike(a);return this.plus(t.negate())}mapUnits(a){if(!this.isValid)return this;const t={};for(const n of Object.keys(this.values))t[n]=Hk(a(this.values[n],n));return ei(this,{values:t},!0)}get(a){return this[st.normalizeUnit(a)]}set(a){if(!this.isValid)return this;const t={...this.values,...Bu(a,st.normalizeUnit)};return ei(this,{values:t})}reconfigure({locale:a,numberingSystem:t,conversionAccuracy:n,matrix:e}={}){const i={loc:this.loc.clone({locale:a,numberingSystem:t}),matrix:e,conversionAccuracy:n};return ei(this,i)}as(a){return this.isValid?this.shiftTo(a).get(a):NaN}normalize(){if(!this.isValid)return this;const a=this.toObject();return E7(this.matrix,a),ei(this,{values:a},!0)}rescale(){if(!this.isValid)return this;const a=x7(this.normalize().shiftToAll().toObject());return ei(this,{values:a},!0)}shiftTo(...a){if(!this.isValid)return this;if(a.length===0)return this;a=a.map(i=>st.normalizeUnit(i));const t={},n={},e=this.toObject();let l;for(const i of Xp)if(a.indexOf(i)>=0){l=i;let p=0;for(const m in n)p+=this.matrix[m][i]*n[m],n[m]=0;dp(e[i])&&(p+=e[i]);const r=Math.trunc(p);t[i]=r,n[i]=(p*1e3-r*1e3)/1e3}else dp(e[i])&&(n[i]=e[i]);for(const i in n)n[i]!==0&&(t[l]+=i===l?n[i]:n[i]/this.matrix[l][i]);return E7(this.matrix,t),ei(this,{values:t},!0)}shiftToAll(){return this.isValid?this.shiftTo("years","months","weeks","days","hours","minutes","seconds","milliseconds"):this}negate(){if(!this.isValid)return this;const a={};for(const t of Object.keys(this.values))a[t]=this.values[t]===0?0:-this.values[t];return ei(this,{values:a},!0)}removeZeros(){if(!this.isValid)return this;const a=x7(this.values);return ei(this,{values:a},!0)}get years(){return this.isValid?this.values.years||0:NaN}get quarters(){return this.isValid?this.values.quarters||0:NaN}get months(){return this.isValid?this.values.months||0:NaN}get weeks(){return this.isValid?this.values.weeks||0:NaN}get days(){return this.isValid?this.values.days||0:NaN}get hours(){return this.isValid?this.values.hours||0:NaN}get minutes(){return this.isValid?this.values.minutes||0:NaN}get seconds(){return this.isValid?this.values.seconds||0:NaN}get milliseconds(){return this.isValid?this.values.milliseconds||0:NaN}get isValid(){return this.invalid===null}get invalidReason(){return this.invalid?this.invalid.reason:null}get invalidExplanation(){return this.invalid?this.invalid.explanation:null}equals(a){if(!this.isValid||!a.isValid||!this.loc.equals(a.loc))return!1;function t(n,e){return n===void 0||n===0?e===void 0||e===0:n===e}for(const n of Xp)if(!t(this.values[n],a.values[n]))return!1;return!0}}const Nr="Invalid Interval";function vG(s,a){return!s||!s.isValid?qt.invalid("missing or invalid start"):!a||!a.isValid?qt.invalid("missing or invalid end"):a<s?qt.invalid("end before start",`The end of an interval must be after its start, but you had start=${s.toISO()} and end=${a.toISO()}`):null}class qt{constructor(a){this.s=a.start,this.e=a.end,this.invalid=a.invalid||null,this.isLuxonInterval=!0}static invalid(a,t=null){if(!a)throw new Rn("need to specify a reason the Interval is invalid");const n=a instanceof pl?a:new pl(a,t);if(jt.throwOnInvalid)throw new HU(n);return new qt({invalid:n})}static fromDateTimes(a,t){const n=wo(a),e=wo(t),l=vG(n,e);return l??new qt({start:n,end:e})}static after(a,t){const n=st.fromDurationLike(t),e=wo(a);return qt.fromDateTimes(e,e.plus(n))}static before(a,t){const n=st.fromDurationLike(t),e=wo(a);return qt.fromDateTimes(e.minus(n),e)}static fromISO(a,t){const[n,e]=(a||"").split("/",2);if(n&&e){let l,i;try{l=La.fromISO(n,t),i=l.isValid}catch{i=!1}let p,r;try{p=La.fromISO(e,t),r=p.isValid}catch{r=!1}if(i&&r)return qt.fromDateTimes(l,p);if(i){const m=st.fromISO(e,t);if(m.isValid)return qt.after(l,m)}else if(r){const m=st.fromISO(n,t);if(m.isValid)return qt.before(p,m)}}return qt.invalid("unparsable",`the input "${a}" can't be parsed as ISO 8601`)}static isInterval(a){return a&&a.isLuxonInterval||!1}get start(){return this.isValid?this.s:null}get end(){return this.isValid?this.e:null}get lastDateTime(){return this.isValid&&this.e?this.e.minus(1):null}get isValid(){return this.invalidReason===null}get invalidReason(){return this.invalid?this.invalid.reason:null}get invalidExplanation(){return this.invalid?this.invalid.explanation:null}length(a="milliseconds"){return this.isValid?this.toDuration(a).get(a):NaN}count(a="milliseconds",t){if(!this.isValid)return NaN;const n=this.start.startOf(a,t);let e;return t?.useLocaleWeeks?e=this.end.reconfigure({locale:n.locale}):e=this.end,e=e.startOf(a,t),Math.floor(e.diff(n,a).get(a))+(e.valueOf()!==this.end.valueOf())}hasSame(a){return this.isValid?this.isEmpty()||this.e.minus(1).hasSame(this.s,a):!1}isEmpty(){return this.s.valueOf()===this.e.valueOf()}isAfter(a){return this.isValid?this.s>a:!1}isBefore(a){return this.isValid?this.e<=a:!1}contains(a){return this.isValid?this.s<=a&&this.e>a:!1}set({start:a,end:t}={}){return this.isValid?qt.fromDateTimes(a||this.s,t||this.e):this}splitAt(...a){if(!this.isValid)return[];const t=a.map(wo).filter(i=>this.contains(i)).sort((i,p)=>i.toMillis()-p.toMillis()),n=[];let{s:e}=this,l=0;for(;e<this.e;){const i=t[l]||this.e,p=+i>+this.e?this.e:i;n.push(qt.fromDateTimes(e,p)),e=p,l+=1}return n}splitBy(a){const t=st.fromDurationLike(a);if(!this.isValid||!t.isValid||t.as("milliseconds")===0)return[];let{s:n}=this,e=1,l;const i=[];for(;n<this.e;){const p=this.start.plus(t.mapUnits(r=>r*e));l=+p>+this.e?this.e:p,i.push(qt.fromDateTimes(n,l)),n=l,e+=1}return i}divideEqually(a){return this.isValid?this.splitBy(this.length()/a).slice(0,a):[]}overlaps(a){return this.e>a.s&&this.s<a.e}abutsStart(a){return this.isValid?+this.e==+a.s:!1}abutsEnd(a){return this.isValid?+a.e==+this.s:!1}engulfs(a){return this.isValid?this.s<=a.s&&this.e>=a.e:!1}equals(a){return!this.isValid||!a.isValid?!1:this.s.equals(a.s)&&this.e.equals(a.e)}intersection(a){if(!this.isValid)return this;const t=this.s>a.s?this.s:a.s,n=this.e<a.e?this.e:a.e;return t>=n?null:qt.fromDateTimes(t,n)}union(a){if(!this.isValid)return this;const t=this.s<a.s?this.s:a.s,n=this.e>a.e?this.e:a.e;return qt.fromDateTimes(t,n)}static merge(a){const[t,n]=a.sort((e,l)=>e.s-l.s).reduce(([e,l],i)=>l?l.overlaps(i)||l.abutsStart(i)?[e,l.union(i)]:[e.concat([l]),i]:[e,i],[[],null]);return n&&t.push(n),t}static xor(a){let t=null,n=0;const e=[],l=a.map(r=>[{time:r.s,type:"s"},{time:r.e,type:"e"}]),i=Array.prototype.concat(...l),p=i.sort((r,m)=>r.time-m.time);for(const r of p)n+=r.type==="s"?1:-1,n===1?t=r.time:(t&&+t!=+r.time&&e.push(qt.fromDateTimes(t,r.time)),t=null);return qt.merge(e)}difference(...a){return qt.xor([this].concat(a)).map(t=>this.intersection(t)).filter(t=>t&&!t.isEmpty())}toString(){return this.isValid?`[${this.s.toISO()} – ${this.e.toISO()})`:Nr}[Symbol.for("nodejs.util.inspect.custom")](){return this.isValid?`Interval { start: ${this.s.toISO()}, end: ${this.e.toISO()} }`:`Interval { Invalid, reason: ${this.invalidReason} }`}toLocaleString(a=Au,t={}){return this.isValid?Vn.create(this.s.loc.clone(t),a).formatInterval(this):Nr}toISO(a){return this.isValid?`${this.s.toISO(a)}/${this.e.toISO(a)}`:Nr}toISODate(){return this.isValid?`${this.s.toISODate()}/${this.e.toISODate()}`:Nr}toISOTime(a){return this.isValid?`${this.s.toISOTime(a)}/${this.e.toISOTime(a)}`:Nr}toFormat(a,{separator:t=" – "}={}){return this.isValid?`${this.s.toFormat(a)}${t}${this.e.toFormat(a)}`:Nr}toDuration(a,t){return this.isValid?this.e.diff(this.s,a,t):st.invalid(this.invalidReason)}mapEndpoints(a){return qt.fromDateTimes(a(this.s),a(this.e))}}class o0{static hasDST(a=jt.defaultZone){const t=La.now().setZone(a).set({month:12});return!a.isUniversal&&t.offset!==t.set({month:6}).offset}static isValidIANAZone(a){return Ti.isValidZone(a)}static normalizeZone(a){return pp(a,jt.defaultZone)}static getStartOfWeek({locale:a=null,locObj:t=null}={}){return(t||dt.create(a)).getStartOfWeek()}static getMinimumDaysInFirstWeek({locale:a=null,locObj:t=null}={}){return(t||dt.create(a)).getMinDaysInFirstWeek()}static getWeekendWeekdays({locale:a=null,locObj:t=null}={}){return(t||dt.create(a)).getWeekendDays().slice()}static months(a="long",{locale:t=null,numberingSystem:n=null,locObj:e=null,outputCalendar:l="gregory"}={}){return(e||dt.create(t,n,l)).months(a)}static monthsFormat(a="long",{locale:t=null,numberingSystem:n=null,locObj:e=null,outputCalendar:l="gregory"}={}){return(e||dt.create(t,n,l)).months(a,!0)}static weekdays(a="long",{locale:t=null,numberingSystem:n=null,locObj:e=null}={}){return(e||dt.create(t,n,null)).weekdays(a)}static weekdaysFormat(a="long",{locale:t=null,numberingSystem:n=null,locObj:e=null}={}){return(e||dt.create(t,n,null)).weekdays(a,!0)}static meridiems({locale:a=null}={}){return dt.create(a).meridiems()}static eras(a="short",{locale:t=null}={}){return dt.create(t,null,"gregory").eras(a)}static features(){return{relative:Rk(),localeWeek:$k()}}}function w7(s,a){const t=e=>e.toUTC(0,{keepLocalTime:!0}).startOf("day").valueOf(),n=t(a)-t(s);return Math.floor(st.fromMillis(n).as("days"))}function EG(s,a,t){const n=[["years",(r,m)=>m.year-r.year],["quarters",(r,m)=>m.quarter-r.quarter+(m.year-r.year)*4],["months",(r,m)=>m.month-r.month+(m.year-r.year)*12],["weeks",(r,m)=>{const o=w7(r,m);return(o-o%7)/7}],["days",w7]],e={},l=s;let i,p;for(const[r,m]of n)t.indexOf(r)>=0&&(i=r,e[r]=m(s,a),p=l.plus(e),p>a?(e[r]--,s=l.plus(e),s>a&&(p=s,e[r]--,s=l.plus(e))):s=p);return[s,e,p,i]}function xG(s,a,t,n){let[e,l,i,p]=EG(s,a,t);const r=a-e,m=t.filter(c=>["hours","minutes","seconds","milliseconds"].indexOf(c)>=0);m.length===0&&(i<a&&(i=e.plus({[p]:1})),i!==e&&(l[p]=(l[p]||0)+r/(i-e)));const o=st.fromObject(l,n);return m.length>0?st.fromMillis(r,n).shiftTo(...m).plus(o):o}const wG="missing Intl.DateTimeFormat.formatToParts support";function lt(s,a=t=>t){return{regex:s,deser:([t])=>a(cj(t))}}const bG=" ",eB=`[ ${bG}]`,lB=new RegExp(eB,"g");function AG(s){return s.replace(/\./g,"\\.?").replace(lB,eB)}function b7(s){return s.replace(/\./g,"").replace(lB," ").toLowerCase()}function al(s,a){return s===null?null:{regex:RegExp(s.map(AG).join("|")),deser:([t])=>s.findIndex(n=>b7(t)===b7(n))+a}}function A7(s,a){return{regex:s,deser:([,t,n])=>Id(t,n),groups:a}}function c0(s){return{regex:s,deser:([a])=>a}}function _G(s){return s.replace(/[\-\[\]{}()*+?.,\\\^$|#\s]/g,"\\$&")}function kG(s,a){const t=sl(a),n=sl(a,"{2}"),e=sl(a,"{3}"),l=sl(a,"{4}"),i=sl(a,"{6}"),p=sl(a,"{1,2}"),r=sl(a,"{1,3}"),m=sl(a,"{1,6}"),o=sl(a,"{1,9}"),c=sl(a,"{2,4}"),h=sl(a,"{4,6}"),u=b=>({regex:RegExp(_G(b.val)),deser:([_])=>_,literal:!0}),x=(b=>{if(s.literal)return u(b);switch(b.val){case"G":return al(a.eras("short"),0);case"GG":return al(a.eras("long"),0);case"y":return lt(m);case"yy":return lt(c,ov);case"yyyy":return lt(l);case"yyyyy":return lt(h);case"yyyyyy":return lt(i);case"M":return lt(p);case"MM":return lt(n);case"MMM":return al(a.months("short",!0),1);case"MMMM":return al(a.months("long",!0),1);case"L":return lt(p);case"LL":return lt(n);case"LLL":return al(a.months("short",!1),1);case"LLLL":return al(a.months("long",!1),1);case"d":return lt(p);case"dd":return lt(n);case"o":return lt(r);case"ooo":return lt(e);case"HH":return lt(n);case"H":return lt(p);case"hh":return lt(n);case"h":return lt(p);case"mm":return lt(n);case"m":return lt(p);case"q":return lt(p);case"qq":return lt(n);case"s":return lt(p);case"ss":return lt(n);case"S":return lt(r);case"SSS":return lt(e);case"u":return c0(o);case"uu":return c0(p);case"uuu":return lt(t);case"a":return al(a.meridiems(),0);case"kkkk":return lt(l);case"kk":return lt(c,ov);case"W":return lt(p);case"WW":return lt(n);case"E":case"c":return lt(t);case"EEE":return al(a.weekdays("short",!1),1);case"EEEE":return al(a.weekdays("long",!1),1);case"ccc":return al(a.weekdays("short",!0),1);case"cccc":return al(a.weekdays("long",!0),1);case"Z":case"ZZ":return A7(new RegExp(`([+-]${p.source})(?::(${n.source}))?`),2);case"ZZZ":return A7(new RegExp(`([+-]${p.source})(${n.source})?`),2);case"z":return c0(/[a-z_+-/]{1,256}?/i);case" ":return c0(/[^\S\n\r]/);default:return u(b)}})(s)||{invalidReason:wG};return x.token=s,x}const BG={year:{"2-digit":"yy",numeric:"yyyyy"},month:{numeric:"M","2-digit":"MM",short:"MMM",long:"MMMM"},day:{numeric:"d","2-digit":"dd"},weekday:{short:"EEE",long:"EEEE"},dayperiod:"a",dayPeriod:"a",hour12:{numeric:"h","2-digit":"hh"},hour24:{numeric:"H","2-digit":"HH"},minute:{numeric:"m","2-digit":"mm"},second:{numeric:"s","2-digit":"ss"},timeZoneName:{long:"ZZZZZ",short:"ZZZ"}};function MG(s,a,t){const{type:n,value:e}=s;if(n==="literal"){const r=/^\s+$/.test(e);return{literal:!r,val:r?" ":e}}const l=a[n];let i=n;n==="hour"&&(a.hour12!=null?i=a.hour12?"hour12":"hour24":a.hourCycle!=null?a.hourCycle==="h11"||a.hourCycle==="h12"?i="hour12":i="hour24":i=t.hour12?"hour12":"hour24");let p=BG[i];if(typeof p=="object"&&(p=p[l]),p)return{literal:!1,val:p}}function TG(s){return[`^${s.map(t=>t.regex).reduce((t,n)=>`${t}(${n.source})`,"")}$`,s]}function CG(s,a,t){const n=s.match(a);if(n){const e={};let l=1;for(const i in t)if(Lm(t,i)){const p=t[i],r=p.groups?p.groups+1:1;!p.literal&&p.token&&(e[p.token.val[0]]=p.deser(n.slice(l,l+r))),l+=r}return[n,e]}else return[n,{}]}function zG(s){const a=l=>{switch(l){case"S":return"millisecond";case"s":return"second";case"m":return"minute";case"h":case"H":return"hour";case"d":return"day";case"o":return"ordinal";case"L":case"M":return"month";case"y":return"year";case"E":case"c":return"weekday";case"W":return"weekNumber";case"k":return"weekYear";case"q":return"quarter";default:return null}};let t=null,n;return Sa(s.z)||(t=Ti.create(s.z)),Sa(s.Z)||(t||(t=new Jn(s.Z)),n=s.Z),Sa(s.q)||(s.M=(s.q-1)*3+1),Sa(s.h)||(s.h<12&&s.a===1?s.h+=12:s.h===12&&s.a===0&&(s.h=0)),s.G===0&&s.y&&(s.y=-s.y),Sa(s.u)||(s.S=w8(s.u)),[Object.keys(s).reduce((l,i)=>{const p=a(i);return p&&(l[p]=s[i]),l},{}),t,n]}let H1=null;function DG(){return H1||(H1=La.fromMillis(1555555555555)),H1}function LG(s,a){if(s.literal)return s;const t=Vn.macroTokenToFormatOpts(s.val),n=mB(t,a);return n==null||n.includes(void 0)?s:n}function iB(s,a){return Array.prototype.concat(...s.map(t=>LG(t,a)))}class pB{constructor(a,t){if(this.locale=a,this.format=t,this.tokens=iB(Vn.parseFormat(t),a),this.units=this.tokens.map(n=>kG(n,a)),this.disqualifyingUnit=this.units.find(n=>n.invalidReason),!this.disqualifyingUnit){const[n,e]=TG(this.units);this.regex=RegExp(n,"i"),this.handlers=e}}explainFromTokens(a){if(this.isValid){const[t,n]=CG(a,this.regex,this.handlers),[e,l,i]=n?zG(n):[null,null,void 0];if(Lm(n,"a")&&Lm(n,"H"))throw new Zr("Can't include meridiem when specifying 24-hour format");return{input:a,tokens:this.tokens,regex:this.regex,rawMatches:t,matches:n,result:e,zone:l,specificOffset:i}}else return{input:a,tokens:this.tokens,invalidReason:this.invalidReason}}get isValid(){return!this.disqualifyingUnit}get invalidReason(){return this.disqualifyingUnit?this.disqualifyingUnit.invalidReason:null}}function rB(s,a,t){return new pB(s,t).explainFromTokens(a)}function SG(s,a,t){const{result:n,zone:e,specificOffset:l,invalidReason:i}=rB(s,a,t);return[n,e,l,i]}function mB(s,a){if(!s)return null;const n=Vn.create(a,s).dtFormatter(DG()),e=n.formatToParts(),l=n.resolvedOptions();return e.map(i=>MG(i,s,l))}const q1="Invalid DateTime",FG=864e13;function Lo(s){return new pl("unsupported zone",`the zone "${s.name}" is not supported`)}function U1(s){return s.weekData===null&&(s.weekData=_u(s.c)),s.weekData}function j1(s){return s.localWeekData===null&&(s.localWeekData=_u(s.c,s.loc.getMinDaysInFirstWeek(),s.loc.getStartOfWeek())),s.localWeekData}function Np(s,a){const t={ts:s.ts,zone:s.zone,c:s.c,o:s.o,loc:s.loc,invalid:s.invalid};return new La({...t,...a,old:t})}function oB(s,a,t){let n=s-a*60*1e3;const e=t.offset(n);if(a===e)return[n,a];n-=(e-a)*60*1e3;const l=t.offset(n);return e===l?[n,e]:[s-Math.min(e,l)*60*1e3,Math.max(e,l)]}function h0(s,a){s+=a*60*1e3;const t=new Date(s);return{year:t.getUTCFullYear(),month:t.getUTCMonth()+1,day:t.getUTCDate(),hour:t.getUTCHours(),minute:t.getUTCMinutes(),second:t.getUTCSeconds(),millisecond:t.getUTCMilliseconds()}}function j0(s,a,t){return oB(Od(s),a,t)}function _7(s,a){const t=s.o,n=s.c.year+Math.trunc(a.years),e=s.c.month+Math.trunc(a.months)+Math.trunc(a.quarters)*3,l={...s.c,year:n,month:e,day:Math.min(s.c.day,ku(n,e))+Math.trunc(a.days)+Math.trunc(a.weeks)*7},i=st.fromObject({years:a.years-Math.trunc(a.years),quarters:a.quarters-Math.trunc(a.quarters),months:a.months-Math.trunc(a.months),weeks:a.weeks-Math.trunc(a.weeks),days:a.days-Math.trunc(a.days),hours:a.hours,minutes:a.minutes,seconds:a.seconds,milliseconds:a.milliseconds}).as("milliseconds"),p=Od(l);let[r,m]=oB(p,t,s.zone);return i!==0&&(r+=i,m=s.zone.offset(r)),{ts:r,o:m}}function Pr(s,a,t,n,e,l){const{setZone:i,zone:p}=t;if(s&&Object.keys(s).length!==0||a){const r=a||p,m=La.fromObject(s,{...t,zone:r,specificOffset:l});return i?m:m.setZone(p)}else return La.invalid(new pl("unparsable",`the input "${e}" can't be parsed as ${n}`))}function u0(s,a,t=!0){return s.isValid?Vn.create(dt.create("en-US"),{allowZ:t,forceSimple:!0}).formatDateTimeFromString(s,a):null}function G1(s,a,t){const n=s.c.year>9999||s.c.year<0;let e="";if(n&&s.c.year>=0&&(e+="+"),e+=tn(s.c.year,n?6:4),t==="year")return e;if(a){if(e+="-",e+=tn(s.c.month),t==="month")return e;e+="-"}else if(e+=tn(s.c.month),t==="month")return e;return e+=tn(s.c.day),e}function k7(s,a,t,n,e,l,i){let p=!t||s.c.millisecond!==0||s.c.second!==0,r="";switch(i){case"day":case"month":case"year":break;default:if(r+=tn(s.c.hour),i==="hour")break;if(a){if(r+=":",r+=tn(s.c.minute),i==="minute")break;p&&(r+=":",r+=tn(s.c.second))}else{if(r+=tn(s.c.minute),i==="minute")break;p&&(r+=tn(s.c.second))}if(i==="second")break;p&&(!n||s.c.millisecond!==0)&&(r+=".",r+=tn(s.c.millisecond,3))}return e&&(s.isOffsetFixed&&s.offset===0&&!l?r+="Z":s.o<0?(r+="-",r+=tn(Math.trunc(-s.o/60)),r+=":",r+=tn(Math.trunc(-s.o%60))):(r+="+",r+=tn(Math.trunc(s.o/60)),r+=":",r+=tn(Math.trunc(s.o%60)))),l&&(r+="["+s.zone.ianaName+"]"),r}const cB={month:1,day:1,hour:0,minute:0,second:0,millisecond:0},OG={weekNumber:1,weekday:1,hour:0,minute:0,second:0,millisecond:0},IG={ordinal:1,hour:0,minute:0,second:0,millisecond:0},G0=["year","month","day","hour","minute","second","millisecond"],NG=["weekYear","weekNumber","weekday","hour","minute","second","millisecond"],PG=["year","ordinal","hour","minute","second","millisecond"];function W0(s){const a={year:"year",years:"year",month:"month",months:"month",day:"day",days:"day",hour:"hour",hours:"hour",minute:"minute",minutes:"minute",quarter:"quarter",quarters:"quarter",second:"second",seconds:"second",millisecond:"millisecond",milliseconds:"millisecond",weekday:"weekday",weekdays:"weekday",weeknumber:"weekNumber",weeksnumber:"weekNumber",weeknumbers:"weekNumber",weekyear:"weekYear",weekyears:"weekYear",ordinal:"ordinal"}[s.toLowerCase()];if(!a)throw new ck(s);return a}function B7(s){switch(s.toLowerCase()){case"localweekday":case"localweekdays":return"localWeekday";case"localweeknumber":case"localweeknumbers":return"localWeekNumber";case"localweekyear":case"localweekyears":return"localWeekYear";default:return W0(s)}}function RG(s){if(So===void 0&&(So=jt.now()),s.type!=="iana")return s.offset(So);const a=s.name;let t=cv.get(a);return t===void 0&&(t=s.offset(So),cv.set(a,t)),t}function M7(s,a){const t=pp(a.zone,jt.defaultZone);if(!t.isValid)return La.invalid(Lo(t));const n=dt.fromObject(a);let e,l;if(Sa(s.year))e=jt.now();else{for(const r of G0)Sa(s[r])&&(s[r]=cB[r]);const i=Nk(s)||Pk(s);if(i)return La.invalid(i);const p=RG(t);[e,l]=j0(s,p,t)}return new La({ts:e,zone:t,loc:n,o:l})}function T7(s,a,t){const n=Sa(t.round)?!0:t.round,e=Sa(t.rounding)?"trunc":t.rounding,l=(p,r)=>(p=b8(p,n||t.calendary?0:2,t.calendary?"round":e),a.loc.clone(t).relFormatter(t).format(p,r)),i=p=>t.calendary?a.hasSame(s,p)?0:a.startOf(p).diff(s.startOf(p),p).get(p):a.diff(s,p).get(p);if(t.unit)return l(i(t.unit),t.unit);for(const p of t.units){const r=i(p);if(Math.abs(r)>=1)return l(r,p)}return l(s>a?-0:0,t.units[t.units.length-1])}function C7(s){let a={},t;return s.length>0&&typeof s[s.length-1]=="object"?(a=s[s.length-1],t=Array.from(s).slice(0,s.length-1)):t=Array.from(s),[a,t]}let So;const cv=new Map;class La{constructor(a){const t=a.zone||jt.defaultZone;let n=a.invalid||(Number.isNaN(a.ts)?new pl("invalid input"):null)||(t.isValid?null:Lo(t));this.ts=Sa(a.ts)?jt.now():a.ts;let e=null,l=null;if(!n)if(a.old&&a.old.ts===this.ts&&a.old.zone.equals(t))[e,l]=[a.old.c,a.old.o];else{const p=dp(a.o)&&!a.old?a.o:t.offset(this.ts);e=h0(this.ts,p),n=Number.isNaN(e.year)?new pl("invalid input"):null,e=n?null:e,l=n?null:p}this._zone=t,this.loc=a.loc||dt.create(),this.invalid=n,this.weekData=null,this.localWeekData=null,this.c=e,this.o=l,this.isLuxonDateTime=!0}static now(){return new La({})}static local(){const[a,t]=C7(arguments),[n,e,l,i,p,r,m]=t;return M7({year:n,month:e,day:l,hour:i,minute:p,second:r,millisecond:m},a)}static utc(){const[a,t]=C7(arguments),[n,e,l,i,p,r,m]=t;return a.zone=Jn.utcInstance,M7({year:n,month:e,day:l,hour:i,minute:p,second:r,millisecond:m},a)}static fromJSDate(a,t={}){const n=fj(a)?a.valueOf():NaN;if(Number.isNaN(n))return La.invalid("invalid input");const e=pp(t.zone,jt.defaultZone);return e.isValid?new La({ts:n,zone:e,loc:dt.fromObject(t)}):La.invalid(Lo(e))}static fromMillis(a,t={}){if(dp(a))return a<-864e13||a>FG?La.invalid("Timestamp out of range"):new La({ts:a,zone:pp(t.zone,jt.defaultZone),loc:dt.fromObject(t)});throw new Rn(`fromMillis requires a numerical input, but received a ${typeof a} with value ${a}`)}static fromSeconds(a,t={}){if(dp(a))return new La({ts:a*1e3,zone:pp(t.zone,jt.defaultZone),loc:dt.fromObject(t)});throw new Rn("fromSeconds requires a numerical input")}static fromObject(a,t={}){a=a||{};const n=pp(t.zone,jt.defaultZone);if(!n.isValid)return La.invalid(Lo(n));const e=dt.fromObject(t),l=Bu(a,B7),{minDaysInFirstWeek:i,startOfWeek:p}=u7(l,e),r=jt.now(),m=Sa(t.specificOffset)?n.offset(r):t.specificOffset,o=!Sa(l.ordinal),c=!Sa(l.year),h=!Sa(l.month)||!Sa(l.day),u=c||h,y=l.weekYear||l.weekNumber;if((u||o)&&y)throw new Zr("Can't mix weekYear/weekNumber units with year/month/day or ordinals");if(h&&o)throw new Zr("Can't mix ordinal dates with month/day");const x=y||l.weekday&&!u;let b,_,T=h0(r,m);x?(b=NG,_=OG,T=_u(T,i,p)):o?(b=PG,_=IG,T=V1(T)):(b=G0,_=cB);let A=!1;for(const G of b){const X=l[G];Sa(X)?A?l[G]=_[G]:l[G]=T[G]:A=!0}const C=x?uj(l,i,p):o?dj(l):Nk(l),I=C||Pk(l);if(I)return La.invalid(I);const $=x?c7(l,i,p):o?h7(l):l,[U,L]=j0($,m,n),P=new La({ts:U,zone:n,o:L,loc:e});return l.weekday&&u&&a.weekday!==P.weekday?La.invalid("mismatched weekday",`you can't specify both a weekday of ${l.weekday} and a date of ${P.toISO()}`):P.isValid?P:La.invalid(P.invalid)}static fromISO(a,t={}){const[n,e]=lG(a);return Pr(n,e,t,"ISO 8601",a)}static fromRFC2822(a,t={}){const[n,e]=iG(a);return Pr(n,e,t,"RFC 2822",a)}static fromHTTP(a,t={}){const[n,e]=pG(a);return Pr(n,e,t,"HTTP",t)}static fromFormat(a,t,n={}){if(Sa(a)||Sa(t))throw new Rn("fromFormat requires an input string and a format");const{locale:e=null,numberingSystem:l=null}=n,i=dt.fromOpts({locale:e,numberingSystem:l,defaultToEN:!0}),[p,r,m,o]=SG(i,a,t);return o?La.invalid(o):Pr(p,r,n,`format ${t}`,a,m)}static fromString(a,t,n={}){return La.fromFormat(a,t,n)}static fromSQL(a,t={}){const[n,e]=dG(a);return Pr(n,e,t,"SQL",a)}static invalid(a,t=null){if(!a)throw new Rn("need to specify a reason the DateTime is invalid");const n=a instanceof pl?a:new pl(a,t);if(jt.throwOnInvalid)throw new VU(n);return new La({invalid:n})}static isDateTime(a){return a&&a.isLuxonDateTime||!1}static parseFormatForOpts(a,t={}){const n=mB(a,dt.fromObject(t));return n?n.map(e=>e?e.val:null).join(""):null}static expandFormat(a,t={}){return iB(Vn.parseFormat(a),dt.fromObject(t)).map(e=>e.val).join("")}static resetCache(){So=void 0,cv.clear()}get(a){return this[a]}get isValid(){return this.invalid===null}get invalidReason(){return this.invalid?this.invalid.reason:null}get invalidExplanation(){return this.invalid?this.invalid.explanation:null}get locale(){return this.isValid?this.loc.locale:null}get numberingSystem(){return this.isValid?this.loc.numberingSystem:null}get outputCalendar(){return this.isValid?this.loc.outputCalendar:null}get zone(){return this._zone}get zoneName(){return this.isValid?this.zone.name:null}get year(){return this.isValid?this.c.year:NaN}get quarter(){return this.isValid?Math.ceil(this.c.month/3):NaN}get month(){return this.isValid?this.c.month:NaN}get day(){return this.isValid?this.c.day:NaN}get hour(){return this.isValid?this.c.hour:NaN}get minute(){return this.isValid?this.c.minute:NaN}get second(){return this.isValid?this.c.second:NaN}get millisecond(){return this.isValid?this.c.millisecond:NaN}get weekYear(){return this.isValid?U1(this).weekYear:NaN}get weekNumber(){return this.isValid?U1(this).weekNumber:NaN}get weekday(){return this.isValid?U1(this).weekday:NaN}get isWeekend(){return this.isValid&&this.loc.getWeekendDays().includes(this.weekday)}get localWeekday(){return this.isValid?j1(this).weekday:NaN}get localWeekNumber(){return this.isValid?j1(this).weekNumber:NaN}get localWeekYear(){return this.isValid?j1(this).weekYear:NaN}get ordinal(){return this.isValid?V1(this.c).ordinal:NaN}get monthShort(){return this.isValid?o0.months("short",{locObj:this.loc})[this.month-1]:null}get monthLong(){return this.isValid?o0.months("long",{locObj:this.loc})[this.month-1]:null}get weekdayShort(){return this.isValid?o0.weekdays("short",{locObj:this.loc})[this.weekday-1]:null}get weekdayLong(){return this.isValid?o0.weekdays("long",{locObj:this.loc})[this.weekday-1]:null}get offset(){return this.isValid?+this.o:NaN}get offsetNameShort(){return this.isValid?this.zone.offsetName(this.ts,{format:"short",locale:this.locale}):null}get offsetNameLong(){return this.isValid?this.zone.offsetName(this.ts,{format:"long",locale:this.locale}):null}get isOffsetFixed(){return this.isValid?this.zone.isUniversal:null}get isInDST(){return this.isOffsetFixed?!1:this.offset>this.set({month:1,day:1}).offset||this.offset>this.set({month:5}).offset}getPossibleOffsets(){if(!this.isValid||this.isOffsetFixed)return[this];const a=864e5,t=6e4,n=Od(this.c),e=this.zone.offset(n-a),l=this.zone.offset(n+a),i=this.zone.offset(n-e*t),p=this.zone.offset(n-l*t);if(i===p)return[this];const r=n-i*t,m=n-p*t,o=h0(r,i),c=h0(m,p);return o.hour===c.hour&&o.minute===c.minute&&o.second===c.second&&o.millisecond===c.millisecond?[Np(this,{ts:r}),Np(this,{ts:m})]:[this]}get isInLeapYear(){return yh(this.year)}get daysInMonth(){return ku(this.year,this.month)}get daysInYear(){return this.isValid?cm(this.year):NaN}get weeksInWeekYear(){return this.isValid?Bc(this.weekYear):NaN}get weeksInLocalWeekYear(){return this.isValid?Bc(this.localWeekYear,this.loc.getMinDaysInFirstWeek(),this.loc.getStartOfWeek()):NaN}resolvedLocaleOptions(a={}){const{locale:t,numberingSystem:n,calendar:e}=Vn.create(this.loc.clone(a),a).resolvedOptions(this);return{locale:t,numberingSystem:n,outputCalendar:e}}toUTC(a=0,t={}){return this.setZone(Jn.instance(a),t)}toLocal(){return this.setZone(jt.defaultZone)}setZone(a,{keepLocalTime:t=!1,keepCalendarTime:n=!1}={}){if(a=pp(a,jt.defaultZone),a.equals(this.zone))return this;if(a.isValid){let e=this.ts;if(t||n){const l=a.offset(this.ts),i=this.toObject();[e]=j0(i,l,a)}return Np(this,{ts:e,zone:a})}else return La.invalid(Lo(a))}reconfigure({locale:a,numberingSystem:t,outputCalendar:n}={}){const e=this.loc.clone({locale:a,numberingSystem:t,outputCalendar:n});return Np(this,{loc:e})}setLocale(a){return this.reconfigure({locale:a})}set(a){if(!this.isValid)return this;const t=Bu(a,B7),{minDaysInFirstWeek:n,startOfWeek:e}=u7(t,this.loc),l=!Sa(t.weekYear)||!Sa(t.weekNumber)||!Sa(t.weekday),i=!Sa(t.ordinal),p=!Sa(t.year),r=!Sa(t.month)||!Sa(t.day),m=p||r,o=t.weekYear||t.weekNumber;if((m||i)&&o)throw new Zr("Can't mix weekYear/weekNumber units with year/month/day or ordinals");if(r&&i)throw new Zr("Can't mix ordinal dates with month/day");let c;l?c=c7({..._u(this.c,n,e),...t},n,e):Sa(t.ordinal)?(c={...this.toObject(),...t},Sa(t.day)&&(c.day=Math.min(ku(c.year,c.month),c.day))):c=h7({...V1(this.c),...t});const[h,u]=j0(c,this.o,this.zone);return Np(this,{ts:h,o:u})}plus(a){if(!this.isValid)return this;const t=st.fromDurationLike(a);return Np(this,_7(this,t))}minus(a){if(!this.isValid)return this;const t=st.fromDurationLike(a).negate();return Np(this,_7(this,t))}startOf(a,{useLocaleWeeks:t=!1}={}){if(!this.isValid)return this;const n={},e=st.normalizeUnit(a);switch(e){case"years":n.month=1;case"quarters":case"months":n.day=1;case"weeks":case"days":n.hour=0;case"hours":n.minute=0;case"minutes":n.second=0;case"seconds":n.millisecond=0;break}if(e==="weeks")if(t){const l=this.loc.getStartOfWeek(),{weekday:i}=this;i<l&&(n.weekNumber=this.weekNumber-1),n.weekday=l}else n.weekday=1;if(e==="quarters"){const l=Math.ceil(this.month/3);n.month=(l-1)*3+1}return this.set(n)}endOf(a,t){return this.isValid?this.plus({[a]:1}).startOf(a,t).minus(1):this}toFormat(a,t={}){return this.isValid?Vn.create(this.loc.redefaultToEN(t)).formatDateTimeFromString(this,a):q1}toLocaleString(a=Au,t={}){return this.isValid?Vn.create(this.loc.clone(t),a).formatDateTime(this):q1}toLocaleParts(a={}){return this.isValid?Vn.create(this.loc.clone(a),a).formatDateTimeParts(this):[]}toISO({format:a="extended",suppressSeconds:t=!1,suppressMilliseconds:n=!1,includeOffset:e=!0,extendedZone:l=!1,precision:i="milliseconds"}={}){if(!this.isValid)return null;i=W0(i);const p=a==="extended";let r=G1(this,p,i);return G0.indexOf(i)>=3&&(r+="T"),r+=k7(this,p,t,n,e,l,i),r}toISODate({format:a="extended",precision:t="day"}={}){return this.isValid?G1(this,a==="extended",W0(t)):null}toISOWeekDate(){return u0(this,"kkkk-'W'WW-c")}toISOTime({suppressMilliseconds:a=!1,suppressSeconds:t=!1,includeOffset:n=!0,includePrefix:e=!1,extendedZone:l=!1,format:i="extended",precision:p="milliseconds"}={}){return this.isValid?(p=W0(p),(e&&G0.indexOf(p)>=3?"T":"")+k7(this,i==="extended",t,a,n,l,p)):null}toRFC2822(){return u0(this,"EEE, dd LLL yyyy HH:mm:ss ZZZ",!1)}toHTTP(){return u0(this.toUTC(),"EEE, dd LLL yyyy HH:mm:ss 'GMT'")}toSQLDate(){return this.isValid?G1(this,!0):null}toSQLTime({includeOffset:a=!0,includeZone:t=!1,includeOffsetSpace:n=!0}={}){let e="HH:mm:ss.SSS";return(t||a)&&(n&&(e+=" "),t?e+="z":a&&(e+="ZZ")),u0(this,e,!0)}toSQL(a={}){return this.isValid?`${this.toSQLDate()} ${this.toSQLTime(a)}`:null}toString(){return this.isValid?this.toISO():q1}[Symbol.for("nodejs.util.inspect.custom")](){return this.isValid?`DateTime { ts: ${this.toISO()}, zone: ${this.zone.name}, locale: ${this.locale} }`:`DateTime { Invalid, reason: ${this.invalidReason} }`}valueOf(){return this.toMillis()}toMillis(){return this.isValid?this.ts:NaN}toSeconds(){return this.isValid?this.ts/1e3:NaN}toUnixInteger(){return this.isValid?Math.floor(this.ts/1e3):NaN}toJSON(){return this.toISO()}toBSON(){return this.toJSDate()}toObject(a={}){if(!this.isValid)return{};const t={...this.c};return a.includeConfig&&(t.outputCalendar=this.outputCalendar,t.numberingSystem=this.loc.numberingSystem,t.locale=this.loc.locale),t}toJSDate(){return new Date(this.isValid?this.ts:NaN)}diff(a,t="milliseconds",n={}){if(!this.isValid||!a.isValid)return st.invalid("created by diffing an invalid DateTime");const e={locale:this.locale,numberingSystem:this.numberingSystem,...n},l=yj(t).map(st.normalizeUnit),i=a.valueOf()>this.valueOf(),p=i?this:a,r=i?a:this,m=xG(p,r,l,e);return i?m.negate():m}diffNow(a="milliseconds",t={}){return this.diff(La.now(),a,t)}until(a){return this.isValid?qt.fromDateTimes(this,a):this}hasSame(a,t,n){if(!this.isValid)return!1;const e=a.valueOf(),l=this.setZone(a.zone,{keepLocalTime:!0});return l.startOf(t,n)<=e&&e<=l.endOf(t,n)}equals(a){return this.isValid&&a.isValid&&this.valueOf()===a.valueOf()&&this.zone.equals(a.zone)&&this.loc.equals(a.loc)}toRelative(a={}){if(!this.isValid)return null;const t=a.base||La.fromObject({},{zone:this.zone}),n=a.padding?this<t?-a.padding:a.padding:0;let e=["years","months","days","hours","minutes","seconds"],l=a.unit;return Array.isArray(a.unit)&&(e=a.unit,l=void 0),T7(t,this.plus(n),{...a,numeric:"always",units:e,unit:l})}toRelativeCalendar(a={}){return this.isValid?T7(a.base||La.fromObject({},{zone:this.zone}),this,{...a,numeric:"auto",units:["years","months","days"],calendary:!0}):null}static min(...a){if(!a.every(La.isDateTime))throw new Rn("min requires all arguments be DateTimes");return d7(a,t=>t.valueOf(),Math.min)}static max(...a){if(!a.every(La.isDateTime))throw new Rn("max requires all arguments be DateTimes");return d7(a,t=>t.valueOf(),Math.max)}static fromFormatExplain(a,t,n={}){const{locale:e=null,numberingSystem:l=null}=n,i=dt.fromOpts({locale:e,numberingSystem:l,defaultToEN:!0});return rB(i,a,t)}static fromStringExplain(a,t,n={}){return La.fromFormatExplain(a,t,n)}static buildFormatParser(a,t={}){const{locale:n=null,numberingSystem:e=null}=t,l=dt.fromOpts({locale:n,numberingSystem:e,defaultToEN:!0});return new pB(l,a)}static fromFormatParser(a,t,n={}){if(Sa(a)||Sa(t))throw new Rn("fromFormatParser requires an input string and a format parser");const{locale:e=null,numberingSystem:l=null}=n,i=dt.fromOpts({locale:e,numberingSystem:l,defaultToEN:!0});if(!i.equals(t.locale))throw new Rn(`fromFormatParser called with a locale of ${i}, but the format parser was created for ${t.locale}`);const{result:p,zone:r,specificOffset:m,invalidReason:o}=t.explainFromTokens(a);return o?La.invalid(o):Pr(p,r,n,`format ${t.format}`,a,m)}static get DATE_SHORT(){return Au}static get DATE_MED(){return hk}static get DATE_MED_WITH_WEEKDAY(){return UU}static get DATE_FULL(){return uk}static get DATE_HUGE(){return dk}static get TIME_SIMPLE(){return gk}static get TIME_WITH_SECONDS(){return fk}static get TIME_WITH_SHORT_OFFSET(){return yk}static get TIME_WITH_LONG_OFFSET(){return vk}static get TIME_24_SIMPLE(){return Ek}static get TIME_24_WITH_SECONDS(){return xk}static get TIME_24_WITH_SHORT_OFFSET(){return wk}static get TIME_24_WITH_LONG_OFFSET(){return bk}static get DATETIME_SHORT(){return Ak}static get DATETIME_SHORT_WITH_SECONDS(){return _k}static get DATETIME_MED(){return kk}static get DATETIME_MED_WITH_SECONDS(){return Bk}static get DATETIME_MED_WITH_WEEKDAY(){return jU}static get DATETIME_FULL(){return Mk}static get DATETIME_FULL_WITH_SECONDS(){return Tk}static get DATETIME_HUGE(){return Ck}static get DATETIME_HUGE_WITH_SECONDS(){return zk}}function wo(s){if(La.isDateTime(s))return s;if(s&&s.valueOf&&dp(s.valueOf()))return La.fromJSDate(s);if(s&&typeof s=="object")return La.fromObject(s);throw new Rn(`Unknown datetime argument: ${s}, of type ${typeof s}`)}fa.factory({id:"list2echarts_tree",name:"ECharts树图",match:/list2echarts_tree(.*)/,detail:"将列表转换为放ECharts树图，可选附加参数: `_tb` 或 `_radial` 或 `_polyline` 等",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=Ga.list2data(t,!1);n=Ga.data2strict(n);let e=hB(n),l=HG(e),i="";const p=a.match(/list2echarts_tree(.*)/);return p&&(p[1].trim().toLowerCase()==="_polyline"&&(i+="edgeShape: 'polyline',"),p[1].trim().toLowerCase()==="_tb"?i+="orient: 'TB',":p[1].trim().toLowerCase()==="_lr"?i+="orient: 'LR',":p[1].trim().toLowerCase()==="_bt"?i+="orient: 'BT',":p[1].trim().toLowerCase()==="_rl"&&(i+="orient: 'RL',"),p[1].trim().toLowerCase()==="_radial"?i+=`layout: 'radial',
symbol: 'emptyCircle',`:i+=`label: {
  position: 'left', // 展开则显示在左侧
  verticalAlign: 'middle',
  align: 'right',
},
leaves: {
  label: {
    position: 'right', // 折叠后则显示在右侧
    verticalAlign: 'middle',
    align: 'left'
  }
},`),`\`\`\`echarts
const option = {
  tooltip: {
    trigger: 'item',
    triggerOn: 'mousemove',
  },
  series: [
    {
      type: 'tree',
      data: [${JSON.stringify(l)}],
      ${i}
      top: '10%',
      bottom: '10%',
      symbolSize: 7,
      initialTreeDepth: 3,
      animationDurationUpdate: 750,
      emphasis: {
        focus: 'descendant',
      },
    },
  ],
}

// const height = 600
\`\`\``}});fa.factory({id:"list2echarts_sunburst",name:"ECharts旭日图",match:"list2echarts_sunburst",detail:"将列表转换为放ECharts旭日图 (平均分配)",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=Ga.list2data(t,!1);n=Ga.data2strict(n);let e=hB(n,!0);return`\`\`\`echarts
const option = {
  series: [
    {
      type: 'sunburst',
      data: ${JSON.stringify(e)},
      radius: [0, '90%'],
      label: {
        rotate: 'radial'
      }
    },
  ],
}
\`\`\``}});fa.factory({id:"list2echarts_gantt",name:"ECharts甘特图",match:"list2echarts_gantt",detail:"将列表转换为放ECharts甘特图 (ECharts的甘特图是用custom模拟的)",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=je.list2c2data(t);const e=$G(n);return VG(e)}});function hB(s,a=!1){let t=[],n=[];for(let e=0;e<s.length;e++){const l=s[e],p={name:l.content,...a?{value:1}:{}};if(n[l.level]=p,l.level>=1&&n.hasOwnProperty(l.level-1)){let r=n[l.level-1];if(typeof r!="object"||Array.isArray(r))return console.error("list数据不合规，父节点的value值不是{}类型"),t;a&&delete r.value,r.hasOwnProperty("children")?r.children.push(p):r.children=[p]}else if(l.level==0)t.push(p);else return console.error(`list数据不合规，没有正规化. level:${l.level}, prev_nodes:${n}`),t}return t}function $G(s){let a=[],t,n=[],e,l,i;for(let p=0;p<s.length;p++){const r=s[p],m=r.content.trim();if(r.level==0){const o=m.split("/"),c=o[0].trim(),h=o[1]?o[1].trim():"P1D";i||(/^\d+$/.test(c)?i="timestamp":La.fromISO(c).isValid?i="ISO8601":i="string");const u={content:"",start:0,end:0,duration:0,group1:0};if(i==="ISO8601"){const y=La.fromISO(c);if(y.isValid)u.start=y.toMillis();else{console.warn("开始时间解析失败1"),t=void 0;continue}const x=La.fromISO(h);if(x.isValid)u.end=x.toMillis();else{const b=st.fromISO(h);if(b.isValid)u.end=y.plus(b).toMillis();else{console.warn("结束时间解析失败1"),t=void 0;continue}}}else if(i==="timestamp"){const y=parseInt(c,10);if(!isNaN(y))u.start=y;else{console.warn("开始时间戳解析失败2"),t=void 0;continue}const x=parseInt(h,10);if(!isNaN(x))u.end=x;else{console.warn("结束时间戳解析失败2"),t=void 0;continue}}else{console.warn("暂不支持非时间格式的甘特图时间解析"),t=void 0;continue}if(e=e===void 0?u.start:Math.min(e,u.start),l=l===void 0?u.end:Math.max(l,u.end),n.length==0)u.group1=0,n.push(u.end);else{let y;for(let x=0;x<n.length;x++)if(!(u.start<n[x])){y=x,u.group1=x,n[x]=u.end;break}y===void 0&&(u.group1=n.length,n.push(u.end))}a.push(u),t=u;continue}else{if(!t||!i)continue;t.content=m,t=void 0}}return{data:a,time_mode:i,max_group1:n.length,min_time:e,max_time:l}}function VG(s){document.body.classList.contains("theme-dark");const{data:a,max_group1:t,min_time:n}=s;let e=[];for(const o of a)e.push({value:[o.content,o.group1,o.start,o.end,o.end-o.start]});const l=Array.from({length:t},(o,c)=>""),i=l.length*40,p=n??0-1440*60*1e3;let r={tooltip:{trigger:"item"},dataZoom:[{type:"slider",xAxisIndex:0,filterMode:"weakFilter",height:20,bottom:10,start:0,end:100,handleIcon:"path://M10.7,11.9v-1.3H9.3v1.3c-4.9,0.3-8.8,4.4-8.8,9.4c0,5,3.9,9.1,8.8,9.4v1.3h1.3v-1.3c4.9-0.3,8.8-4.4,8.8-9.4C19.5,16.3,15.6,12.2,10.7,11.9z M13.3,24.4H6.7V23h6.6V24.4z M13.3,19.6H6.7v-1.4h6.6V19.6z",handleSize:"80%",showDetail:!1},{type:"inside",xAxisIndex:0,filterMode:"weakFilter",start:0,end:100}],grid:{height:i,top:40,bottom:40,left:15,right:15},xAxis:{position:"top",min:p,scale:!0,splitNumber:30,axisLabel:{fontSize:10}},yAxis:{data:l,inverse:!0},series:[{type:"custom",itemStyle:{opacity:.8},encode:{x:[2,3],y:1},data:e}]};return`\`\`\`echarts
height = ${i+80}
const option = ${JSON.stringify(r,null,2)}

// 自定义渲染函数 - x轴
let lastMonth = -1; // 用来记录上一个刻度的月份
option.xAxis.axisLabel.formatter = function (val) {
  let date = new Date(val);
  let currentMonth = date.getMonth();
  if (lastMonth !== currentMonth) { // 如果月份与上一个不同，或者这是第一个刻度
    lastMonth = currentMonth;
    return echarts.format.formatTime('yyyy-MM', val) + '\\n' +
      echarts.format.formatTime('dd', val);
  } else {
    return '\\n' +   // (头部空换行保持高度不变，避免缩放时图表跳动)
      echarts.format.formatTime('dd', val);
  }
}

// 自定义渲染函数 - tooltip, 支持html
option.tooltip.formatter = function (params) {
  // let categoryName = categories[params.value[1]];
  return echarts.format.formatTime('MM-dd', params.value[2]) +
    '/' +
    echarts.format.formatTime('MM-dd', params.value[3]) +
    '<br />' +
    params.value[0].replaceAll('\\n', '<br />');
},

// 自定义渲染函数 - 图表内容
option.series[0].renderItem = function renderItem(params, api) {
  var categoryIndex = api.value(1); // 分类索引
  var start = api.coord([api.value(2), categoryIndex]);
  var end = api.coord([api.value(3), categoryIndex]);
  var height = api.size([0, 1])[1] * 1 - 5;
  var rectShape = echarts.graphic.clipRectByRect( // 绘制矩形，并裁减
    {
      x: start[0],
      y: start[1] - height / 2,
      width: end[0] - start[0],
      height: height,
    },
    {
      x: params.coordSys.x,
      y: params.coordSys.y,
      width: params.coordSys.width,
      height: params.coordSys.height
    }
  );
  var text_array = api.value(0).split('\\n');
  const textColor = myChart.getOption().textStyle.color ?? "black";
  return (
    rectShape && {
      type: 'group', // group 元素，包含矩形和文本
      children: [
        {
          type: 'rect', // 矩形元素
          transition: ['shape'],
          shape: {
            ...rectShape,
            r: 5, // 圆角半径
          },
          style: api.style()
        },
        {
          type: 'text', // 文本元素
          transition: ['style'], // 对样式应用过渡动画
          style: {
            x: start[0] + 5, // 将文本放在矩形内部，并向右偏移 5px
            y: start[1], // 垂直居中
            text: (text_array.length > 1 ? text_array[0] + '...' : text_array[0]),
            fill: textColor, // 文本颜色
            textVerticalAlign: 'middle', // 文本垂直对齐方式
            textAlign: 'left' // 文本水平对齐方式
          }
        }
      ]
    }
  );
}
\`\`\``}function HG(s){return s.length==1?s[0]:{name:"root",children:s}}var qG=typeof globalThis<"u"?globalThis:typeof window<"u"?window:typeof global<"u"?global:typeof self<"u"?self:{};function B8(s){return s&&s.__esModule&&Object.prototype.hasOwnProperty.call(s,"default")?s.default:s}function mp(s){throw new Error('Could not dynamically require "'+s+'". 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p<10?String.fromCharCode(48+p):(p-=10,p<26?String.fromCharCode(65+p):(p-=26,p<26?String.fromCharCode(97+p):(p-=26,p===0?"-":p===1?"_":"?")))}function i(p,r,m){var o=p>>2,c=(p&3)<<4|r>>4,h=(r&15)<<2|m>>6,u=m&63,y="";return y+=l(o&63),y+=l(c&63),y+=l(h&63),y+=l(u&63),y}n.exports=function(p){for(var r="",m=0;m<p.length;m+=3)m+2===p.length?r+=i(p.charCodeAt(m),p.charCodeAt(m+1),0):m+1===p.length?r+=i(p.charCodeAt(m),0,0):r+=i(p.charCodeAt(m),p.charCodeAt(m+1),p.charCodeAt(m+2));return r}},{}],3:[function(t,n,e){var l=t("./deflate"),i=t("./encode64");n.exports.encode=function(p){var r=l(p);return i(r)}},{"./deflate":1,"./encode64":2}],4:[function(t,n,e){var l=t("./zlib/deflate"),i=t("./utils/common"),p=t("./utils/strings"),r=t("./zlib/messages"),m=t("./zlib/zstream"),o=Object.prototype.toString,c=0,h=4,u=0,y=1,x=2,b=-1,_=0,T=8;function A(U){if(!(this instanceof A))return new A(U);this.options=i.assign({level:b,method:T,chunkSize:16384,windowBits:15,memLevel:8,strategy:_,to:""},U||{});var L=this.options;L.raw&&L.windowBits>0?L.windowBits=-L.windowBits:L.gzip&&L.windowBits>0&&L.windowBits<16&&(L.windowBits+=16),this.err=0,this.msg="",this.ended=!1,this.chunks=[],this.strm=new m,this.strm.avail_out=0;var P=l.deflateInit2(this.strm,L.level,L.method,L.windowBits,L.memLevel,L.strategy);if(P!==u)throw new Error(r[P]);if(L.header&&l.deflateSetHeader(this.strm,L.header),L.dictionary){var G;if(typeof L.dictionary=="string"?G=p.string2buf(L.dictionary):o.call(L.dictionary)==="[object ArrayBuffer]"?G=new Uint8Array(L.dictionary):G=L.dictionary,P=l.deflateSetDictionary(this.strm,G),P!==u)throw new Error(r[P]);this._dict_set=!0}}A.prototype.push=function(U,L){var P=this.strm,G=this.options.chunkSize,X,S;if(this.ended)return!1;S=L===~~L?L:L===!0?h:c,typeof U=="string"?P.input=p.string2buf(U):o.call(U)==="[object ArrayBuffer]"?P.input=new Uint8Array(U):P.input=U,P.next_in=0,P.avail_in=P.input.length;do{if(P.avail_out===0&&(P.output=new i.Buf8(G),P.next_out=0,P.avail_out=G),X=l.deflate(P,S),X!==y&&X!==u)return this.onEnd(X),this.ended=!0,!1;(P.avail_out===0||P.avail_in===0&&(S===h||S===x))&&(this.options.to==="string"?this.onData(p.buf2binstring(i.shrinkBuf(P.output,P.next_out))):this.onData(i.shrinkBuf(P.output,P.next_out)))}while((P.avail_in>0||P.avail_out===0)&&X!==y);return S===h?(X=l.deflateEnd(this.strm),this.onEnd(X),this.ended=!0,X===u):(S===x&&(this.onEnd(u),P.avail_out=0),!0)},A.prototype.onData=function(U){this.chunks.push(U)},A.prototype.onEnd=function(U){U===u&&(this.options.to==="string"?this.result=this.chunks.join(""):this.result=i.flattenChunks(this.chunks)),this.chunks=[],this.err=U,this.msg=this.strm.msg};function C(U,L){var P=new A(L);if(P.push(U,!0),P.err)throw P.msg||r[P.err];return P.result}function I(U,L){return L=L||{},L.raw=!0,C(U,L)}function $(U,L){return L=L||{},L.gzip=!0,C(U,L)}e.Deflate=A,e.deflate=C,e.deflateRaw=I,e.gzip=$},{"./utils/common":5,"./utils/strings":6,"./zlib/deflate":9,"./zlib/messages":10,"./zlib/zstream":12}],5:[function(t,n,e){var l=typeof Uint8Array<"u"&&typeof Uint16Array<"u"&&typeof Int32Array<"u";function i(m,o){return Object.prototype.hasOwnProperty.call(m,o)}e.assign=function(m){for(var o=Array.prototype.slice.call(arguments,1);o.length;){var c=o.shift();if(c){if(typeof c!="object")throw new TypeError(c+"must be non-object");for(var h in c)i(c,h)&&(m[h]=c[h])}}return m},e.shrinkBuf=function(m,o){return m.length===o?m:m.subarray?m.subarray(0,o):(m.length=o,m)};var p={arraySet:function(m,o,c,h,u){if(o.subarray&&m.subarray){m.set(o.subarray(c,c+h),u);return}for(var y=0;y<h;y++)m[u+y]=o[c+y]},flattenChunks:function(m){var o,c,h,u,y,x;for(h=0,o=0,c=m.length;o<c;o++)h+=m[o].length;for(x=new Uint8Array(h),u=0,o=0,c=m.length;o<c;o++)y=m[o],x.set(y,u),u+=y.length;return x}},r={arraySet:function(m,o,c,h,u){for(var y=0;y<h;y++)m[u+y]=o[c+y]},flattenChunks:function(m){return[].concat.apply([],m)}};e.setTyped=function(m){m?(e.Buf8=Uint8Array,e.Buf16=Uint16Array,e.Buf32=Int32Array,e.assign(e,p)):(e.Buf8=Array,e.Buf16=Array,e.Buf32=Array,e.assign(e,r))},e.setTyped(l)},{}],6:[function(t,n,e){var l=t("./common"),i=!0,p=!0;try{String.fromCharCode.apply(null,[0])}catch{i=!1}try{String.fromCharCode.apply(null,new Uint8Array(1))}catch{p=!1}for(var r=new l.Buf8(256),m=0;m<256;m++)r[m]=m>=252?6:m>=248?5:m>=240?4:m>=224?3:m>=192?2:1;r[254]=r[254]=1,e.string2buf=function(c){var h,u,y,x,b,_=c.length,T=0;for(x=0;x<_;x++)u=c.charCodeAt(x),(u&64512)===55296&&x+1<_&&(y=c.charCodeAt(x+1),(y&64512)===56320&&(u=65536+(u-55296<<10)+(y-56320),x++)),T+=u<128?1:u<2048?2:u<65536?3:4;for(h=new l.Buf8(T),b=0,x=0;b<T;x++)u=c.charCodeAt(x),(u&64512)===55296&&x+1<_&&(y=c.charCodeAt(x+1),(y&64512)===56320&&(u=65536+(u-55296<<10)+(y-56320),x++)),u<128?h[b++]=u:u<2048?(h[b++]=192|u>>>6,h[b++]=128|u&63):u<65536?(h[b++]=224|u>>>12,h[b++]=128|u>>>6&63,h[b++]=128|u&63):(h[b++]=240|u>>>18,h[b++]=128|u>>>12&63,h[b++]=128|u>>>6&63,h[b++]=128|u&63);return h};function o(c,h){if(h<65534&&(c.subarray&&p||!c.subarray&&i))return String.fromCharCode.apply(null,l.shrinkBuf(c,h));for(var u="",y=0;y<h;y++)u+=String.fromCharCode(c[y]);return u}e.buf2binstring=function(c){return o(c,c.length)},e.binstring2buf=function(c){for(var h=new l.Buf8(c.length),u=0,y=h.length;u<y;u++)h[u]=c.charCodeAt(u);return h},e.buf2string=function(c,h){var u,y,x,b,_=h||c.length,T=new 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l=t("../utils/common"),i=t("./trees"),p=t("./adler32"),r=t("./crc32"),m=t("./messages"),o=0,c=1,h=3,u=4,y=5,x=0,b=1,_=-2,T=-3,A=-5,C=-1,I=1,$=2,U=3,L=4,P=0,G=2,X=8,S=9,is=15,R=8,ss=29,Y=256,os=Y+1+ss,ps=30,bs=19,Is=2*os+1,Ys=15,vs=3,gs=258,xs=gs+vs+1,ys=32,ks=42,rs=69,Ms=73,Ns=91,Gs=103,zs=113,Bs=666,H=1,N=2,V=3,Z=4,as=3;function es(E,W){return E.msg=m[W],W}function Os(E){return(E<<1)-(E>4?9:0)}function Ds(E){for(var W=E.length;--W>=0;)E[W]=0}function Ls(E){var W=E.state,Ss=W.pending;Ss>E.avail_out&&(Ss=E.avail_out),Ss!==0&&(l.arraySet(E.output,W.pending_buf,W.pending_out,Ss,E.next_out),E.next_out+=Ss,W.pending_out+=Ss,E.total_out+=Ss,E.avail_out-=Ss,W.pending-=Ss,W.pending===0&&(W.pending_out=0))}function As(E,W){i._tr_flush_block(E,E.block_start>=0?E.block_start:-1,E.strstart-E.block_start,W),E.block_start=E.strstart,Ls(E.strm)}function Vs(E,W){E.pending_buf[E.pending++]=W}function Ks(E,W){E.pending_buf[E.pending++]=W>>>8&255,E.pending_buf[E.pending++]=W&255}function aa(E,W,Ss,q){var ds=E.avail_in;return ds>q&&(ds=q),ds===0?0:(E.avail_in-=ds,l.arraySet(W,E.input,E.next_in,ds,Ss),E.state.wrap===1?E.adler=p(E.adler,W,ds,Ss):E.state.wrap===2&&(E.adler=r(E.adler,W,ds,Ss)),E.next_in+=ds,E.total_in+=ds,ds)}function ya(E,W){var Ss=E.max_chain_length,q=E.strstart,ds,Fs,Js=E.prev_length,ia=E.nice_match,oa=E.strstart>E.w_size-xs?E.strstart-(E.w_size-xs):0,Ha=E.window,ms=E.w_mask,Ja=E.prev,us=E.strstart+gs,Rt=Ha[q+Js-1],cn=Ha[q+Js];E.prev_length>=E.good_match&&(Ss>>=2),ia>E.lookahead&&(ia=E.lookahead);do if(ds=W,!(Ha[ds+Js]!==cn||Ha[ds+Js-1]!==Rt||Ha[ds]!==Ha[q]||Ha[++ds]!==Ha[q+1])){q+=2,ds++;do;while(Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&Ha[++q]===Ha[++ds]&&q<us);if(Fs=gs-(us-q),q=us-gs,Fs>Js){if(E.match_start=W,Js=Fs,Fs>=ia)break;Rt=Ha[q+Js-1],cn=Ha[q+Js]}}while((W=Ja[W&ms])>oa&&--Ss!==0);return Js<=E.lookahead?Js:E.lookahead}function Aa(E){var W=E.w_size,Ss,q,ds,Fs,Js;do{if(Fs=E.window_size-E.lookahead-E.strstart,E.strstart>=W+(W-xs)){l.arraySet(E.window,E.window,W,W,0),E.match_start-=W,E.strstart-=W,E.block_start-=W,q=E.hash_size,Ss=q;do ds=E.head[--Ss],E.head[Ss]=ds>=W?ds-W:0;while(--q);q=W,Ss=q;do ds=E.prev[--Ss],E.prev[Ss]=ds>=W?ds-W:0;while(--q);Fs+=W}if(E.strm.avail_in===0)break;if(q=aa(E.strm,E.window,E.strstart+E.lookahead,Fs),E.lookahead+=q,E.lookahead+E.insert>=vs)for(Js=E.strstart-E.insert,E.ins_h=E.window[Js],E.ins_h=(E.ins_h<<E.hash_shift^E.window[Js+1])&E.hash_mask;E.insert&&(E.ins_h=(E.ins_h<<E.hash_shift^E.window[Js+vs-1])&E.hash_mask,E.prev[Js&E.w_mask]=E.head[E.ins_h],E.head[E.ins_h]=Js,Js++,E.insert--,!(E.lookahead+E.insert<vs)););}while(E.lookahead<xs&&E.strm.avail_in!==0)}function Ra(E,W){var Ss=65535;for(Ss>E.pending_buf_size-5&&(Ss=E.pending_buf_size-5);;){if(E.lookahead<=1){if(Aa(E),E.lookahead===0&&W===o)return H;if(E.lookahead===0)break}E.strstart+=E.lookahead,E.lookahead=0;var q=E.block_start+Ss;if((E.strstart===0||E.strstart>=q)&&(E.lookahead=E.strstart-q,E.strstart=q,As(E,!1),E.strm.avail_out===0)||E.strstart-E.block_start>=E.w_size-xs&&(As(E,!1),E.strm.avail_out===0))return H}return E.insert=0,W===u?(As(E,!0),E.strm.avail_out===0?V:Z):(E.strstart>E.block_start&&(As(E,!1),E.strm.avail_out===0),H)}function $a(E,W){for(var Ss,q;;){if(E.lookahead<xs){if(Aa(E),E.lookahead<xs&&W===o)return H;if(E.lookahead===0)break}if(Ss=0,E.lookahead>=vs&&(E.ins_h=(E.ins_h<<E.hash_shift^E.window[E.strstart+vs-1])&E.hash_mask,Ss=E.prev[E.strstart&E.w_mask]=E.head[E.ins_h],E.head[E.ins_h]=E.strstart),Ss!==0&&E.strstart-Ss<=E.w_size-xs&&(E.match_length=ya(E,Ss)),E.match_length>=vs)if(q=i._tr_tally(E,E.strstart-E.match_start,E.match_length-vs),E.lookahead-=E.match_length,E.match_length<=E.max_lazy_match&&E.lookahead>=vs){E.match_length--;do E.strstart++,E.ins_h=(E.ins_h<<E.hash_shift^E.window[E.strstart+vs-1])&E.hash_mask,Ss=E.prev[E.strstart&E.w_mask]=E.head[E.ins_h],E.head[E.ins_h]=E.strstart;while(--E.match_length!==0);E.strstart++}else E.strstart+=E.match_length,E.match_length=0,E.ins_h=E.window[E.strstart],E.ins_h=(E.ins_h<<E.hash_shift^E.window[E.strstart+1])&E.hash_mask;else q=i._tr_tally(E,0,E.window[E.strstart]),E.lookahead--,E.strstart++;if(q&&(As(E,!1),E.strm.avail_out===0))return H}return E.insert=E.strstart<vs-1?E.strstart:vs-1,W===u?(As(E,!0),E.strm.avail_out===0?V:Z):E.last_lit&&(As(E,!1),E.strm.avail_out===0)?H:N}function Wa(E,W){for(var Ss,q,ds;;){if(E.lookahead<xs){if(Aa(E),E.lookahead<xs&&W===o)return H;if(E.lookahead===0)break}if(Ss=0,E.lookahead>=vs&&(E.ins_h=(E.ins_h<<E.hash_shift^E.window[E.strstart+vs-1])&E.hash_mask,Ss=E.prev[E.strstart&E.w_mask]=E.head[E.ins_h],E.head[E.ins_h]=E.strstart),E.prev_length=E.match_length,E.prev_match=E.match_start,E.match_length=vs-1,Ss!==0&&E.prev_length<E.max_lazy_match&&E.strstart-Ss<=E.w_size-xs&&(E.match_length=ya(E,Ss),E.match_length<=5&&(E.strategy===I||E.match_length===vs&&E.strstart-E.match_start>4096)&&(E.match_length=vs-1)),E.prev_length>=vs&&E.match_length<=E.prev_length){ds=E.strstart+E.lookahead-vs,q=i._tr_tally(E,E.strstart-1-E.prev_match,E.prev_length-vs),E.lookahead-=E.prev_length-1,E.prev_length-=2;do++E.strstart<=ds&&(E.ins_h=(E.ins_h<<E.hash_shift^E.window[E.strstart+vs-1])&E.hash_mask,Ss=E.prev[E.strstart&E.w_mask]=E.head[E.ins_h],E.head[E.ins_h]=E.strstart);while(--E.prev_length!==0);if(E.match_available=0,E.match_length=vs-1,E.strstart++,q&&(As(E,!1),E.strm.avail_out===0))return H}else if(E.match_available){if(q=i._tr_tally(E,0,E.window[E.strstart-1]),q&&As(E,!1),E.strstart++,E.lookahead--,E.strm.avail_out===0)return H}else E.match_available=1,E.strstart++,E.lookahead--}return E.match_available&&(q=i._tr_tally(E,0,E.window[E.strstart-1]),E.match_available=0),E.insert=E.strstart<vs-1?E.strstart:vs-1,W===u?(As(E,!0),E.strm.avail_out===0?V:Z):E.last_lit&&(As(E,!1),E.strm.avail_out===0)?H:N}function Qa(E,W){for(var Ss,q,ds,Fs,Js=E.window;;){if(E.lookahead<=gs){if(Aa(E),E.lookahead<=gs&&W===o)return H;if(E.lookahead===0)break}if(E.match_length=0,E.lookahead>=vs&&E.strstart>0&&(ds=E.strstart-1,q=Js[ds],q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds])){Fs=E.strstart+gs;do;while(q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&q===Js[++ds]&&ds<Fs);E.match_length=gs-(Fs-ds),E.match_length>E.lookahead&&(E.match_length=E.lookahead)}if(E.match_length>=vs?(Ss=i._tr_tally(E,1,E.match_length-vs),E.lookahead-=E.match_length,E.strstart+=E.match_length,E.match_length=0):(Ss=i._tr_tally(E,0,E.window[E.strstart]),E.lookahead--,E.strstart++),Ss&&(As(E,!1),E.strm.avail_out===0))return H}return E.insert=0,W===u?(As(E,!0),E.strm.avail_out===0?V:Z):E.last_lit&&(As(E,!1),E.strm.avail_out===0)?H:N}function kt(E,W){for(var Ss;;){if(E.lookahead===0&&(Aa(E),E.lookahead===0)){if(W===o)return H;break}if(E.match_length=0,Ss=i._tr_tally(E,0,E.window[E.strstart]),E.lookahead--,E.strstart++,Ss&&(As(E,!1),E.strm.avail_out===0))return H}return E.insert=0,W===u?(As(E,!0),E.strm.avail_out===0?V:Z):E.last_lit&&(As(E,!1),E.strm.avail_out===0)?H:N}function tt(E,W,Ss,q,ds){this.good_length=E,this.max_lazy=W,this.nice_length=Ss,this.max_chain=q,this.func=ds}var gn;gn=[new tt(0,0,0,0,Ra),new tt(4,4,8,4,$a),new tt(4,5,16,8,$a),new tt(4,6,32,32,$a),new tt(4,4,16,16,Wa),new tt(8,16,32,32,Wa),new tt(8,16,128,128,Wa),new tt(8,32,128,256,Wa),new tt(32,128,258,1024,Wa),new tt(32,258,258,4096,Wa)];function xe(E){E.window_size=2*E.w_size,Ds(E.head),E.max_lazy_match=gn[E.level].max_lazy,E.good_match=gn[E.level].good_length,E.nice_match=gn[E.level].nice_length,E.max_chain_length=gn[E.level].max_chain,E.strstart=0,E.block_start=0,E.lookahead=0,E.insert=0,E.match_length=E.prev_length=vs-1,E.match_available=0,E.ins_h=0}function J(){this.strm=null,this.status=0,this.pending_buf=null,this.pending_buf_size=0,this.pending_out=0,this.pending=0,this.wrap=0,this.gzhead=null,this.gzindex=0,this.method=X,this.last_flush=-1,this.w_size=0,this.w_bits=0,this.w_mask=0,this.window=null,this.window_size=0,this.prev=null,this.head=null,this.ins_h=0,this.hash_size=0,this.hash_bits=0,this.hash_mask=0,this.hash_shift=0,this.block_start=0,this.match_length=0,this.prev_match=0,this.match_available=0,this.strstart=0,this.match_start=0,this.lookahead=0,this.prev_length=0,this.max_chain_length=0,this.max_lazy_match=0,this.level=0,this.strategy=0,this.good_match=0,this.nice_match=0,this.dyn_ltree=new l.Buf16(Is*2),this.dyn_dtree=new l.Buf16((2*ps+1)*2),this.bl_tree=new l.Buf16((2*bs+1)*2),Ds(this.dyn_ltree),Ds(this.dyn_dtree),Ds(this.bl_tree),this.l_desc=null,this.d_desc=null,this.bl_desc=null,this.bl_count=new l.Buf16(Ys+1),this.heap=new l.Buf16(2*os+1),Ds(this.heap),this.heap_len=0,this.heap_max=0,this.depth=new l.Buf16(2*os+1),Ds(this.depth),this.l_buf=0,this.lit_bufsize=0,this.last_lit=0,this.d_buf=0,this.opt_len=0,this.static_len=0,this.matches=0,this.insert=0,this.bi_buf=0,this.bi_valid=0}function Zs(E){var W;return!E||!E.state?es(E,_):(E.total_in=E.total_out=0,E.data_type=G,W=E.state,W.pending=0,W.pending_out=0,W.wrap<0&&(W.wrap=-W.wrap),W.status=W.wrap?ks:zs,E.adler=W.wrap===2?0:1,W.last_flush=o,i._tr_init(W),x)}function la(E){var W=Zs(E);return W===x&&xe(E.state),W}function va(E,W){return!E||!E.state||E.state.wrap!==2?_:(E.state.gzhead=W,x)}function ws(E,W,Ss,q,ds,Fs){if(!E)return _;var Js=1;if(W===C&&(W=6),q<0?(Js=0,q=-q):q>15&&(Js=2,q-=16),ds<1||ds>S||Ss!==X||q<8||q>15||W<0||W>9||Fs<0||Fs>L)return es(E,_);q===8&&(q=9);var ia=new J;return E.state=ia,ia.strm=E,ia.wrap=Js,ia.gzhead=null,ia.w_bits=q,ia.w_size=1<<ia.w_bits,ia.w_mask=ia.w_size-1,ia.hash_bits=ds+7,ia.hash_size=1<<ia.hash_bits,ia.hash_mask=ia.hash_size-1,ia.hash_shift=~~((ia.hash_bits+vs-1)/vs),ia.window=new l.Buf8(ia.w_size*2),ia.head=new l.Buf16(ia.hash_size),ia.prev=new l.Buf16(ia.w_size),ia.lit_bufsize=1<<ds+6,ia.pending_buf_size=ia.lit_bufsize*4,ia.pending_buf=new l.Buf8(ia.pending_buf_size),ia.d_buf=1*ia.lit_bufsize,ia.l_buf=3*ia.lit_bufsize,ia.level=W,ia.strategy=Fs,ia.method=Ss,la(E)}function Rs(E,W){return ws(E,W,X,is,R,P)}function v(E,W){var Ss,q,ds,Fs;if(!E||!E.state||W>y||W<0)return E?es(E,_):_;if(q=E.state,!E.output||!E.input&&E.avail_in!==0||q.status===Bs&&W!==u)return es(E,E.avail_out===0?A:_);if(q.strm=E,Ss=q.last_flush,q.last_flush=W,q.status===ks)if(q.wrap===2)E.adler=0,Vs(q,31),Vs(q,139),Vs(q,8),q.gzhead?(Vs(q,(q.gzhead.text?1:0)+(q.gzhead.hcrc?2:0)+(q.gzhead.extra?4:0)+(q.gzhead.name?8:0)+(q.gzhead.comment?16:0)),Vs(q,q.gzhead.time&255),Vs(q,q.gzhead.time>>8&255),Vs(q,q.gzhead.time>>16&255),Vs(q,q.gzhead.time>>24&255),Vs(q,q.level===9?2:q.strategy>=$||q.level<2?4:0),Vs(q,q.gzhead.os&255),q.gzhead.extra&&q.gzhead.extra.length&&(Vs(q,q.gzhead.extra.length&255),Vs(q,q.gzhead.extra.length>>8&255)),q.gzhead.hcrc&&(E.adler=r(E.adler,q.pending_buf,q.pending,0)),q.gzindex=0,q.status=rs):(Vs(q,0),Vs(q,0),Vs(q,0),Vs(q,0),Vs(q,0),Vs(q,q.level===9?2:q.strategy>=$||q.level<2?4:0),Vs(q,as),q.status=zs);else{var Js=X+(q.w_bits-8<<4)<<8,ia=-1;q.strategy>=$||q.level<2?ia=0:q.level<6?ia=1:q.level===6?ia=2:ia=3,Js|=ia<<6,q.strstart!==0&&(Js|=ys),Js+=31-Js%31,q.status=zs,Ks(q,Js),q.strstart!==0&&(Ks(q,E.adler>>>16),Ks(q,E.adler&65535)),E.adler=1}if(q.status===rs)if(q.gzhead.extra){for(ds=q.pending;q.gzindex<(q.gzhead.extra.length&65535)&&!(q.pending===q.pending_buf_size&&(q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),Ls(E),ds=q.pending,q.pending===q.pending_buf_size));)Vs(q,q.gzhead.extra[q.gzindex]&255),q.gzindex++;q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),q.gzindex===q.gzhead.extra.length&&(q.gzindex=0,q.status=Ms)}else q.status=Ms;if(q.status===Ms)if(q.gzhead.name){ds=q.pending;do{if(q.pending===q.pending_buf_size&&(q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),Ls(E),ds=q.pending,q.pending===q.pending_buf_size)){Fs=1;break}q.gzindex<q.gzhead.name.length?Fs=q.gzhead.name.charCodeAt(q.gzindex++)&255:Fs=0,Vs(q,Fs)}while(Fs!==0);q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),Fs===0&&(q.gzindex=0,q.status=Ns)}else q.status=Ns;if(q.status===Ns)if(q.gzhead.comment){ds=q.pending;do{if(q.pending===q.pending_buf_size&&(q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),Ls(E),ds=q.pending,q.pending===q.pending_buf_size)){Fs=1;break}q.gzindex<q.gzhead.comment.length?Fs=q.gzhead.comment.charCodeAt(q.gzindex++)&255:Fs=0,Vs(q,Fs)}while(Fs!==0);q.gzhead.hcrc&&q.pending>ds&&(E.adler=r(E.adler,q.pending_buf,q.pending-ds,ds)),Fs===0&&(q.status=Gs)}else q.status=Gs;if(q.status===Gs&&(q.gzhead.hcrc?(q.pending+2>q.pending_buf_size&&Ls(E),q.pending+2<=q.pending_buf_size&&(Vs(q,E.adler&255),Vs(q,E.adler>>8&255),E.adler=0,q.status=zs)):q.status=zs),q.pending!==0){if(Ls(E),E.avail_out===0)return q.last_flush=-1,x}else if(E.avail_in===0&&Os(W)<=Os(Ss)&&W!==u)return es(E,A);if(q.status===Bs&&E.avail_in!==0)return es(E,A);if(E.avail_in!==0||q.lookahead!==0||W!==o&&q.status!==Bs){var oa=q.strategy===$?kt(q,W):q.strategy===U?Qa(q,W):gn[q.level].func(q,W);if((oa===V||oa===Z)&&(q.status=Bs),oa===H||oa===V)return E.avail_out===0&&(q.last_flush=-1),x;if(oa===N&&(W===c?i._tr_align(q):W!==y&&(i._tr_stored_block(q,0,0,!1),W===h&&(Ds(q.head),q.lookahead===0&&(q.strstart=0,q.block_start=0,q.insert=0))),Ls(E),E.avail_out===0))return q.last_flush=-1,x}return W!==u?x:q.wrap<=0?b:(q.wrap===2?(Vs(q,E.adler&255),Vs(q,E.adler>>8&255),Vs(q,E.adler>>16&255),Vs(q,E.adler>>24&255),Vs(q,E.total_in&255),Vs(q,E.total_in>>8&255),Vs(q,E.total_in>>16&255),Vs(q,E.total_in>>24&255)):(Ks(q,E.adler>>>16),Ks(q,E.adler&65535)),Ls(E),q.wrap>0&&(q.wrap=-q.wrap),q.pending!==0?x:b)}function D(E){var W;return!E||!E.state?_:(W=E.state.status,W!==ks&&W!==rs&&W!==Ms&&W!==Ns&&W!==Gs&&W!==zs&&W!==Bs?es(E,_):(E.state=null,W===zs?es(E,T):x))}function Ps(E,W){var Ss=W.length,q,ds,Fs,Js,ia,oa,Ha,ms;if(!E||!E.state||(q=E.state,Js=q.wrap,Js===2||Js===1&&q.status!==ks||q.lookahead))return _;for(Js===1&&(E.adler=p(E.adler,W,Ss,0)),q.wrap=0,Ss>=q.w_size&&(Js===0&&(Ds(q.head),q.strstart=0,q.block_start=0,q.insert=0),ms=new l.Buf8(q.w_size),l.arraySet(ms,W,Ss-q.w_size,q.w_size,0),W=ms,Ss=q.w_size),ia=E.avail_in,oa=E.next_in,Ha=E.input,E.avail_in=Ss,E.next_in=0,E.input=W,Aa(q);q.lookahead>=vs;){ds=q.strstart,Fs=q.lookahead-(vs-1);do q.ins_h=(q.ins_h<<q.hash_shift^q.window[ds+vs-1])&q.hash_mask,q.prev[ds&q.w_mask]=q.head[q.ins_h],q.head[q.ins_h]=ds,ds++;while(--Fs);q.strstart=ds,q.lookahead=vs-1,Aa(q)}return q.strstart+=q.lookahead,q.block_start=q.strstart,q.insert=q.lookahead,q.lookahead=0,q.match_length=q.prev_length=vs-1,q.match_available=0,E.next_in=oa,E.input=Ha,E.avail_in=ia,q.wrap=Js,x}e.deflateInit=Rs,e.deflateInit2=ws,e.deflateReset=la,e.deflateResetKeep=Zs,e.deflateSetHeader=va,e.deflate=v,e.deflateEnd=D,e.deflateSetDictionary=Ps,e.deflateInfo="pako deflate (from Nodeca project)"},{"../utils/common":5,"./adler32":7,"./crc32":8,"./messages":10,"./trees":11}],10:[function(t,n,e){n.exports={2:"need dictionary",1:"stream end",0:"","-1":"file error","-2":"stream error","-3":"data error","-4":"insufficient memory","-5":"buffer error","-6":"incompatible version"}},{}],11:[function(t,n,e){var l=t("../utils/common"),i=4,p=0,r=1,m=2;function o(J){for(var Zs=J.length;--Zs>=0;)J[Zs]=0}var c=0,h=1,u=2,y=3,x=258,b=29,_=256,T=_+1+b,A=30,C=19,I=2*T+1,$=15,U=16,L=7,P=256,G=16,X=17,S=18,is=[0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,0],R=[0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13],ss=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,7],Y=[16,17,18,0,8,7,9,6,10,5,11,4,12,3,13,2,14,1,15],os=512,ps=new Array((T+2)*2);o(ps);var bs=new Array(A*2);o(bs);var Is=new Array(os);o(Is);var Ys=new Array(x-y+1);o(Ys);var vs=new Array(b);o(vs);var gs=new Array(A);o(gs);function xs(J,Zs,la,va,ws){this.static_tree=J,this.extra_bits=Zs,this.extra_base=la,this.elems=va,this.max_length=ws,this.has_stree=J&&J.length}var ys,ks,rs;function Ms(J,Zs){this.dyn_tree=J,this.max_code=0,this.stat_desc=Zs}function Ns(J){return J<256?Is[J]:Is[256+(J>>>7)]}function Gs(J,Zs){J.pending_buf[J.pending++]=Zs&255,J.pending_buf[J.pending++]=Zs>>>8&255}function zs(J,Zs,la){J.bi_valid>U-la?(J.bi_buf|=Zs<<J.bi_valid&65535,Gs(J,J.bi_buf),J.bi_buf=Zs>>U-J.bi_valid,J.bi_valid+=la-U):(J.bi_buf|=Zs<<J.bi_valid&65535,J.bi_valid+=la)}function Bs(J,Zs,la){zs(J,la[Zs*2],la[Zs*2+1])}function H(J,Zs){var la=0;do la|=J&1,J>>>=1,la<<=1;while(--Zs>0);return la>>>1}function N(J){J.bi_valid===16?(Gs(J,J.bi_buf),J.bi_buf=0,J.bi_valid=0):J.bi_valid>=8&&(J.pending_buf[J.pending++]=J.bi_buf&255,J.bi_buf>>=8,J.bi_valid-=8)}function V(J,Zs){var la=Zs.dyn_tree,va=Zs.max_code,ws=Zs.stat_desc.static_tree,Rs=Zs.stat_desc.has_stree,v=Zs.stat_desc.extra_bits,D=Zs.stat_desc.extra_base,Ps=Zs.stat_desc.max_length,E,W,Ss,q,ds,Fs,Js=0;for(q=0;q<=$;q++)J.bl_count[q]=0;for(la[J.heap[J.heap_max]*2+1]=0,E=J.heap_max+1;E<I;E++)W=J.heap[E],q=la[la[W*2+1]*2+1]+1,q>Ps&&(q=Ps,Js++),la[W*2+1]=q,!(W>va)&&(J.bl_count[q]++,ds=0,W>=D&&(ds=v[W-D]),Fs=la[W*2],J.opt_len+=Fs*(q+ds),Rs&&(J.static_len+=Fs*(ws[W*2+1]+ds)));if(Js!==0){do{for(q=Ps-1;J.bl_count[q]===0;)q--;J.bl_count[q]--,J.bl_count[q+1]+=2,J.bl_count[Ps]--,Js-=2}while(Js>0);for(q=Ps;q!==0;q--)for(W=J.bl_count[q];W!==0;)Ss=J.heap[--E],!(Ss>va)&&(la[Ss*2+1]!==q&&(J.opt_len+=(q-la[Ss*2+1])*la[Ss*2],la[Ss*2+1]=q),W--)}}function Z(J,Zs,la){var va=new Array($+1),ws=0,Rs,v;for(Rs=1;Rs<=$;Rs++)va[Rs]=ws=ws+la[Rs-1]<<1;for(v=0;v<=Zs;v++){var D=J[v*2+1];D!==0&&(J[v*2]=H(va[D]++,D))}}function as(){var J,Zs,la,va,ws,Rs=new Array($+1);for(la=0,va=0;va<b-1;va++)for(vs[va]=la,J=0;J<1<<is[va];J++)Ys[la++]=va;for(Ys[la-1]=va,ws=0,va=0;va<16;va++)for(gs[va]=ws,J=0;J<1<<R[va];J++)Is[ws++]=va;for(ws>>=7;va<A;va++)for(gs[va]=ws<<7,J=0;J<1<<R[va]-7;J++)Is[256+ws++]=va;for(Zs=0;Zs<=$;Zs++)Rs[Zs]=0;for(J=0;J<=143;)ps[J*2+1]=8,J++,Rs[8]++;for(;J<=255;)ps[J*2+1]=9,J++,Rs[9]++;for(;J<=279;)ps[J*2+1]=7,J++,Rs[7]++;for(;J<=287;)ps[J*2+1]=8,J++,Rs[8]++;for(Z(ps,T+1,Rs),J=0;J<A;J++)bs[J*2+1]=5,bs[J*2]=H(J,5);ys=new xs(ps,is,_+1,T,$),ks=new xs(bs,R,0,A,$),rs=new xs(new Array(0),ss,0,C,L)}function es(J){var Zs;for(Zs=0;Zs<T;Zs++)J.dyn_ltree[Zs*2]=0;for(Zs=0;Zs<A;Zs++)J.dyn_dtree[Zs*2]=0;for(Zs=0;Zs<C;Zs++)J.bl_tree[Zs*2]=0;J.dyn_ltree[P*2]=1,J.opt_len=J.static_len=0,J.last_lit=J.matches=0}function Os(J){J.bi_valid>8?Gs(J,J.bi_buf):J.bi_valid>0&&(J.pending_buf[J.pending++]=J.bi_buf),J.bi_buf=0,J.bi_valid=0}function Ds(J,Zs,la,va){Os(J),Gs(J,la),Gs(J,~la),l.arraySet(J.pending_buf,J.window,Zs,la,J.pending),J.pending+=la}function Ls(J,Zs,la,va){var ws=Zs*2,Rs=la*2;return J[ws]<J[Rs]||J[ws]===J[Rs]&&va[Zs]<=va[la]}function As(J,Zs,la){for(var va=J.heap[la],ws=la<<1;ws<=J.heap_len&&(ws<J.heap_len&&Ls(Zs,J.heap[ws+1],J.heap[ws],J.depth)&&ws++,!Ls(Zs,va,J.heap[ws],J.depth));)J.heap[la]=J.heap[ws],la=ws,ws<<=1;J.heap[la]=va}function Vs(J,Zs,la){var va,ws,Rs=0,v,D;if(J.last_lit!==0)do va=J.pending_buf[J.d_buf+Rs*2]<<8|J.pending_buf[J.d_buf+Rs*2+1],ws=J.pending_buf[J.l_buf+Rs],Rs++,va===0?Bs(J,ws,Zs):(v=Ys[ws],Bs(J,v+_+1,Zs),D=is[v],D!==0&&(ws-=vs[v],zs(J,ws,D)),va--,v=Ns(va),Bs(J,v,la),D=R[v],D!==0&&(va-=gs[v],zs(J,va,D)));while(Rs<J.last_lit);Bs(J,P,Zs)}function Ks(J,Zs){var la=Zs.dyn_tree,va=Zs.stat_desc.static_tree,ws=Zs.stat_desc.has_stree,Rs=Zs.stat_desc.elems,v,D,Ps=-1,E;for(J.heap_len=0,J.heap_max=I,v=0;v<Rs;v++)la[v*2]!==0?(J.heap[++J.heap_len]=Ps=v,J.depth[v]=0):la[v*2+1]=0;for(;J.heap_len<2;)E=J.heap[++J.heap_len]=Ps<2?++Ps:0,la[E*2]=1,J.depth[E]=0,J.opt_len--,ws&&(J.static_len-=va[E*2+1]);for(Zs.max_code=Ps,v=J.heap_len>>1;v>=1;v--)As(J,la,v);E=Rs;do v=J.heap[1],J.heap[1]=J.heap[J.heap_len--],As(J,la,1),D=J.heap[1],J.heap[--J.heap_max]=v,J.heap[--J.heap_max]=D,la[E*2]=la[v*2]+la[D*2],J.depth[E]=(J.depth[v]>=J.depth[D]?J.depth[v]:J.depth[D])+1,la[v*2+1]=la[D*2+1]=E,J.heap[1]=E++,As(J,la,1);while(J.heap_len>=2);J.heap[--J.heap_max]=J.heap[1],V(J,Zs),Z(la,Ps,J.bl_count)}function aa(J,Zs,la){var 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ws;for(zs(J,Zs-257,5),zs(J,la-1,5),zs(J,va-4,4),ws=0;ws<va;ws++)zs(J,J.bl_tree[Y[ws]*2+1],3);ya(J,J.dyn_ltree,Zs-1),ya(J,J.dyn_dtree,la-1)}function $a(J){var Zs=4093624447,la;for(la=0;la<=31;la++,Zs>>>=1)if(Zs&1&&J.dyn_ltree[la*2]!==0)return p;if(J.dyn_ltree[18]!==0||J.dyn_ltree[20]!==0||J.dyn_ltree[26]!==0)return r;for(la=32;la<_;la++)if(J.dyn_ltree[la*2]!==0)return r;return p}var Wa=!1;function Qa(J){Wa||(as(),Wa=!0),J.l_desc=new Ms(J.dyn_ltree,ys),J.d_desc=new Ms(J.dyn_dtree,ks),J.bl_desc=new Ms(J.bl_tree,rs),J.bi_buf=0,J.bi_valid=0,es(J)}function kt(J,Zs,la,va){zs(J,(c<<1)+(va?1:0),3),Ds(J,Zs,la)}function tt(J){zs(J,h<<1,3),Bs(J,P,ps),N(J)}function gn(J,Zs,la,va){var ws,Rs,v=0;J.level>0?(J.strm.data_type===m&&(J.strm.data_type=$a(J)),Ks(J,J.l_desc),Ks(J,J.d_desc),v=Aa(J),ws=J.opt_len+3+7>>>3,Rs=J.static_len+3+7>>>3,Rs<=ws&&(ws=Rs)):ws=Rs=la+5,la+4<=ws&&Zs!==-1?kt(J,Zs,la,va):J.strategy===i||Rs===ws?(zs(J,(h<<1)+(va?1:0),3),Vs(J,ps,bs)):(zs(J,(u<<1)+(va?1:0),3),Ra(J,J.l_desc.max_code+1,J.d_desc.max_code+1,v+1),Vs(J,J.dyn_ltree,J.dyn_dtree)),es(J),va&&Os(J)}function xe(J,Zs,la){return J.pending_buf[J.d_buf+J.last_lit*2]=Zs>>>8&255,J.pending_buf[J.d_buf+J.last_lit*2+1]=Zs&255,J.pending_buf[J.l_buf+J.last_lit]=la&255,J.last_lit++,Zs===0?J.dyn_ltree[la*2]++:(J.matches++,Zs--,J.dyn_ltree[(Ys[la]+_+1)*2]++,J.dyn_dtree[Ns(Zs)*2]++),J.last_lit===J.lit_bufsize-1}e._tr_init=Qa,e._tr_stored_block=kt,e._tr_flush_block=gn,e._tr_tally=xe,e._tr_align=tt},{"../utils/common":5}],12:[function(t,n,e){function 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p==="_"?63:p==="-"?62:r>=97?r-61:r>=65?r-55:r>=48?r-48:"?"}function i(p){var r=l(p[0]),m=l(p[1]),o=l(p[2]),c=l(p[3]),h=r<<2|m>>4&63,u=m<<4&240|o>>2&15,y=o<<6&192|c&63;return[h,u,y]}n.exports=function(p){var r="",m=0;for(m=0;m<p.length;m+=4){var o=i(p.substring(m,m+4));r=r+String.fromCharCode(o[0]),r=r+String.fromCharCode(o[1]),r=r+String.fromCharCode(o[2])}return r}},{}],3:[function(t,n,e){var l=t("./inflate"),i=t("./decode64");n.exports.decode=function(p){var r=i(p);return l(r)}},{"./decode64":2,"./inflate":1}],4:[function(t,n,e){var l=t("./zlib/inflate"),i=t("./utils/common"),p=t("./utils/strings"),r=t("./zlib/constants"),m=t("./zlib/messages"),o=t("./zlib/zstream"),c=t("./zlib/gzheader"),h=Object.prototype.toString;function u(b){if(!(this instanceof u))return new u(b);this.options=i.assign({chunkSize:16384,windowBits:0,to:""},b||{});var 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b=="string"?T.input=p.binstring2buf(b):h.call(b)==="[object ArrayBuffer]"?T.input=new Uint8Array(b):T.input=b,T.next_in=0,T.avail_in=T.input.length;do{if(T.avail_out===0&&(T.output=new i.Buf8(A),T.next_out=0,T.avail_out=A),I=l.inflate(T,r.Z_NO_FLUSH),I===r.Z_NEED_DICT&&C&&(I=l.inflateSetDictionary(this.strm,C)),I===r.Z_BUF_ERROR&&G===!0&&(I=r.Z_OK,G=!1),I!==r.Z_STREAM_END&&I!==r.Z_OK)return this.onEnd(I),this.ended=!0,!1;T.next_out&&(T.avail_out===0||I===r.Z_STREAM_END||T.avail_in===0&&($===r.Z_FINISH||$===r.Z_SYNC_FLUSH))&&(this.options.to==="string"?(U=p.utf8border(T.output,T.next_out),L=T.next_out-U,P=p.buf2string(T.output,U),T.next_out=L,T.avail_out=A-L,L&&i.arraySet(T.output,T.output,U,L,0),this.onData(P)):this.onData(i.shrinkBuf(T.output,T.next_out))),T.avail_in===0&&T.avail_out===0&&(G=!0)}while((T.avail_in>0||T.avail_out===0)&&I!==r.Z_STREAM_END);return I===r.Z_STREAM_END&&($=r.Z_FINISH),$===r.Z_FINISH?(I=l.inflateEnd(this.strm),this.onEnd(I),this.ended=!0,I===r.Z_OK):($===r.Z_SYNC_FLUSH&&(this.onEnd(r.Z_OK),T.avail_out=0),!0)},u.prototype.onData=function(b){this.chunks.push(b)},u.prototype.onEnd=function(b){b===r.Z_OK&&(this.options.to==="string"?this.result=this.chunks.join(""):this.result=i.flattenChunks(this.chunks)),this.chunks=[],this.err=b,this.msg=this.strm.msg};function y(b,_){var T=new u(_);if(T.push(b,!0),T.err)throw T.msg||m[T.err];return T.result}function x(b,_){return _=_||{},_.raw=!0,y(b,_)}e.Inflate=u,e.inflate=y,e.inflateRaw=x,e.ungzip=y},{"./utils/common":5,"./utils/strings":6,"./zlib/constants":8,"./zlib/gzheader":10,"./zlib/inflate":12,"./zlib/messages":14,"./zlib/zstream":15}],5:[function(t,n,e){var l=typeof Uint8Array<"u"&&typeof Uint16Array<"u"&&typeof Int32Array<"u";function i(m,o){return Object.prototype.hasOwnProperty.call(m,o)}e.assign=function(m){for(var o=Array.prototype.slice.call(arguments,1);o.length;){var c=o.shift();if(c){if(typeof c!="object")throw new TypeError(c+"must be non-object");for(var h in c)i(c,h)&&(m[h]=c[h])}}return m},e.shrinkBuf=function(m,o){return m.length===o?m:m.subarray?m.subarray(0,o):(m.length=o,m)};var p={arraySet:function(m,o,c,h,u){if(o.subarray&&m.subarray){m.set(o.subarray(c,c+h),u);return}for(var y=0;y<h;y++)m[u+y]=o[c+y]},flattenChunks:function(m){var o,c,h,u,y,x;for(h=0,o=0,c=m.length;o<c;o++)h+=m[o].length;for(x=new Uint8Array(h),u=0,o=0,c=m.length;o<c;o++)y=m[o],x.set(y,u),u+=y.length;return x}},r={arraySet:function(m,o,c,h,u){for(var y=0;y<h;y++)m[u+y]=o[c+y]},flattenChunks:function(m){return[].concat.apply([],m)}};e.setTyped=function(m){m?(e.Buf8=Uint8Array,e.Buf16=Uint16Array,e.Buf32=Int32Array,e.assign(e,p)):(e.Buf8=Array,e.Buf16=Array,e.Buf32=Array,e.assign(e,r))},e.setTyped(l)},{}],6:[function(t,n,e){var l=t("./common"),i=!0,p=!0;try{String.fromCharCode.apply(null,[0])}catch{i=!1}try{String.fromCharCode.apply(null,new Uint8Array(1))}catch{p=!1}for(var r=new l.Buf8(256),m=0;m<256;m++)r[m]=m>=252?6:m>=248?5:m>=240?4:m>=224?3:m>=192?2:1;r[254]=r[254]=1,e.string2buf=function(c){var h,u,y,x,b,_=c.length,T=0;for(x=0;x<_;x++)u=c.charCodeAt(x),(u&64512)===55296&&x+1<_&&(y=c.charCodeAt(x+1),(y&64512)===56320&&(u=65536+(u-55296<<10)+(y-56320),x++)),T+=u<128?1:u<2048?2:u<65536?3:4;for(h=new l.Buf8(T),b=0,x=0;b<T;x++)u=c.charCodeAt(x),(u&64512)===55296&&x+1<_&&(y=c.charCodeAt(x+1),(y&64512)===56320&&(u=65536+(u-55296<<10)+(y-56320),x++)),u<128?h[b++]=u:u<2048?(h[b++]=192|u>>>6,h[b++]=128|u&63):u<65536?(h[b++]=224|u>>>12,h[b++]=128|u>>>6&63,h[b++]=128|u&63):(h[b++]=240|u>>>18,h[b++]=128|u>>>12&63,h[b++]=128|u>>>6&63,h[b++]=128|u&63);return h};function o(c,h){if(h<65534&&(c.subarray&&p||!c.subarray&&i))return String.fromCharCode.apply(null,l.shrinkBuf(c,h));for(var u="",y=0;y<h;y++)u+=String.fromCharCode(c[y]);return u}e.buf2binstring=function(c){return o(c,c.length)},e.binstring2buf=function(c){for(var h=new l.Buf8(c.length),u=0,y=h.length;u<y;u++)h[u]=c.charCodeAt(u);return h},e.buf2string=function(c,h){var u,y,x,b,_=h||c.length,T=new Array(_*2);for(y=0,u=0;u<_;){if(x=c[u++],x<128){T[y++]=x;continue}if(b=r[x],b>4){T[y++]=65533,u+=b-1;continue}for(x&=b===2?31:b===3?15:7;b>1&&u<_;)x=x<<6|c[u++]&63,b--;if(b>1){T[y++]=65533;continue}x<65536?T[y++]=x:(x-=65536,T[y++]=55296|x>>10&1023,T[y++]=56320|x&1023)}return o(T,y)},e.utf8border=function(c,h){var u;for(h=h||c.length,h>c.length&&(h=c.length),u=h-1;u>=0&&(c[u]&192)===128;)u--;return u<0||u===0?h:u+r[c[u]]>h?u:h}},{"./common":5}],7:[function(t,n,e){function l(i,p,r,m){for(var o=i&65535|0,c=i>>>16&65535|0,h=0;r!==0;){h=r>2e3?2e3:r,r-=h;do o=o+p[m++]|0,c=c+o|0;while(--h);o%=65521,c%=65521}return o|c<<16|0}n.exports=l},{}],8:[function(t,n,e){n.exports={Z_NO_FLUSH:0,Z_PARTIAL_FLUSH:1,Z_SYNC_FLUSH:2,Z_FULL_FLUSH:3,Z_FINISH:4,Z_BLOCK:5,Z_TREES:6,Z_OK:0,Z_STREAM_END:1,Z_NEED_DICT:2,Z_ERRNO:-1,Z_STREAM_ERROR:-2,Z_DATA_ERROR:-3,Z_BUF_ERROR:-5,Z_NO_COMPRESSION:0,Z_BEST_SPEED:1,Z_BEST_COMPRESSION:9,Z_DEFAULT_COMPRESSION:-1,Z_FILTERED:1,Z_HUFFMAN_ONLY:2,Z_RLE:3,Z_FIXED:4,Z_DEFAULT_STRATEGY:0,Z_BINARY:0,Z_TEXT:1,Z_UNKNOWN:2,Z_DEFLATED:8}},{}],9:[function(t,n,e){function l(){for(var r,m=[],o=0;o<256;o++){r=o;for(var c=0;c<8;c++)r=r&1?3988292384^r>>>1:r>>>1;m[o]=r}return m}var i=l();function p(r,m,o,c){var h=i,u=c+o;r^=-1;for(var y=c;y<u;y++)r=r>>>8^h[(r^m[y])&255];return r^-1}n.exports=p},{}],10:[function(t,n,e){function l(){this.text=0,this.time=0,this.xflags=0,this.os=0,this.extra=null,this.extra_len=0,this.name="",this.comment="",this.hcrc=0,this.done=!1}n.exports=l},{}],11:[function(t,n,e){var l=30,i=12;n.exports=function(r,m){var o,c,h,u,y,x,b,_,T,A,C,I,$,U,L,P,G,X,S,is,R,ss,Y,os,ps;o=r.state,c=r.next_in,os=r.input,h=c+(r.avail_in-5),u=r.next_out,ps=r.output,y=u-(m-r.avail_out),x=u+(r.avail_out-257),b=o.dmax,_=o.wsize,T=o.whave,A=o.wnext,C=o.window,I=o.hold,$=o.bits,U=o.lencode,L=o.distcode,P=(1<<o.lenbits)-1,G=(1<<o.distbits)-1;s:do{$<15&&(I+=os[c++]<<$,$+=8,I+=os[c++]<<$,$+=8),X=U[I&P];a:for(;;){if(S=X>>>24,I>>>=S,$-=S,S=X>>>16&255,S===0)ps[u++]=X&65535;else if(S&16){is=X&65535,S&=15,S&&($<S&&(I+=os[c++]<<$,$+=8),is+=I&(1<<S)-1,I>>>=S,$-=S),$<15&&(I+=os[c++]<<$,$+=8,I+=os[c++]<<$,$+=8),X=L[I&G];t:for(;;){if(S=X>>>24,I>>>=S,$-=S,S=X>>>16&255,S&16){if(R=X&65535,S&=15,$<S&&(I+=os[c++]<<$,$+=8,$<S&&(I+=os[c++]<<$,$+=8)),R+=I&(1<<S)-1,R>b){r.msg="invalid distance too far back",o.mode=l;break s}if(I>>>=S,$-=S,S=u-y,R>S){if(S=R-S,S>T&&o.sane){r.msg="invalid distance too far back",o.mode=l;break s}if(ss=0,Y=C,A===0){if(ss+=_-S,S<is){is-=S;do ps[u++]=C[ss++];while(--S);ss=u-R,Y=ps}}else 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l=t("../utils/common"),i=t("./adler32"),p=t("./crc32"),r=t("./inffast"),m=t("./inftrees"),o=0,c=1,h=2,u=4,y=5,x=6,b=0,_=1,T=2,A=-2,C=-3,I=-4,$=-5,U=8,L=1,P=2,G=3,X=4,S=5,is=6,R=7,ss=8,Y=9,os=10,ps=11,bs=12,Is=13,Ys=14,vs=15,gs=16,xs=17,ys=18,ks=19,rs=20,Ms=21,Ns=22,Gs=23,zs=24,Bs=25,H=26,N=27,V=28,Z=29,as=30,es=31,Os=32,Ds=852,Ls=592,As=15,Vs=As;function Ks(ws){return(ws>>>24&255)+(ws>>>8&65280)+((ws&65280)<<8)+((ws&255)<<24)}function aa(){this.mode=0,this.last=!1,this.wrap=0,this.havedict=!1,this.flags=0,this.dmax=0,this.check=0,this.total=0,this.head=null,this.wbits=0,this.wsize=0,this.whave=0,this.wnext=0,this.window=null,this.hold=0,this.bits=0,this.length=0,this.offset=0,this.extra=0,this.lencode=null,this.distcode=null,this.lenbits=0,this.distbits=0,this.ncode=0,this.nlen=0,this.ndist=0,this.have=0,this.next=null,this.lens=new l.Buf16(320),this.work=new l.Buf16(288),this.lendyn=null,this.distdyn=null,this.sane=0,this.back=0,this.was=0}function ya(ws){var Rs;return!ws||!ws.state?A:(Rs=ws.state,ws.total_in=ws.total_out=Rs.total=0,ws.msg="",Rs.wrap&&(ws.adler=Rs.wrap&1),Rs.mode=L,Rs.last=0,Rs.havedict=0,Rs.dmax=32768,Rs.head=null,Rs.hold=0,Rs.bits=0,Rs.lencode=Rs.lendyn=new l.Buf32(Ds),Rs.distcode=Rs.distdyn=new l.Buf32(Ls),Rs.sane=1,Rs.back=-1,b)}function Aa(ws){var Rs;return!ws||!ws.state?A:(Rs=ws.state,Rs.wsize=0,Rs.whave=0,Rs.wnext=0,ya(ws))}function Ra(ws,Rs){var v,D;return!ws||!ws.state||(D=ws.state,Rs<0?(v=0,Rs=-Rs):(v=(Rs>>4)+1,Rs<48&&(Rs&=15)),Rs&&(Rs<8||Rs>15))?A:(D.window!==null&&D.wbits!==Rs&&(D.window=null),D.wrap=v,D.wbits=Rs,Aa(ws))}function $a(ws,Rs){var v,D;return ws?(D=new aa,ws.state=D,D.window=null,v=Ra(ws,Rs),v!==b&&(ws.state=null),v):A}function Wa(ws){return $a(ws,Vs)}var Qa=!0,kt,tt;function gn(ws){if(Qa){var Rs;for(kt=new l.Buf32(512),tt=new 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A;v=ws.state,v.mode===bs&&(v.mode=Is),W=ws.next_out,Ps=ws.output,q=ws.avail_out,E=ws.next_in,D=ws.input,Ss=ws.avail_in,ds=v.hold,Fs=v.bits,Js=Ss,ia=q,Un=b;s:for(;;)switch(v.mode){case L:if(v.wrap===0){v.mode=Is;break}for(;Fs<16;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(v.wrap&2&&ds===35615){v.check=0,$t[0]=ds&255,$t[1]=ds>>>8&255,v.check=p(v.check,$t,2,0),ds=0,Fs=0,v.mode=P;break}if(v.flags=0,v.head&&(v.head.done=!1),!(v.wrap&1)||(((ds&255)<<8)+(ds>>8))%31){ws.msg="incorrect header check",v.mode=as;break}if((ds&15)!==U){ws.msg="unknown compression method",v.mode=as;break}if(ds>>>=4,Fs-=4,zt=(ds&15)+8,v.wbits===0)v.wbits=zt;else if(zt>v.wbits){ws.msg="invalid window size",v.mode=as;break}v.dmax=1<<zt,ws.adler=v.check=1,v.mode=ds&512?os:bs,ds=0,Fs=0;break;case P:for(;Fs<16;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(v.flags=ds,(v.flags&255)!==U){ws.msg="unknown compression method",v.mode=as;break}if(v.flags&57344){ws.msg="unknown header flags set",v.mode=as;break}v.head&&(v.head.text=ds>>8&1),v.flags&512&&($t[0]=ds&255,$t[1]=ds>>>8&255,v.check=p(v.check,$t,2,0)),ds=0,Fs=0,v.mode=G;case G:for(;Fs<32;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.head&&(v.head.time=ds),v.flags&512&&($t[0]=ds&255,$t[1]=ds>>>8&255,$t[2]=ds>>>16&255,$t[3]=ds>>>24&255,v.check=p(v.check,$t,4,0)),ds=0,Fs=0,v.mode=X;case X:for(;Fs<16;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.head&&(v.head.xflags=ds&255,v.head.os=ds>>8),v.flags&512&&($t[0]=ds&255,$t[1]=ds>>>8&255,v.check=p(v.check,$t,2,0)),ds=0,Fs=0,v.mode=S;case S:if(v.flags&1024){for(;Fs<16;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.length=ds,v.head&&(v.head.extra_len=ds),v.flags&512&&($t[0]=ds&255,$t[1]=ds>>>8&255,v.check=p(v.check,$t,2,0)),ds=0,Fs=0}else v.head&&(v.head.extra=null);v.mode=is;case is:if(v.flags&1024&&(oa=v.length,oa>Ss&&(oa=Ss),oa&&(v.head&&(zt=v.head.extra_len-v.length,v.head.extra||(v.head.extra=new Array(v.head.extra_len)),l.arraySet(v.head.extra,D,E,oa,zt)),v.flags&512&&(v.check=p(v.check,D,oa,E)),Ss-=oa,E+=oa,v.length-=oa),v.length))break s;v.length=0,v.mode=R;case R:if(v.flags&2048){if(Ss===0)break s;oa=0;do zt=D[E+oa++],v.head&&zt&&v.length<65536&&(v.head.name+=String.fromCharCode(zt));while(zt&&oa<Ss);if(v.flags&512&&(v.check=p(v.check,D,oa,E)),Ss-=oa,E+=oa,zt)break s}else v.head&&(v.head.name=null);v.length=0,v.mode=ss;case ss:if(v.flags&4096){if(Ss===0)break s;oa=0;do zt=D[E+oa++],v.head&&zt&&v.length<65536&&(v.head.comment+=String.fromCharCode(zt));while(zt&&oa<Ss);if(v.flags&512&&(v.check=p(v.check,D,oa,E)),Ss-=oa,E+=oa,zt)break s}else v.head&&(v.head.comment=null);v.mode=Y;case Y:if(v.flags&512){for(;Fs<16;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(ds!==(v.check&65535)){ws.msg="header crc mismatch",v.mode=as;break}ds=0,Fs=0}v.head&&(v.head.hcrc=v.flags>>9&1,v.head.done=!0),ws.adler=v.check=0,v.mode=bs;break;case os:for(;Fs<32;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}ws.adler=v.check=Ks(ds),ds=0,Fs=0,v.mode=ps;case ps:if(v.havedict===0)return ws.next_out=W,ws.avail_out=q,ws.next_in=E,ws.avail_in=Ss,v.hold=ds,v.bits=Fs,T;ws.adler=v.check=1,v.mode=bs;case bs:if(Rs===y||Rs===x)break s;case Is:if(v.last){ds>>>=Fs&7,Fs-=Fs&7,v.mode=N;break}for(;Fs<3;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}switch(v.last=ds&1,ds>>>=1,Fs-=1,ds&3){case 0:v.mode=Ys;break;case 1:if(gn(v),v.mode=rs,Rs===x){ds>>>=2,Fs-=2;break s}break;case 2:v.mode=xs;break;case 3:ws.msg="invalid block type",v.mode=as}ds>>>=2,Fs-=2;break;case Ys:for(ds>>>=Fs&7,Fs-=Fs&7;Fs<32;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if((ds&65535)!==(ds>>>16^65535)){ws.msg="invalid stored block lengths",v.mode=as;break}if(v.length=ds&65535,ds=0,Fs=0,v.mode=vs,Rs===x)break s;case vs:v.mode=gs;case gs:if(oa=v.length,oa){if(oa>Ss&&(oa=Ss),oa>q&&(oa=q),oa===0)break s;l.arraySet(Ps,D,E,oa,W),Ss-=oa,E+=oa,q-=oa,W+=oa,v.length-=oa;break}v.mode=bs;break;case xs:for(;Fs<14;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(v.nlen=(ds&31)+257,ds>>>=5,Fs-=5,v.ndist=(ds&31)+1,ds>>>=5,Fs-=5,v.ncode=(ds&15)+4,ds>>>=4,Fs-=4,v.nlen>286||v.ndist>30){ws.msg="too many length or distance symbols",v.mode=as;break}v.have=0,v.mode=ys;case ys:for(;v.have<v.ncode;){for(;Fs<3;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.lens[Da[v.have++]]=ds&7,ds>>>=3,Fs-=3}for(;v.have<19;)v.lens[Da[v.have++]]=0;if(v.lencode=v.lendyn,v.lenbits=7,we={bits:v.lenbits},Un=m(o,v.lens,0,19,v.lencode,0,v.work,we),v.lenbits=we.bits,Un){ws.msg="invalid code lengths set",v.mode=as;break}v.have=0,v.mode=ks;case ks:for(;v.have<v.nlen+v.ndist;){for(;Ja=v.lencode[ds&(1<<v.lenbits)-1],us=Ja>>>24,Rt=Ja>>>16&255,cn=Ja&65535,!(us<=Fs);){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(cn<16)ds>>>=us,Fs-=us,v.lens[v.have++]=cn;else{if(cn===16){for(jn=us+2;Fs<jn;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(ds>>>=us,Fs-=us,v.have===0){ws.msg="invalid bit length repeat",v.mode=as;break}zt=v.lens[v.have-1],oa=3+(ds&3),ds>>>=2,Fs-=2}else if(cn===17){for(jn=us+3;Fs<jn;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}ds>>>=us,Fs-=us,zt=0,oa=3+(ds&7),ds>>>=3,Fs-=3}else{for(jn=us+7;Fs<jn;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}ds>>>=us,Fs-=us,zt=0,oa=11+(ds&127),ds>>>=7,Fs-=7}if(v.have+oa>v.nlen+v.ndist){ws.msg="invalid bit length repeat",v.mode=as;break}for(;oa--;)v.lens[v.have++]=zt}}if(v.mode===as)break;if(v.lens[256]===0){ws.msg="invalid code -- missing end-of-block",v.mode=as;break}if(v.lenbits=9,we={bits:v.lenbits},Un=m(c,v.lens,0,v.nlen,v.lencode,0,v.work,we),v.lenbits=we.bits,Un){ws.msg="invalid literal/lengths set",v.mode=as;break}if(v.distbits=6,v.distcode=v.distdyn,we={bits:v.distbits},Un=m(h,v.lens,v.nlen,v.ndist,v.distcode,0,v.work,we),v.distbits=we.bits,Un){ws.msg="invalid distances set",v.mode=as;break}if(v.mode=rs,Rs===x)break s;case rs:v.mode=Ms;case Ms:if(Ss>=6&&q>=258){ws.next_out=W,ws.avail_out=q,ws.next_in=E,ws.avail_in=Ss,v.hold=ds,v.bits=Fs,r(ws,ia),W=ws.next_out,Ps=ws.output,q=ws.avail_out,E=ws.next_in,D=ws.input,Ss=ws.avail_in,ds=v.hold,Fs=v.bits,v.mode===bs&&(v.back=-1);break}for(v.back=0;Ja=v.lencode[ds&(1<<v.lenbits)-1],us=Ja>>>24,Rt=Ja>>>16&255,cn=Ja&65535,!(us<=Fs);){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(Rt&&(Rt&240)===0){for(fn=us,Bp=Rt,Mp=cn;Ja=v.lencode[Mp+((ds&(1<<fn+Bp)-1)>>fn)],us=Ja>>>24,Rt=Ja>>>16&255,cn=Ja&65535,!(fn+us<=Fs);){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}ds>>>=fn,Fs-=fn,v.back+=fn}if(ds>>>=us,Fs-=us,v.back+=us,v.length=cn,Rt===0){v.mode=H;break}if(Rt&32){v.back=-1,v.mode=bs;break}if(Rt&64){ws.msg="invalid literal/length code",v.mode=as;break}v.extra=Rt&15,v.mode=Ns;case Ns:if(v.extra){for(jn=v.extra;Fs<jn;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.length+=ds&(1<<v.extra)-1,ds>>>=v.extra,Fs-=v.extra,v.back+=v.extra}v.was=v.length,v.mode=Gs;case Gs:for(;Ja=v.distcode[ds&(1<<v.distbits)-1],us=Ja>>>24,Rt=Ja>>>16&255,cn=Ja&65535,!(us<=Fs);){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if((Rt&240)===0){for(fn=us,Bp=Rt,Mp=cn;Ja=v.distcode[Mp+((ds&(1<<fn+Bp)-1)>>fn)],us=Ja>>>24,Rt=Ja>>>16&255,cn=Ja&65535,!(fn+us<=Fs);){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}ds>>>=fn,Fs-=fn,v.back+=fn}if(ds>>>=us,Fs-=us,v.back+=us,Rt&64){ws.msg="invalid distance code",v.mode=as;break}v.offset=cn,v.extra=Rt&15,v.mode=zs;case zs:if(v.extra){for(jn=v.extra;Fs<jn;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}v.offset+=ds&(1<<v.extra)-1,ds>>>=v.extra,Fs-=v.extra,v.back+=v.extra}if(v.offset>v.dmax){ws.msg="invalid distance too far back",v.mode=as;break}v.mode=Bs;case Bs:if(q===0)break s;if(oa=ia-q,v.offset>oa){if(oa=v.offset-oa,oa>v.whave&&v.sane){ws.msg="invalid distance too far back",v.mode=as;break}oa>v.wnext?(oa-=v.wnext,Ha=v.wsize-oa):Ha=v.wnext-oa,oa>v.length&&(oa=v.length),ms=v.window}else ms=Ps,Ha=W-v.offset,oa=v.length;oa>q&&(oa=q),q-=oa,v.length-=oa;do Ps[W++]=ms[Ha++];while(--oa);v.length===0&&(v.mode=Ms);break;case H:if(q===0)break s;Ps[W++]=v.length,q--,v.mode=Ms;break;case N:if(v.wrap){for(;Fs<32;){if(Ss===0)break s;Ss--,ds|=D[E++]<<Fs,Fs+=8}if(ia-=q,ws.total_out+=ia,v.total+=ia,ia&&(ws.adler=v.check=v.flags?p(v.check,Ps,ia,W-ia):i(v.check,Ps,ia,W-ia)),ia=q,(v.flags?ds:Ks(ds))!==v.check){ws.msg="incorrect data check",v.mode=as;break}ds=0,Fs=0}v.mode=V;case V:if(v.wrap&&v.flags){for(;Fs<32;){if(Ss===0)break s;Ss--,ds+=D[E++]<<Fs,Fs+=8}if(ds!==(v.total&4294967295)){ws.msg="incorrect length check",v.mode=as;break}ds=0,Fs=0}v.mode=Z;case Z:Un=_;break s;case as:Un=C;break s;case es:return I;case Os:default:return A}return ws.next_out=W,ws.avail_out=q,ws.next_in=E,ws.avail_in=Ss,v.hold=ds,v.bits=Fs,(v.wsize||ia!==ws.avail_out&&v.mode<as&&(v.mode<N||Rs!==u))&&xe(ws,ws.output,ws.next_out,ia-ws.avail_out),Js-=ws.avail_in,ia-=ws.avail_out,ws.total_in+=Js,ws.total_out+=ia,v.total+=ia,v.wrap&&ia&&(ws.adler=v.check=v.flags?p(v.check,Ps,ia,ws.next_out-ia):i(v.check,Ps,ia,ws.next_out-ia)),ws.data_type=v.bits+(v.last?64:0)+(v.mode===bs?128:0)+(v.mode===rs||v.mode===vs?256:0),(Js===0&&ia===0||Rs===u)&&Un===b&&(Un=$),Un}function Zs(ws){if(!ws||!ws.state)return A;var Rs=ws.state;return Rs.window&&(Rs.window=null),ws.state=null,b}function la(ws,Rs){var v;return!ws||!ws.state||(v=ws.state,(v.wrap&2)===0)?A:(v.head=Rs,Rs.done=!1,b)}function va(ws,Rs){var v=Rs.length,D,Ps,E;return!ws||!ws.state||(D=ws.state,D.wrap!==0&&D.mode!==ps)?A:D.mode===ps&&(Ps=1,Ps=i(Ps,Rs,v,0),Ps!==D.check)?C:(E=xe(ws,Rs,v,v),E?(D.mode=es,I):(D.havedict=1,b))}e.inflateReset=Aa,e.inflateReset2=Ra,e.inflateResetKeep=ya,e.inflateInit=Wa,e.inflateInit2=$a,e.inflate=J,e.inflateEnd=Zs,e.inflateGetHeader=la,e.inflateSetDictionary=va,e.inflateInfo="pako inflate (from Nodeca project)"},{"../utils/common":5,"./adler32":7,"./crc32":9,"./inffast":11,"./inftrees":13}],13:[function(t,n,e){var l=t("../utils/common"),i=15,p=852,r=592,m=0,o=1,c=2,h=[3,4,5,6,7,8,9,10,11,13,15,17,19,23,27,31,35,43,51,59,67,83,99,115,131,163,195,227,258,0,0],u=[16,16,16,16,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19,20,20,20,20,21,21,21,21,16,72,78],y=[1,2,3,4,5,7,9,13,17,25,33,49,65,97,129,193,257,385,513,769,1025,1537,2049,3073,4097,6145,8193,12289,16385,24577,0,0],x=[16,16,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,27,27,28,28,29,29,64,64];n.exports=function(_,T,A,C,I,$,U,L){var P=L.bits,G=0,X=0,S=0,is=0,R=0,ss=0,Y=0,os=0,ps=0,bs=0,Is,Ys,vs,gs,xs,ys=null,ks=0,rs,Ms=new l.Buf16(i+1),Ns=new l.Buf16(i+1),Gs=null,zs=0,Bs,H,N;for(G=0;G<=i;G++)Ms[G]=0;for(X=0;X<C;X++)Ms[T[A+X]]++;for(R=P,is=i;is>=1&&Ms[is]===0;is--);if(R>is&&(R=is),is===0)return I[$++]=1<<24|64<<16|0,I[$++]=1<<24|64<<16|0,L.bits=1,0;for(S=1;S<is&&Ms[S]===0;S++);for(R<S&&(R=S),os=1,G=1;G<=i;G++)if(os<<=1,os-=Ms[G],os<0)return-1;if(os>0&&(_===m||is!==1))return-1;for(Ns[1]=0,G=1;G<i;G++)Ns[G+1]=Ns[G]+Ms[G];for(X=0;X<C;X++)T[A+X]!==0&&(U[Ns[T[A+X]]++]=X);if(_===m?(ys=Gs=U,rs=19):_===o?(ys=h,ks-=257,Gs=u,zs-=257,rs=256):(ys=y,Gs=x,rs=-1),bs=0,X=0,G=S,xs=$,ss=R,Y=0,vs=-1,ps=1<<R,gs=ps-1,_===o&&ps>p||_===c&&ps>r)return 1;for(;;){Bs=G-Y,U[X]<rs?(H=0,N=U[X]):U[X]>rs?(H=Gs[zs+U[X]],N=ys[ks+U[X]]):(H=96,N=0),Is=1<<G-Y,Ys=1<<ss,S=Ys;do Ys-=Is,I[xs+(bs>>Y)+Ys]=Bs<<24|H<<16|N|0;while(Ys!==0);for(Is=1<<G-1;bs&Is;)Is>>=1;if(Is!==0?(bs&=Is-1,bs+=Is):bs=0,X++,--Ms[G]===0){if(G===is)break;G=T[A+U[X]]}if(G>R&&(bs&gs)!==vs){for(Y===0&&(Y=R),xs+=S,ss=G-Y,os=1<<ss;ss+Y<is&&(os-=Ms[ss+Y],!(os<=0));)ss++,os<<=1;if(ps+=1<<ss,_===o&&ps>p||_===c&&ps>r)return 1;vs=bs&gs,I[vs]=R<<24|ss<<16|xs-$|0}}return bs!==0&&(I[xs+bs]=G-Y<<24|64<<16|0),L.bits=R,0}},{"../utils/common":5}],14:[function(t,n,e){n.exports={2:"need dictionary",1:"stream end",0:"","-1":"file error","-2":"stream error","-3":"data error","-4":"insufficient memory","-5":"buffer error","-6":"incompatible version"}},{}],15:[function(t,n,e){function l(){this.input=null,this.next_in=0,this.avail_in=0,this.total_in=0,this.output=null,this.next_out=0,this.avail_out=0,this.total_out=0,this.msg="",this.state=null,this.data_type=2,this.adler=0}n.exports=l},{}]},{},[3])(3)})})(dB);var jG=dB.exports,GG={encode:UG.encode,decode:jG.decode};const WG=B8(GG),hv="if ",YG="switch ",XG="match ",uv="while ",dv="group ",gv="partition ",gB="lane ",W1="elif ",Y1="else if ",z7="else",D7="default",KG="case _",X1="case ",L7=":",ZG="  ";class QG{constructor(a,t){this.content=a.trim(),this.level=t}isReservedWord(){return this.content==="start"||this.content==="stop"||this.content==="kill"||this.content==="detach"||this.content==="break"||this.content==="end"||this.content==="fork"||this.content==="fork again"||this.content==="end fork"||this.content==="end merge"}isStatementOfType(...a){return a.some(t=>this.content.startsWith(t))&&this.content.endsWith(L7)}takeTagOfStat(a){return this.content.substring(a.length,this.content.length-L7.length).trim()}}const JG={[hv]:sW,[YG]:S7,[XG]:S7,[uv]:aW,[dv]:tW,[gv]:nW,[gB]:eW};function Ep(s,a,t){let n="",e=a;for(;e<s.length&&(t===-1||s[e].level>t);){const l=s[e];if(l.isReservedWord()){n+=l.content+`
`,e++;continue}let i=!1;for(const[p,r]of Object.entries(JG))if(l.isStatementOfType(p)){const{result:m,nextIndex:o}=r(s,e,p);n+=m,e=o,i=!0;break}i||(l.content.length>0&&(n+=`:${l.content};
`),e++)}return{result:n,nextIndex:e}}function sW(s,a){let t=hv;const n=s[a],e=n.takeTagOfStat(hv);let l=a+1;if(l<s.length&&s[l].level>n.level){t+=`(${e}) then (yes)
`;const{result:i,nextIndex:p}=Ep(s,l,n.level);t+=cr(i),l=p}for(;l<s.length&&s[l].level===n.level&&s[l].isStatementOfType(W1,Y1,z7);){const i=s[l];if(i.isStatementOfType(Y1)){const m=i.takeTagOfStat(Y1);t+=`else if(${m}) then (yes)
`}else if(i.isStatementOfType(W1)){const m=i.takeTagOfStat(W1);t+=`else if(${m}) then (yes)
`}else i.isStatementOfType(z7)&&(t+=`else (else)
`);const{result:p,nextIndex:r}=Ep(s,l+1,n.level);t+=cr(p),l=r}return t+=`endif
`,{result:t,nextIndex:l}}function S7(s,a,t){let n="switch ";const e=s[a],l=e.takeTagOfStat(t);let i=a+1;n+=`(${l})
`;let p=!1;for(;i<s.length&&s[i].level>=e.level&&s[i].isStatementOfType(X1,D7);){const r=s[i];if(i++,r.isStatementOfType(D7)||r.isStatementOfType(KG)){n+=`case (default)
`,p=!0;const{result:m,nextIndex:o}=Ep(s,i,r.level);n+=cr(m),i=o}else if(r.isStatementOfType(X1)){const m=r.takeTagOfStat(X1);n+=`case (${m})
`;const{result:o,nextIndex:c}=Ep(s,i,r.level);n+=cr(o),i=c}}return p||(n+=`case (default)
`),n+=`endswitch
`,{result:n,nextIndex:i}}function aW(s,a){const t=s[a],n=t.takeTagOfStat(uv);let e=`${uv}(${n}) is (true)
`,l=a+1;const{result:i,nextIndex:p}=Ep(s,l,t.level);return e+=cr(i),l=p,e+=`endwhile
`,{result:e,nextIndex:l}}function tW(s,a){const t=s[a],n=t.takeTagOfStat(dv);let e=`${dv}${n}
`,l=a+1;const{result:i,nextIndex:p}=Ep(s,l,t.level);return e+=cr(i),l=p,e+=`endgroup
`,{result:e,nextIndex:l}}function nW(s,a){const t=s[a],n=t.takeTagOfStat(gv);let e=`${gv}${n} {
`,l=a+1;const{result:i,nextIndex:p}=Ep(s,l,t.level);return e+=cr(i),l=p,e+=`}
`,{result:e,nextIndex:l}}function eW(s,a){return{result:"|"+s[a].takeTagOfStat(gB)+`|
`,nextIndex:a+1}}function cr(s){return s.split(`
`).filter(a=>a!=="").map(a=>ZG+a).join(`
`)+`
`}function fB(s){let a=`@startuml
`;const t=s.map(e=>new QG(e.content.trim(),e.level)),{result:n}=Ep(t,0,-1);return a+=n,a+="@enduml",a}fa.factory({id:"json2pumlJson",name:"json到可视化",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>(t=`@startjson
`+t+`
@endjson
`,Pd(t,s),s)});fa.factory({id:"list2pumlWBS",name:"列表到puml工作分解结构",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{let n=Ga.list2data(t);n=Ga.data2strict(n);let e=`@startwbs
`;for(let l of n)l.content.startsWith("< ")?e+="*".repeat(l.level+1)+"< "+l.content.slice(2)+`
`:e+="*".repeat(l.level+1)+" "+l.content+`
`;return e+="@endwbs",Pd(e,s),s}});fa.factory({id:"list2pumlMindmap",name:"列表到puml思维导图",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{let n=Ga.list2data(t);n=Ga.data2strict(n);let e=`@startmindmap
`;for(let l of n)e+="*".repeat(l.level+1)+" "+l.content+`
`;return e+="@endmindmap",Pd(e,s),s}});fa.factory({id:"list2pumlActivityDiagramText",name:"列表到puml活动图文本",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>fB(Ga.data2strict(Ga.list2data(t)))});fa.factory({id:"list2pumlActivityDiagram",name:"列表到puml活动图",process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{const n=fB(Ga.data2strict(Ga.list2data(t)));return Pd(n,s),s}});async function Pd(s,a){var t=WG.encode(s);let n="http://www.plantuml.com/plantuml/img/"+t;return a.innerHTML=`<img src="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2F%24%7Bn%7D">`,a}fa.factory({id:"title2mindmap",name:"标题到脑图",process_param:Cs.text,process_return:Cs.el,process:async(s,a,t)=>{const n=Ga.title2data(t);return await vB(n,s)}});fa.factory({id:"list2mindmap",name:"列表转mermaid思维导图",process_param:Cs.text,process_return:Cs.el,process:async(s,a,t)=>{const n=Ga.list2data(t);return await vB(n,s)}});fa.factory({id:"list2mermaid",name:"列表转mermaid流程图",match:/^list2mermaid(\((.*)\))?$/,process_param:Cs.text,process_return:Cs.el,process:(s,a,t)=>{let n=a.match(/^list2mermaid(\((.*)\))?$/);if(!n)return console.error("no match",n),s;let e="graph LR";n[2]&&(e=n[2]);const l=Ga.list2data(t),i=e+`
`+yB(l);return M8(i,s),s}});fa.factory({id:"list2mermaidText",name:"列表转mermaid文本",match:/^list2mermaidText(\((.*)\))?$/,detail:"列表转mermaid文本",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=a.match(/^list2mermaidText(\((.*)\))?$/);if(!n)return console.error("no match",n),"error, no match";let e="graph LR";n[2]&&(e=n[2]);const l=Ga.list2data(t);return e+`
`+yB(l)}});fa.factory({id:"list2mehrmaidText",name:"列表转mehrmaid文本",match:/^list2mehrmaidText(\((.*)\))?$/,detail:"需要配合mehrmaid插件和code(mehrmaid)使用，或使用别名简化",process_param:Cs.text,process_return:Cs.text,process:(s,a,t)=>{let n=a.match(/^list2mehrmaidText(\((.*)\))?$/);if(!n)return console.error("no match",n),"error, no match";let e="flowchart LR";n[2]&&(e=n[2]);const l=Ga.list2data(t);return e+`
`+lW(l)}});fa.factory({id:"mermaid-with",name:"新mermaid",match:/^mermaid(\((.*)\))?$/,default:"mermaid(graph TB)",detail:"由于需要兼容脑图，这里会使用插件内置的最新版mermaid",process_param:Cs.text,process_return:Cs.el,process:async(s,a,t)=>{let n=a.match(/^mermaid(\((.*)\))?$/);return n?(n[2]&&(t=n[2]+`
`+t),M8(t,s)):s}});function yB(s){let a=[],t="",n=999;for(let l=0;l<s.length;l++){if(s[l].level>n)t=t+" --> "+s[l].content;else{a.push(t),t="";for(let i=l;i>=0;i--)if(s[i].level<s[l].level){t=s[i].content;break}t&&(t=t+" --> "),t=t+s[l].content}n=s[l].level}return a.push(t),a.join(`
`)}function lW(s){const a=[];for(let i=0;i<s.length;i++)a[i]=s[i].content,s[i].content=i.toString();let t=[],n="",e=999;for(let i=0;i<s.length;i++){if(s[i].level>e)n=n+" --> "+s[i].content;else{t.push(n),n="";for(let p=i;p>=0;p--)if(s[p].level<s[i].level){n=s[p].content;break}n&&(n=n+" --> "),n=n+s[i].content}e=s[i].level}t.push(n);let l=t.join(`
`);l+=`

`;for(let i=0;i<a.length;i++)l+=`${i}(("${a[i]}"))
`;return l}async function vB(s,a){let t=[];for(let e of s){let l="";for(let i=0;i<e.level;i++)l+=" ";t.push(l+e.content.replace(`
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Uint16Array('ᵁ<Õıʊҝջאٵ۞ޢߖࠏ੊ઑඡ๭༉༦჊ረዡᐕᒝᓃᓟᔥ\0\0\0\0\0\0ᕫᛍᦍᰒᷝ὾⁠↰⊍⏀⏻⑂⠤⤒ⴈ⹈⿎〖㊺㘹㞬㣾㨨㩱㫠㬮ࠀEMabcfglmnoprstu\\bfms„‹•˜¦³¹ÈÏlig耻Æ䃆P耻&䀦cute耻Á䃁reve;䄂Āiyx}rc耻Â䃂;䐐r;쀀𝔄rave耻À䃀pha;䎑acr;䄀d;橓Āgp¡on;䄄f;쀀𝔸plyFunction;恡ing耻Å䃅Ācs¾Ãr;쀀𝒜ign;扔ilde耻Ã䃃ml耻Ä䃄ЀaceforsuåûþėĜĢħĪĀcrêòkslash;或Ŷöø;櫧ed;挆y;䐑ƀcrtąċĔause;戵noullis;愬a;䎒r;쀀𝔅pf;쀀𝔹eve;䋘còēmpeq;扎܀HOacdefhilorsuōőŖƀƞƢƵƷƺǜȕɳɸɾcy;䐧PY耻©䂩ƀcpyŝŢźute;䄆Ā;iŧŨ拒talDifferentialD;慅leys;愭ȀaeioƉƎƔƘron;䄌dil耻Ç䃇rc;䄈nint;戰ot;䄊ĀdnƧƭilla;䂸terDot;䂷òſi;䎧rcleȀDMPTLJNjǑǖot;抙inus;抖lus;投imes;抗oĀcsǢǸkwiseContourIntegral;戲eCurlyĀDQȃȏoubleQuote;思uote;怙ȀlnpuȞȨɇɕonĀ;eȥȦ户;橴ƀgitȯȶȺruent;扡nt;戯ourIntegral;戮ĀfrɌɎ;愂oduct;成nterClockwiseContourIntegral;戳oss;樯cr;쀀𝒞pĀ;Cʄʅ拓ap;才րDJSZacefiosʠʬʰʴʸˋ˗ˡ˦̳ҍĀ;oŹʥtrahd;椑cy;䐂cy;䐅cy;䐏ƀgrsʿ˄ˇger;怡r;憡hv;櫤Āayː˕ron;䄎;䐔lĀ;t˝˞戇a;䎔r;쀀𝔇Āaf˫̧Ācm˰̢riticalȀADGT̖̜̀̆cute;䂴oŴ̋̍;䋙bleAcute;䋝rave;䁠ilde;䋜ond;拄ferentialD;慆Ѱ̽\0\0\0͔͂\0Ѕf;쀀𝔻ƀ;DE͈͉͍䂨ot;惜qual;扐blèCDLRUVͣͲ΂ϏϢϸontourIntegraìȹoɴ͹\0\0ͻ»͉nArrow;懓Āeo·ΤftƀARTΐΖΡrrow;懐ightArrow;懔eåˊngĀLRΫτeftĀARγιrrow;柸ightArrow;柺ightArrow;柹ightĀATϘϞrrow;懒ee;抨pɁϩ\0\0ϯrrow;懑ownArrow;懕erticalBar;戥ǹABLRTaВЪаўѿͼrrowƀ;BUНОТ憓ar;椓pArrow;懵reve;䌑eft˒к\0ц\0ѐightVector;楐eeVector;楞ectorĀ;Bљњ憽ar;楖ightǔѧ\0ѱeeVector;楟ectorĀ;BѺѻ懁ar;楗eeĀ;A҆҇护rrow;憧ĀctҒҗr;쀀𝒟rok;䄐ࠀNTacdfglmopqstuxҽӀӄӋӞӢӧӮӵԡԯԶՒ՝ՠեG;䅊H耻Ð䃐cute耻É䃉ƀaiyӒӗӜron;䄚rc耻Ê䃊;䐭ot;䄖r;쀀𝔈rave耻È䃈ement;戈ĀapӺӾcr;䄒tyɓԆ\0\0ԒmallSquare;旻erySmallSquare;斫ĀgpԦԪon;䄘f;쀀𝔼silon;䎕uĀaiԼՉlĀ;TՂՃ橵ilde;扂librium;懌Āci՗՚r;愰m;橳a;䎗ml耻Ë䃋Āipժկsts;戃onentialE;慇ʀcfiosօֈ֍ֲ׌y;䐤r;쀀𝔉lledɓ֗\0\0֣mallSquare;旼erySmallSquare;斪Ͱֺ\0ֿ\0\0ׄf;쀀𝔽All;戀riertrf;愱cò׋؀JTabcdfgorstר׬ׯ׺؀ؒؖ؛؝أ٬ٲcy;䐃耻>䀾mmaĀ;d׷׸䎓;䏜reve;䄞ƀeiy؇،ؐdil;䄢rc;䄜;䐓ot;䄠r;쀀𝔊;拙pf;쀀𝔾eater̀EFGLSTصلَٖٛ٦qualĀ;Lؾؿ扥ess;招ullEqual;执reater;檢ess;扷lantEqual;橾ilde;扳cr;쀀𝒢;扫ЀAacfiosuڅڋږڛڞڪھۊRDcy;䐪Āctڐڔek;䋇;䁞irc;䄤r;愌lbertSpace;愋ǰگ\0ڲf;愍izontalLine;攀Āctۃۅòکrok;䄦mpńېۘownHumðįqual;扏܀EJOacdfgmnostuۺ۾܃܇܎ܚܞܡܨ݄ݸދޏޕcy;䐕lig;䄲cy;䐁cute耻Í䃍Āiyܓܘrc耻Î䃎;䐘ot;䄰r;愑rave耻Ì䃌ƀ;apܠܯܿĀcgܴܷr;䄪inaryI;慈lieóϝǴ݉\0ݢĀ;eݍݎ戬Āgrݓݘral;戫section;拂isibleĀCTݬݲomma;恣imes;恢ƀgptݿރވon;䄮f;쀀𝕀a;䎙cr;愐ilde;䄨ǫޚ\0ޞcy;䐆l耻Ï䃏ʀcfosuެ޷޼߂ߐĀiyޱ޵rc;䄴;䐙r;쀀𝔍pf;쀀𝕁ǣ߇\0ߌr;쀀𝒥rcy;䐈kcy;䐄΀HJacfosߤߨ߽߬߱ࠂࠈcy;䐥cy;䐌ppa;䎚Āey߶߻dil;䄶;䐚r;쀀𝔎pf;쀀𝕂cr;쀀𝒦րJTaceflmostࠥࠩࠬࡐࡣ঳সে্਷ੇcy;䐉耻<䀼ʀcmnpr࠷࠼ࡁࡄࡍute;䄹bda;䎛g;柪lacetrf;愒r;憞ƀaeyࡗ࡜ࡡron;䄽dil;䄻;䐛Āfsࡨ॰tԀACDFRTUVarࡾࢩࢱࣦ࣠ࣼयज़ΐ४Ānrࢃ࢏gleBracket;柨rowƀ;BR࢙࢚࢞憐ar;懤ightArrow;懆eiling;挈oǵࢷ\0ࣃbleBracket;柦nǔࣈ\0࣒eeVector;楡ectorĀ;Bࣛࣜ懃ar;楙loor;挊ightĀAV࣯ࣵrrow;憔ector;楎Āerँगeƀ;AVउऊऐ抣rrow;憤ector;楚iangleƀ;BEतथऩ抲ar;槏qual;抴pƀDTVषूौownVector;楑eeVector;楠ectorĀ;Bॖॗ憿ar;楘ectorĀ;B॥०憼ar;楒ightáΜs̀EFGLSTॾঋকঝঢভqualGreater;拚ullEqual;扦reater;扶ess;檡lantEqual;橽ilde;扲r;쀀𝔏Ā;eঽা拘ftarrow;懚idot;䄿ƀnpw৔ਖਛgȀLRlr৞৷ਂਐeftĀAR০৬rrow;柵ightArrow;柷ightArrow;柶eftĀarγਊightáοightáϊf;쀀𝕃erĀLRਢਬeftArrow;憙ightArrow;憘ƀchtਾੀੂòࡌ;憰rok;䅁;扪Ѐacefiosuਗ਼੝੠੷੼અઋ઎p;椅y;䐜Ādl੥੯iumSpace;恟lintrf;愳r;쀀𝔐nusPlus;戓pf;쀀𝕄cò੶;䎜ҀJacefostuણધભીଔଙඑ඗ඞcy;䐊cute;䅃ƀaey઴હાron;䅇dil;䅅;䐝ƀgswે૰଎ativeƀMTV૓૟૨ediumSpace;怋hiĀcn૦૘ë૙eryThiî૙tedĀGL૸ଆreaterGreateòٳessLesóੈLine;䀊r;쀀𝔑ȀBnptଢନଷ଺reak;恠BreakingSpace;䂠f;愕ڀ;CDEGHLNPRSTV୕ୖ୪୼஡௫ఄ౞಄ದ೘ൡඅ櫬Āou୛୤ngruent;扢pCap;扭oubleVerticalBar;戦ƀlqxஃஊ஛ement;戉ualĀ;Tஒஓ扠ilde;쀀≂̸ists;戄reater΀;EFGLSTஶஷ஽௉௓௘௥扯qual;扱ullEqual;쀀≧̸reater;쀀≫̸ess;批lantEqual;쀀⩾̸ilde;扵umpń௲௽ownHump;쀀≎̸qual;쀀≏̸eĀfsఊధtTriangleƀ;BEచఛడ拪ar;쀀⧏̸qual;括s̀;EGLSTవశ఼ౄోౘ扮qual;扰reater;扸ess;쀀≪̸lantEqual;쀀⩽̸ilde;扴estedĀGL౨౹reaterGreater;쀀⪢̸essLess;쀀⪡̸recedesƀ;ESಒಓಛ技qual;쀀⪯̸lantEqual;拠ĀeiಫಹverseElement;戌ghtTriangleƀ;BEೋೌ೒拫ar;쀀⧐̸qual;拭ĀquೝഌuareSuĀbp೨೹setĀ;E೰ೳ쀀⊏̸qual;拢ersetĀ;Eഃആ쀀⊐̸qual;拣ƀbcpഓതൎsetĀ;Eഛഞ쀀⊂⃒qual;抈ceedsȀ;ESTലള഻െ抁qual;쀀⪰̸lantEqual;拡ilde;쀀≿̸ersetĀ;E൘൛쀀⊃⃒qual;抉ildeȀ;EFT൮൯൵ൿ扁qual;扄ullEqual;扇ilde;扉erticalBar;戤cr;쀀𝒩ilde耻Ñ䃑;䎝܀Eacdfgmoprstuvලෂ෉෕ෛ෠෧෼ขภยา฿ไlig;䅒cute耻Ó䃓Āiy෎ීrc耻Ô䃔;䐞blac;䅐r;쀀𝔒rave耻Ò䃒ƀaei෮ෲ෶cr;䅌ga;䎩cron;䎟pf;쀀𝕆enCurlyĀDQฎบoubleQuote;怜uote;怘;橔Āclวฬr;쀀𝒪ash耻Ø䃘iŬื฼de耻Õ䃕es;樷ml耻Ö䃖erĀBP๋๠Āar๐๓r;怾acĀek๚๜;揞et;掴arenthesis;揜Ҁacfhilors๿ງຊຏຒດຝະ໼rtialD;戂y;䐟r;쀀𝔓i;䎦;䎠usMinus;䂱Āipຢອncareplanåڝf;愙Ȁ;eio຺ູ໠໤檻cedesȀ;EST່້໏໚扺qual;檯lantEqual;扼ilde;找me;怳Ādp໩໮uct;戏ortionĀ;aȥ໹l;戝Āci༁༆r;쀀𝒫;䎨ȀUfos༑༖༛༟OT耻"䀢r;쀀𝔔pf;愚cr;쀀𝒬؀BEacefhiorsu༾གྷཇའཱིྦྷྪྭ႖ႩႴႾarr;椐G耻®䂮ƀcnrཎནབute;䅔g;柫rĀ;tཛྷཝ憠l;椖ƀaeyཧཬཱron;䅘dil;䅖;䐠Ā;vླྀཹ愜erseĀEUྂྙĀlq྇ྎement;戋uilibrium;懋pEquilibrium;楯r»ཹo;䎡ghtЀACDFTUVa࿁࿫࿳ဢဨၛႇϘĀnr࿆࿒gleBracket;柩rowƀ;BL࿜࿝࿡憒ar;懥eftArrow;懄eiling;按oǵ࿹\0စbleBracket;柧nǔည\0နeeVector;楝ectorĀ;Bဝသ懂ar;楕loor;挋Āerိ၃eƀ;AVဵံြ抢rrow;憦ector;楛iangleƀ;BEၐၑၕ抳ar;槐qual;抵pƀDTVၣၮၸownVector;楏eeVector;楜ectorĀ;Bႂႃ憾ar;楔ectorĀ;B႑႒懀ar;楓Āpuႛ႞f;愝ndImplies;楰ightarrow;懛ĀchႹႼr;愛;憱leDelayed;槴ڀHOacfhimoqstuფჱჷჽᄙᄞᅑᅖᅡᅧᆵᆻᆿĀCcჩხHcy;䐩y;䐨FTcy;䐬cute;䅚ʀ;aeiyᄈᄉᄎᄓᄗ檼ron;䅠dil;䅞rc;䅜;䐡r;쀀𝔖ortȀDLRUᄪᄴᄾᅉownArrow»ОeftArrow»࢚ightArrow»࿝pArrow;憑gma;䎣allCircle;战pf;쀀𝕊ɲᅭ\0\0ᅰt;戚areȀ;ISUᅻᅼᆉᆯ斡ntersection;抓uĀbpᆏᆞsetĀ;Eᆗᆘ抏qual;抑ersetĀ;Eᆨᆩ抐qual;抒nion;抔cr;쀀𝒮ar;拆ȀbcmpᇈᇛሉላĀ;sᇍᇎ拐etĀ;Eᇍᇕqual;抆ĀchᇠህeedsȀ;ESTᇭᇮᇴᇿ扻qual;檰lantEqual;扽ilde;承Tháྌ;我ƀ;esሒሓሣ拑rsetĀ;Eሜም抃qual;抇et»ሓրHRSacfhiorsሾቄ቉ቕ቞ቱቶኟዂወዑORN耻Þ䃞ADE;愢ĀHc቎ቒcy;䐋y;䐦Ābuቚቜ;䀉;䎤ƀaeyብቪቯron;䅤dil;䅢;䐢r;쀀𝔗Āeiቻ኉Dzኀ\0ኇefore;戴a;䎘Ācn኎ኘkSpace;쀀  Space;怉ldeȀ;EFTካኬኲኼ戼qual;扃ullEqual;扅ilde;扈pf;쀀𝕋ipleDot;惛Āctዖዛr;쀀𝒯rok;䅦ૡዷጎጚጦ\0ጬጱ\0\0\0\0\0ጸጽ፷ᎅ\0᏿ᐄᐊᐐĀcrዻጁute耻Ú䃚rĀ;oጇገ憟cir;楉rǣጓ\0጖y;䐎ve;䅬Āiyጞጣrc耻Û䃛;䐣blac;䅰r;쀀𝔘rave耻Ù䃙acr;䅪Ādiፁ፩erĀBPፈ፝Āarፍፐr;䁟acĀekፗፙ;揟et;掵arenthesis;揝onĀ;P፰፱拃lus;抎Āgp፻፿on;䅲f;쀀𝕌ЀADETadps᎕ᎮᎸᏄϨᏒᏗᏳrrowƀ;BDᅐᎠᎤar;椒ownArrow;懅ownArrow;憕quilibrium;楮eeĀ;AᏋᏌ报rrow;憥ownáϳerĀLRᏞᏨeftArrow;憖ightArrow;憗iĀ;lᏹᏺ䏒on;䎥ing;䅮cr;쀀𝒰ilde;䅨ml耻Ü䃜ҀDbcdefosvᐧᐬᐰᐳᐾᒅᒊᒐᒖash;披ar;櫫y;䐒ashĀ;lᐻᐼ抩;櫦Āerᑃᑅ;拁ƀbtyᑌᑐᑺar;怖Ā;iᑏᑕcalȀBLSTᑡᑥᑪᑴar;戣ine;䁼eparator;杘ilde;所ThinSpace;怊r;쀀𝔙pf;쀀𝕍cr;쀀𝒱dash;抪ʀcefosᒧᒬᒱᒶᒼirc;䅴dge;拀r;쀀𝔚pf;쀀𝕎cr;쀀𝒲Ȁfiosᓋᓐᓒᓘr;쀀𝔛;䎞pf;쀀𝕏cr;쀀𝒳ҀAIUacfosuᓱᓵᓹᓽᔄᔏᔔᔚᔠcy;䐯cy;䐇cy;䐮cute耻Ý䃝Āiyᔉᔍrc;䅶;䐫r;쀀𝔜pf;쀀𝕐cr;쀀𝒴ml;䅸ЀHacdefosᔵᔹᔿᕋᕏᕝᕠᕤcy;䐖cute;䅹Āayᕄᕉron;䅽;䐗ot;䅻Dzᕔ\0ᕛoWidtè૙a;䎖r;愨pf;愤cr;쀀𝒵௡ᖃᖊᖐ\0ᖰᖶᖿ\0\0\0\0ᗆᗛᗫᙟ᙭\0ᚕ᚛ᚲᚹ\0ᚾcute耻á䃡reve;䄃̀;Ediuyᖜᖝᖡᖣᖨᖭ戾;쀀∾̳;房rc耻â䃢te肻´̆;䐰lig耻æ䃦Ā;r²ᖺ;쀀𝔞rave耻à䃠ĀepᗊᗖĀfpᗏᗔsym;愵èᗓha;䎱ĀapᗟcĀclᗤᗧr;䄁g;樿ɤᗰ\0\0ᘊʀ;adsvᗺᗻᗿᘁᘇ戧nd;橕;橜lope;橘;橚΀;elmrszᘘᘙᘛᘞᘿᙏᙙ戠;榤e»ᘙsdĀ;aᘥᘦ戡ѡᘰᘲᘴᘶᘸᘺᘼᘾ;榨;榩;榪;榫;榬;榭;榮;榯tĀ;vᙅᙆ戟bĀ;dᙌᙍ抾;榝Āptᙔᙗh;戢»¹arr;捼Āgpᙣᙧon;䄅f;쀀𝕒΀;Eaeiop዁ᙻᙽᚂᚄᚇᚊ;橰cir;橯;扊d;手s;䀧roxĀ;e዁ᚒñᚃing耻å䃥ƀctyᚡᚦᚨr;쀀𝒶;䀪mpĀ;e዁ᚯñʈilde耻ã䃣ml耻ä䃤Āciᛂᛈoninôɲnt;樑ࠀNabcdefiklnoprsu᛭ᛱᜰ᜼ᝃᝈ᝸᝽០៦ᠹᡐᜍ᤽᥈ᥰot;櫭Ācrᛶ᜞kȀcepsᜀᜅᜍᜓong;扌psilon;䏶rime;怵imĀ;e᜚᜛戽q;拍Ŷᜢᜦee;抽edĀ;gᜬᜭ挅e»ᜭrkĀ;t፜᜷brk;掶Āoyᜁᝁ;䐱quo;怞ʀcmprtᝓ᝛ᝡᝤᝨausĀ;eĊĉptyv;榰séᜌnoõēƀahwᝯ᝱ᝳ;䎲;愶een;扬r;쀀𝔟g΀costuvwឍឝឳេ៕៛៞ƀaiuបពរðݠrc;旯p»፱ƀdptឤឨឭot;樀lus;樁imes;樂ɱឹ\0\0ើcup;樆ar;昅riangleĀdu៍្own;施p;斳plus;樄eåᑄåᒭarow;植ƀako៭ᠦᠵĀcn៲ᠣkƀlst៺֫᠂ozenge;槫riangleȀ;dlr᠒᠓᠘᠝斴own;斾eft;旂ight;斸k;搣Ʊᠫ\0ᠳƲᠯ\0ᠱ;斒;斑4;斓ck;斈ĀeoᠾᡍĀ;qᡃᡆ쀀=⃥uiv;쀀≡⃥t;挐Ȁptwxᡙᡞᡧᡬf;쀀𝕓Ā;tᏋᡣom»Ꮜtie;拈؀DHUVbdhmptuvᢅᢖᢪᢻᣗᣛᣬ᣿ᤅᤊᤐᤡȀLRlrᢎᢐᢒᢔ;敗;敔;敖;敓ʀ;DUduᢡᢢᢤᢦᢨ敐;敦;敩;敤;敧ȀLRlrᢳᢵᢷᢹ;敝;敚;敜;教΀;HLRhlrᣊᣋᣍᣏᣑᣓᣕ救;敬;散;敠;敫;敢;敟ox;槉ȀLRlrᣤᣦᣨᣪ;敕;敒;攐;攌ʀ;DUduڽ᣷᣹᣻᣽;敥;敨;攬;攴inus;抟lus;択imes;抠ȀLRlrᤙᤛᤝ᤟;敛;敘;攘;攔΀;HLRhlrᤰᤱᤳᤵᤷ᤻᤹攂;敪;敡;敞;攼;攤;攜Āevģ᥂bar耻¦䂦Ȁceioᥑᥖᥚᥠr;쀀𝒷mi;恏mĀ;e᜚᜜lƀ;bhᥨᥩᥫ䁜;槅sub;柈Ŭᥴ᥾lĀ;e᥹᥺怢t»᥺pƀ;Eeįᦅᦇ;檮Ā;qۜۛೡᦧ\0᧨ᨑᨕᨲ\0ᨷᩐ\0\0᪴\0\0᫁\0\0ᬡᬮ᭍᭒\0᯽\0ᰌƀcpr᦭ᦲ᧝ute;䄇̀;abcdsᦿᧀᧄ᧊᧕᧙戩nd;橄rcup;橉Āau᧏᧒p;橋p;橇ot;橀;쀀∩︀Āeo᧢᧥t;恁îړȀaeiu᧰᧻ᨁᨅǰ᧵\0᧸s;橍on;䄍dil耻ç䃧rc;䄉psĀ;sᨌᨍ橌m;橐ot;䄋ƀdmnᨛᨠᨦil肻¸ƭptyv;榲t脀¢;eᨭᨮ䂢räƲr;쀀𝔠ƀceiᨽᩀᩍy;䑇ckĀ;mᩇᩈ朓ark»ᩈ;䏇r΀;Ecefms᩟᩠ᩢᩫ᪤᪪᪮旋;槃ƀ;elᩩᩪᩭ䋆q;扗eɡᩴ\0\0᪈rrowĀlr᩼᪁eft;憺ight;憻ʀRSacd᪒᪔᪖᪚᪟»ཇ;擈st;抛irc;抚ash;抝nint;樐id;櫯cir;槂ubsĀ;u᪻᪼晣it»᪼ˬ᫇᫔᫺\0ᬊonĀ;eᫍᫎ䀺Ā;qÇÆɭ᫙\0\0᫢aĀ;t᫞᫟䀬;䁀ƀ;fl᫨᫩᫫戁îᅠeĀmx᫱᫶ent»᫩eóɍǧ᫾\0ᬇĀ;dኻᬂot;橭nôɆƀfryᬐᬔᬗ;쀀𝕔oäɔ脀©;sŕᬝr;愗Āaoᬥᬩrr;憵ss;朗Ācuᬲᬷr;쀀𝒸Ābpᬼ᭄Ā;eᭁᭂ櫏;櫑Ā;eᭉᭊ櫐;櫒dot;拯΀delprvw᭠᭬᭷ᮂᮬᯔ᯹arrĀlr᭨᭪;椸;椵ɰ᭲\0\0᭵r;拞c;拟arrĀ;p᭿ᮀ憶;椽̀;bcdosᮏᮐᮖᮡᮥᮨ截rcap;橈Āauᮛᮞp;橆p;橊ot;抍r;橅;쀀∪︀Ȁalrv᮵ᮿᯞᯣrrĀ;mᮼᮽ憷;椼yƀevwᯇᯔᯘqɰᯎ\0\0ᯒreã᭳uã᭵ee;拎edge;拏en耻¤䂤earrowĀlrᯮ᯳eft»ᮀight»ᮽeäᯝĀciᰁᰇoninôǷnt;戱lcty;挭ঀAHabcdefhijlorstuwz᰸᰻᰿ᱝᱩᱵᲊᲞᲬᲷ᳻᳿ᴍᵻᶑᶫᶻ᷆᷍rò΁ar;楥Ȁglrs᱈ᱍ᱒᱔ger;怠eth;愸òᄳhĀ;vᱚᱛ怐»ऊūᱡᱧarow;椏aã̕Āayᱮᱳron;䄏;䐴ƀ;ao̲ᱼᲄĀgrʿᲁr;懊tseq;橷ƀglmᲑᲔᲘ耻°䂰ta;䎴ptyv;榱ĀirᲣᲨsht;楿;쀀𝔡arĀlrᲳᲵ»ࣜ»သʀaegsv᳂͸᳖᳜᳠mƀ;oș᳊᳔ndĀ;ș᳑uit;晦amma;䏝in;拲ƀ;io᳧᳨᳸䃷de脀÷;o᳧ᳰntimes;拇nø᳷cy;䑒cɯᴆ\0\0ᴊrn;挞op;挍ʀlptuwᴘᴝᴢᵉᵕlar;䀤f;쀀𝕕ʀ;emps̋ᴭᴷᴽᵂqĀ;d͒ᴳot;扑inus;戸lus;戔quare;抡blebarwedgåúnƀadhᄮᵝᵧownarrowóᲃarpoonĀlrᵲᵶefôᲴighôᲶŢᵿᶅkaro÷གɯᶊ\0\0ᶎrn;挟op;挌ƀcotᶘᶣᶦĀryᶝᶡ;쀀𝒹;䑕l;槶rok;䄑Ādrᶰᶴot;拱iĀ;fᶺ᠖斿Āah᷀᷃ròЩaòྦangle;榦Āci᷒ᷕy;䑟grarr;柿ऀDacdefglmnopqrstuxḁḉḙḸոḼṉṡṾấắẽỡἪἷὄ὎὚ĀDoḆᴴoôᲉĀcsḎḔute耻é䃩ter;橮ȀaioyḢḧḱḶron;䄛rĀ;cḭḮ扖耻ê䃪lon;払;䑍ot;䄗ĀDrṁṅot;扒;쀀𝔢ƀ;rsṐṑṗ檚ave耻è䃨Ā;dṜṝ檖ot;檘Ȁ;ilsṪṫṲṴ檙nters;揧;愓Ā;dṹṺ檕ot;檗ƀapsẅẉẗcr;䄓tyƀ;svẒẓẕ戅et»ẓpĀ1;ẝẤijạả;怄;怅怃ĀgsẪẬ;䅋p;怂ĀgpẴẸon;䄙f;쀀𝕖ƀalsỄỎỒrĀ;sỊị拕l;槣us;橱iƀ;lvỚớở䎵on»ớ;䏵ȀcsuvỪỳἋἣĀioữḱrc»Ḯɩỹ\0\0ỻíՈantĀglἂἆtr»ṝess»Ṻƀaeiἒ἖Ἒls;䀽st;扟vĀ;DȵἠD;橸parsl;槥ĀDaἯἳot;打rr;楱ƀcdiἾὁỸr;愯oô͒ĀahὉὋ;䎷耻ð䃰Āmrὓὗl耻ë䃫o;悬ƀcipὡὤὧl;䀡sôծĀeoὬὴctatioîՙnentialåչৡᾒ\0ᾞ\0ᾡᾧ\0\0ῆῌ\0ΐ\0ῦῪ \0 ⁚llingdotseñṄy;䑄male;晀ƀilrᾭᾳ῁lig;耀ffiɩᾹ\0\0᾽g;耀ffig;耀ffl;쀀𝔣lig;耀filig;쀀fjƀaltῙ῜ῡt;晭ig;耀flns;斱of;䆒ǰ΅\0ῳf;쀀𝕗ĀakֿῷĀ;vῼ´拔;櫙artint;樍Āao‌⁕Ācs‑⁒ႉ‸⁅⁈\0⁐β•‥‧‪‬\0‮耻½䂽;慓耻¼䂼;慕;慙;慛Ƴ‴\0‶;慔;慖ʴ‾⁁\0\0⁃耻¾䂾;慗;慜5;慘ƶ⁌\0⁎;慚;慝8;慞l;恄wn;挢cr;쀀𝒻ࢀEabcdefgijlnorstv₂₉₟₥₰₴⃰⃵⃺⃿℃ℒℸ̗ℾ⅒↞Ā;lٍ₇;檌ƀcmpₐₕ₝ute;䇵maĀ;dₜ᳚䎳;檆reve;䄟Āiy₪₮rc;䄝;䐳ot;䄡Ȁ;lqsؾق₽⃉ƀ;qsؾٌ⃄lanô٥Ȁ;cdl٥⃒⃥⃕c;檩otĀ;o⃜⃝檀Ā;l⃢⃣檂;檄Ā;e⃪⃭쀀⋛︀s;檔r;쀀𝔤Ā;gٳ؛mel;愷cy;䑓Ȁ;Eajٚℌℎℐ;檒;檥;檤ȀEaesℛℝ℩ℴ;扩pĀ;p℣ℤ檊rox»ℤĀ;q℮ℯ檈Ā;q℮ℛim;拧pf;쀀𝕘Āci⅃ⅆr;愊mƀ;el٫ⅎ⅐;檎;檐茀>;cdlqr׮ⅠⅪⅮⅳⅹĀciⅥⅧ;檧r;橺ot;拗Par;榕uest;橼ʀadelsↄⅪ←ٖ↛ǰ↉\0↎proø₞r;楸qĀlqؿ↖lesó₈ií٫Āen↣↭rtneqq;쀀≩︀Å↪ԀAabcefkosy⇄⇇⇱⇵⇺∘∝∯≨≽ròΠȀilmr⇐⇔⇗⇛rsðᒄf»․ilôکĀdr⇠⇤cy;䑊ƀ;cwࣴ⇫⇯ir;楈;憭ar;意irc;䄥ƀalr∁∎∓rtsĀ;u∉∊晥it»∊lip;怦con;抹r;쀀𝔥sĀew∣∩arow;椥arow;椦ʀamopr∺∾≃≞≣rr;懿tht;戻kĀlr≉≓eftarrow;憩ightarrow;憪f;쀀𝕙bar;怕ƀclt≯≴≸r;쀀𝒽asè⇴rok;䄧Ābp⊂⊇ull;恃hen»ᱛૡ⊣\0⊪\0⊸⋅⋎\0⋕⋳\0\0⋸⌢⍧⍢⍿\0⎆⎪⎴cute耻í䃭ƀ;iyݱ⊰⊵rc耻î䃮;䐸Ācx⊼⊿y;䐵cl耻¡䂡ĀfrΟ⋉;쀀𝔦rave耻ì䃬Ȁ;inoܾ⋝⋩⋮Āin⋢⋦nt;樌t;戭fin;槜ta;愩lig;䄳ƀaop⋾⌚⌝ƀcgt⌅⌈⌗r;䄫ƀelpܟ⌏⌓inåގarôܠh;䄱f;抷ed;䆵ʀ;cfotӴ⌬⌱⌽⍁are;愅inĀ;t⌸⌹戞ie;槝doô⌙ʀ;celpݗ⍌⍐⍛⍡al;抺Āgr⍕⍙eróᕣã⍍arhk;樗rod;樼Ȁcgpt⍯⍲⍶⍻y;䑑on;䄯f;쀀𝕚a;䎹uest耻¿䂿Āci⎊⎏r;쀀𝒾nʀ;EdsvӴ⎛⎝⎡ӳ;拹ot;拵Ā;v⎦⎧拴;拳Ā;iݷ⎮lde;䄩ǫ⎸\0⎼cy;䑖l耻ï䃯̀cfmosu⏌⏗⏜⏡⏧⏵Āiy⏑⏕rc;䄵;䐹r;쀀𝔧ath;䈷pf;쀀𝕛ǣ⏬\0⏱r;쀀𝒿rcy;䑘kcy;䑔Ѐacfghjos␋␖␢␧␭␱␵␻ppaĀ;v␓␔䎺;䏰Āey␛␠dil;䄷;䐺r;쀀𝔨reen;䄸cy;䑅cy;䑜pf;쀀𝕜cr;쀀𝓀஀ABEHabcdefghjlmnoprstuv⑰⒁⒆⒍⒑┎┽╚▀♎♞♥♹♽⚚⚲⛘❝❨➋⟀⠁⠒ƀart⑷⑺⑼rò৆òΕail;椛arr;椎Ā;gঔ⒋;檋ar;楢ॣ⒥\0⒪\0⒱\0\0\0\0\0⒵Ⓔ\0ⓆⓈⓍ\0⓹ute;䄺mptyv;榴raîࡌbda;䎻gƀ;dlࢎⓁⓃ;榑åࢎ;檅uo耻«䂫rЀ;bfhlpst࢙ⓞⓦⓩ⓫⓮⓱⓵Ā;f࢝ⓣs;椟s;椝ë≒p;憫l;椹im;楳l;憢ƀ;ae⓿─┄檫il;椙Ā;s┉┊檭;쀀⪭︀ƀabr┕┙┝rr;椌rk;杲Āak┢┬cĀek┨┪;䁻;䁛Āes┱┳;榋lĀdu┹┻;榏;榍Ȁaeuy╆╋╖╘ron;䄾Ādi═╔il;䄼ìࢰâ┩;䐻Ȁcqrs╣╦╭╽a;椶uoĀ;rนᝆĀdu╲╷har;楧shar;楋h;憲ʀ;fgqs▋▌উ◳◿扤tʀahlrt▘▤▷◂◨rrowĀ;t࢙□aé⓶arpoonĀdu▯▴own»њp»०eftarrows;懇ightƀahs◍◖◞rrowĀ;sࣴࢧarpoonó྘quigarro÷⇰hreetimes;拋ƀ;qs▋ও◺lanôবʀ;cdgsব☊☍☝☨c;檨otĀ;o☔☕橿Ā;r☚☛檁;檃Ā;e☢☥쀀⋚︀s;檓ʀadegs☳☹☽♉♋pproøⓆot;拖qĀgq♃♅ôউgtò⒌ôছiíলƀilr♕࣡♚sht;楼;쀀𝔩Ā;Eজ♣;檑š♩♶rĀdu▲♮Ā;l॥♳;楪lk;斄cy;䑙ʀ;achtੈ⚈⚋⚑⚖rò◁orneòᴈard;楫ri;旺Āio⚟⚤dot;䅀ustĀ;a⚬⚭掰che»⚭ȀEaes⚻⚽⛉⛔;扨pĀ;p⛃⛄檉rox»⛄Ā;q⛎⛏檇Ā;q⛎⚻im;拦Ѐa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di㩀㩑Ābg㩅㩉ar;機eĀ;qᗺ㩏;扙erp;愘r;쀀𝔴pf;쀀𝕨Ā;eᑹ㩦atèᑹcr;쀀𝓌ૣណ㪇\0㪋\0㪐㪛\0\0㪝㪨㪫㪯\0\0㫃㫎\0㫘ៜ៟tré៑r;쀀𝔵ĀAa㪔㪗ròσrò৶;䎾ĀAa㪡㪤ròθrò৫að✓is;拻ƀdptឤ㪵㪾Āfl㪺ឩ;쀀𝕩imåឲĀAa㫇㫊ròώròਁĀcq㫒ីr;쀀𝓍Āpt៖㫜ré។Ѐacefiosu㫰㫽㬈㬌㬑㬕㬛㬡cĀuy㫶㫻te耻ý䃽;䑏Āiy㬂㬆rc;䅷;䑋n耻¥䂥r;쀀𝔶cy;䑗pf;쀀𝕪cr;쀀𝓎Ācm㬦㬩y;䑎l耻ÿ䃿Ԁacdefhiosw㭂㭈㭔㭘㭤㭩㭭㭴㭺㮀cute;䅺Āay㭍㭒ron;䅾;䐷ot;䅼Āet㭝㭡træᕟa;䎶r;쀀𝔷cy;䐶grarr;懝pf;쀀𝕫cr;쀀𝓏Ājn㮅㮇;怍j;怌'.split("").map(s=>s.charCodeAt(0))),CB=new 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Us;(function(s){s.controlCharacterInInputStream="control-character-in-input-stream",s.noncharacterInInputStream="noncharacter-in-input-stream",s.surrogateInInputStream="surrogate-in-input-stream",s.nonVoidHtmlElementStartTagWithTrailingSolidus="non-void-html-element-start-tag-with-trailing-solidus",s.endTagWithAttributes="end-tag-with-attributes",s.endTagWithTrailingSolidus="end-tag-with-trailing-solidus",s.unexpectedSolidusInTag="unexpected-solidus-in-tag",s.unexpectedNullCharacter="unexpected-null-character",s.unexpectedQuestionMarkInsteadOfTagName="unexpected-question-mark-instead-of-tag-name",s.invalidFirstCharacterOfTagName="invalid-first-character-of-tag-name",s.unexpectedEqualsSignBeforeAttributeName="unexpected-equals-sign-before-attribute-name",s.missingEndTagName="missing-end-tag-name",s.unexpectedCharacterInAttributeName="unexpected-character-in-attribute-name",s.unknownNamedCharacterReference="unknown-named-character-reference",s.missingSemicolonAfterCharacterReference="missing-semicolon-after-character-reference",s.unexpectedCharacterAfterDoctypeSystemIdentifier="unexpected-character-after-doctype-system-identifier",s.unexpectedCharacterInUnquotedAttributeValue="unexpected-character-in-unquoted-attribute-value",s.eofBeforeTagName="eof-before-tag-name",s.eofInTag="eof-in-tag",s.missingAttributeValue="missing-attribute-value",s.missingWhitespaceBetweenAttributes="missing-whitespace-between-attributes",s.missingWhitespaceAfterDoctypePublicKeyword="missing-whitespace-after-doctype-public-keyword",s.missingWhitespaceBetweenDoctypePublicAndSystemIdentifiers="missing-whitespace-between-doctype-public-and-system-identifiers",s.missingWhitespaceAfterDoctypeSystemKeyword="missing-whitespace-after-doctype-system-keyword",s.missingQuoteBeforeDoctypePublicIdentifier="missing-quote-before-doctype-public-identifier",s.missingQuoteBeforeDoctypeSystemIdentifier="missing-quote-before-doctype-system-identifier",s.missingDoctypePublicIdentifier="missing-doctype-public-identifier",s.missingDoctypeSystemIdentifier="missing-doctype-system-identifier",s.abruptDoctypePublicIdentifier="abrupt-doctype-public-identifier",s.abruptDoctypeSystemIdentifier="abrupt-doctype-system-identifier",s.cdataInHtmlContent="cdata-in-html-content",s.incorrectlyOpenedComment="incorrectly-opened-comment",s.eofInScriptHtmlCommentLikeText="eof-in-script-html-comment-like-text",s.eofInDoctype="eof-in-doctype",s.nestedComment="nested-comment",s.abruptClosingOfEmptyComment="abrupt-closing-of-empty-comment",s.eofInComment="eof-in-comment",s.incorrectlyClosedComment="incorrectly-closed-comment",s.eofInCdata="eof-in-cdata",s.absenceOfDigitsInNumericCharacterReference="absence-of-digits-in-numeric-character-reference",s.nullCharacterReference="null-character-reference",s.surrogateCharacterReference="surrogate-character-reference",s.characterReferenceOutsideUnicodeRange="character-reference-outside-unicode-range",s.controlCharacterReference="control-character-reference",s.noncharacterCharacterReference="noncharacter-character-reference",s.missingWhitespaceBeforeDoctypeName="missing-whitespace-before-doctype-name",s.missingDoctypeName="missing-doctype-name",s.invalidCharacterSequenceAfterDoctypeName="invalid-character-sequence-after-doctype-name",s.duplicateAttribute="duplicate-attribute",s.nonConformingDoctype="non-conforming-doctype",s.missingDoctype="missing-doctype",s.misplacedDoctype="misplaced-doctype",s.endTagWithoutMatchingOpenElement="end-tag-without-matching-open-element",s.closingOfElementWithOpenChildElements="closing-of-element-with-open-child-elements",s.disallowedContentInNoscriptInHead="disallowed-content-in-noscript-in-head",s.openElementsLeftAfterEof="open-elements-left-after-eof",s.abandonedHeadElementChild="abandoned-head-element-child",s.misplacedStartTagForHeadElement="misplaced-start-tag-for-head-element",s.nestedNoscriptInHead="nested-noscript-in-head",s.eofInElementThatCanContainOnlyText="eof-in-element-that-can-contain-only-text"})(Us||(Us={}));const jK=65536;class GK{constructor(a){this.handler=a,this.html="",this.pos=-1,this.lastGapPos=-2,this.gapStack=[],this.skipNextNewLine=!1,this.lastChunkWritten=!1,this.endOfChunkHit=!1,this.bufferWaterline=jK,this.isEol=!1,this.lineStartPos=0,this.droppedBufferSize=0,this.line=1,this.lastErrOffset=-1}get col(){return this.pos-this.lineStartPos+ +(this.lastGapPos!==this.pos)}get offset(){return this.droppedBufferSize+this.pos}getError(a,t){const{line:n,col:e,offset:l}=this,i=e+t,p=l+t;return{code:a,startLine:n,endLine:n,startCol:i,endCol:i,startOffset:p,endOffset:p}}_err(a){this.handler.onParseError&&this.lastErrOffset!==this.offset&&(this.lastErrOffset=this.offset,this.handler.onParseError(this.getError(a,0)))}_addGap(){this.gapStack.push(this.lastGapPos),this.lastGapPos=this.pos}_processSurrogate(a){if(this.pos!==this.html.length-1){const t=this.html.charCodeAt(this.pos+1);if(qK(t))return this.pos++,this._addGap(),UK(a,t)}else if(!this.lastChunkWritten)return this.endOfChunkHit=!0,ts.EOF;return this._err(Us.surrogateInInputStream),a}willDropParsedChunk(){return this.pos>this.bufferWaterline}dropParsedChunk(){this.willDropParsedChunk()&&(this.html=this.html.substring(this.pos),this.lineStartPos-=this.pos,this.droppedBufferSize+=this.pos,this.pos=0,this.lastGapPos=-2,this.gapStack.length=0)}write(a,t){this.html.length>0?this.html+=a:this.html=a,this.endOfChunkHit=!1,this.lastChunkWritten=t}insertHtmlAtCurrentPos(a){this.html=this.html.substring(0,this.pos+1)+a+this.html.substring(this.pos+1),this.endOfChunkHit=!1}startsWith(a,t){if(this.pos+a.length>this.html.length)return this.endOfChunkHit=!this.lastChunkWritten,!1;if(t)return this.html.startsWith(a,this.pos);for(let n=0;n<a.length;n++)if((this.html.charCodeAt(this.pos+n)|32)!==a.charCodeAt(n))return!1;return!0}peek(a){const t=this.pos+a;if(t>=this.html.length)return this.endOfChunkHit=!this.lastChunkWritten,ts.EOF;const n=this.html.charCodeAt(t);return 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Uint16Array('ᵁ<Õıʊҝջאٵ۞ޢߖࠏ੊ઑඡ๭༉༦჊ረዡᐕᒝᓃᓟᔥ\0\0\0\0\0\0ᕫᛍᦍᰒᷝ὾⁠↰⊍⏀⏻⑂⠤⤒ⴈ⹈⿎〖㊺㘹㞬㣾㨨㩱㫠㬮ࠀEMabcfglmnoprstu\\bfms„‹•˜¦³¹ÈÏlig耻Æ䃆P耻&䀦cute耻Á䃁reve;䄂Āiyx}rc耻Â䃂;䐐r;쀀𝔄rave耻À䃀pha;䎑acr;䄀d;橓Āgp¡on;䄄f;쀀𝔸plyFunction;恡ing耻Å䃅Ācs¾Ãr;쀀𝒜ign;扔ilde耻Ã䃃ml耻Ä䃄ЀaceforsuåûþėĜĢħĪĀcrêòkslash;或Ŷöø;櫧ed;挆y;䐑ƀcrtąċĔause;戵noullis;愬a;䎒r;쀀𝔅pf;쀀𝔹eve;䋘còēmpeq;扎܀HOacdefhilorsuōőŖƀƞƢƵƷƺǜȕɳɸɾcy;䐧PY耻©䂩ƀcpyŝŢźute;䄆Ā;iŧŨ拒talDifferentialD;慅leys;愭ȀaeioƉƎƔƘron;䄌dil耻Ç䃇rc;䄈nint;戰ot;䄊ĀdnƧƭilla;䂸terDot;䂷òſi;䎧rcleȀDMPTLJNjǑǖot;抙inus;抖lus;投imes;抗oĀcsǢǸkwiseContourIntegral;戲eCurlyĀDQȃȏoubleQuote;思uote;怙ȀlnpuȞȨɇɕonĀ;eȥȦ户;橴ƀgitȯȶȺruent;扡nt;戯ourIntegral;戮ĀfrɌɎ;愂oduct;成nterClockwiseContourIntegral;戳oss;樯cr;쀀𝒞pĀ;Cʄʅ拓ap;才րDJSZacefiosʠʬʰʴʸˋ˗ˡ˦̳ҍĀ;oŹʥtrahd;椑cy;䐂cy;䐅cy;䐏ƀgrsʿ˄ˇger;怡r;憡hv;櫤Āayː˕ron;䄎;䐔lĀ;t˝˞戇a;䎔r;쀀𝔇Āaf˫̧Ācm˰̢riticalȀADGT̖̜̀̆cute;䂴oŴ̋̍;䋙bleAcute;䋝rave;䁠ilde;䋜ond;拄ferentialD;慆Ѱ̽\0\0\0͔͂\0Ѕf;쀀𝔻ƀ;DE͈͉͍䂨ot;惜qual;扐blèCDLRUVͣͲ΂ϏϢϸontourIntegraìȹoɴ͹\0\0ͻ»͉nArrow;懓Āeo·ΤftƀARTΐΖΡrrow;懐ightArrow;懔eåˊngĀLRΫτeftĀARγιrrow;柸ightArrow;柺ightArrow;柹ightĀATϘϞrrow;懒ee;抨pɁϩ\0\0ϯrrow;懑ownArrow;懕erticalBar;戥ǹABLRTaВЪаўѿͼrrowƀ;BUНОТ憓ar;椓pArrow;懵reve;䌑eft˒к\0ц\0ѐightVector;楐eeVector;楞ectorĀ;Bљњ憽ar;楖ightǔѧ\0ѱeeVector;楟ectorĀ;BѺѻ懁ar;楗eeĀ;A҆҇护rrow;憧ĀctҒҗr;쀀𝒟rok;䄐ࠀNTacdfglmopqstuxҽӀӄӋӞӢӧӮӵԡԯԶՒ՝ՠեG;䅊H耻Ð䃐cute耻É䃉ƀaiyӒӗӜron;䄚rc耻Ê䃊;䐭ot;䄖r;쀀𝔈rave耻È䃈ement;戈ĀapӺӾcr;䄒tyɓԆ\0\0ԒmallSquare;旻erySmallSquare;斫ĀgpԦԪon;䄘f;쀀𝔼silon;䎕uĀaiԼՉlĀ;TՂՃ橵ilde;扂librium;懌Āci՗՚r;愰m;橳a;䎗ml耻Ë䃋Āipժկsts;戃onentialE;慇ʀcfiosօֈ֍ֲ׌y;䐤r;쀀𝔉lledɓ֗\0\0֣mallSquare;旼erySmallSquare;斪Ͱֺ\0ֿ\0\0ׄf;쀀𝔽All;戀riertrf;愱cò׋؀JTabcdfgorstר׬ׯ׺؀ؒؖ؛؝أ٬ٲcy;䐃耻>䀾mmaĀ;d׷׸䎓;䏜reve;䄞ƀeiy؇،ؐdil;䄢rc;䄜;䐓ot;䄠r;쀀𝔊;拙pf;쀀𝔾eater̀EFGLSTصلَٖٛ٦qualĀ;Lؾؿ扥ess;招ullEqual;执reater;檢ess;扷lantEqual;橾ilde;扳cr;쀀𝒢;扫ЀAacfiosuڅڋږڛڞڪھۊRDcy;䐪Āctڐڔek;䋇;䁞irc;䄤r;愌lbertSpace;愋ǰگ\0ڲf;愍izontalLine;攀Āctۃۅòکrok;䄦mpńېۘownHumðįqual;扏܀EJOacdfgmnostuۺ۾܃܇܎ܚܞܡܨ݄ݸދޏޕcy;䐕lig;䄲cy;䐁cute耻Í䃍Āiyܓܘrc耻Î䃎;䐘ot;䄰r;愑rave耻Ì䃌ƀ;apܠܯܿĀcgܴܷr;䄪inaryI;慈lieóϝǴ݉\0ݢĀ;eݍݎ戬Āgrݓݘral;戫section;拂isibleĀCTݬݲomma;恣imes;恢ƀgptݿރވon;䄮f;쀀𝕀a;䎙cr;愐ilde;䄨ǫޚ\0ޞcy;䐆l耻Ï䃏ʀcfosuެ޷޼߂ߐĀiyޱ޵rc;䄴;䐙r;쀀𝔍pf;쀀𝕁ǣ߇\0ߌr;쀀𝒥rcy;䐈kcy;䐄΀HJacfosߤߨ߽߬߱ࠂࠈcy;䐥cy;䐌ppa;䎚Āey߶߻dil;䄶;䐚r;쀀𝔎pf;쀀𝕂cr;쀀𝒦րJTaceflmostࠥࠩࠬࡐࡣ঳সে্਷ੇcy;䐉耻<䀼ʀcmnpr࠷࠼ࡁࡄࡍute;䄹bda;䎛g;柪lacetrf;愒r;憞ƀaeyࡗ࡜ࡡron;䄽dil;䄻;䐛Āfsࡨ॰tԀACDFRTUVarࡾࢩࢱࣦ࣠ࣼयज़ΐ४Ānrࢃ࢏gleBracket;柨rowƀ;BR࢙࢚࢞憐ar;懤ightArrow;懆eiling;挈oǵࢷ\0ࣃbleBracket;柦nǔࣈ\0࣒eeVector;楡ectorĀ;Bࣛࣜ懃ar;楙loor;挊ightĀAV࣯ࣵrrow;憔ector;楎Āerँगeƀ;AVउऊऐ抣rrow;憤ector;楚iangleƀ;BEतथऩ抲ar;槏qual;抴pƀDTVषूौownVector;楑eeVector;楠ectorĀ;Bॖॗ憿ar;楘ectorĀ;B॥०憼ar;楒ightáΜs̀EFGLSTॾঋকঝঢভqualGreater;拚ullEqual;扦reater;扶ess;檡lantEqual;橽ilde;扲r;쀀𝔏Ā;eঽা拘ftarrow;懚idot;䄿ƀnpw৔ਖਛgȀLRlr৞৷ਂਐeftĀAR০৬rrow;柵ightArrow;柷ightArrow;柶eftĀarγਊightáοightáϊf;쀀𝕃erĀLRਢਬeftArrow;憙ightArrow;憘ƀchtਾੀੂòࡌ;憰rok;䅁;扪Ѐacefiosuਗ਼੝੠੷੼અઋ઎p;椅y;䐜Ādl੥੯iumSpace;恟lintrf;愳r;쀀𝔐nusPlus;戓pf;쀀𝕄cò੶;䎜ҀJacefostuણધભીଔଙඑ඗ඞcy;䐊cute;䅃ƀaey઴હાron;䅇dil;䅅;䐝ƀgswે૰଎ativeƀMTV૓૟૨ediumSpace;怋hiĀcn૦૘ë૙eryThiî૙tedĀGL૸ଆreaterGreateòٳessLesóੈLine;䀊r;쀀𝔑ȀBnptଢନଷ଺reak;恠BreakingSpace;䂠f;愕ڀ;CDEGHLNPRSTV୕ୖ୪୼஡௫ఄ౞಄ದ೘ൡඅ櫬Āou୛୤ngruent;扢pCap;扭oubleVerticalBar;戦ƀlqxஃஊ஛ement;戉ualĀ;Tஒஓ扠ilde;쀀≂̸ists;戄reater΀;EFGLSTஶஷ஽௉௓௘௥扯qual;扱ullEqual;쀀≧̸reater;쀀≫̸ess;批lantEqual;쀀⩾̸ilde;扵umpń௲௽ownHump;쀀≎̸qual;쀀≏̸eĀfsఊధtTriangleƀ;BEచఛడ拪ar;쀀⧏̸qual;括s̀;EGLSTవశ఼ౄోౘ扮qual;扰reater;扸ess;쀀≪̸lantEqual;쀀⩽̸ilde;扴estedĀGL౨౹reaterGreater;쀀⪢̸essLess;쀀⪡̸recedesƀ;ESಒಓಛ技qual;쀀⪯̸lantEqual;拠ĀeiಫಹverseElement;戌ghtTriangleƀ;BEೋೌ೒拫ar;쀀⧐̸qual;拭ĀquೝഌuareSuĀbp೨೹setĀ;E೰ೳ쀀⊏̸qual;拢ersetĀ;Eഃആ쀀⊐̸qual;拣ƀbcpഓതൎsetĀ;Eഛഞ쀀⊂⃒qual;抈ceedsȀ;ESTലള഻െ抁qual;쀀⪰̸lantEqual;拡ilde;쀀≿̸ersetĀ;E൘൛쀀⊃⃒qual;抉ildeȀ;EFT൮൯൵ൿ扁qual;扄ullEqual;扇ilde;扉erticalBar;戤cr;쀀𝒩ilde耻Ñ䃑;䎝܀Eacdfgmoprstuvලෂ෉෕ෛ෠෧෼ขภยา฿ไlig;䅒cute耻Ó䃓Āiy෎ීrc耻Ô䃔;䐞blac;䅐r;쀀𝔒rave耻Ò䃒ƀaei෮ෲ෶cr;䅌ga;䎩cron;䎟pf;쀀𝕆enCurlyĀDQฎบoubleQuote;怜uote;怘;橔Āclวฬr;쀀𝒪ash耻Ø䃘iŬื฼de耻Õ䃕es;樷ml耻Ö䃖erĀBP๋๠Āar๐๓r;怾acĀek๚๜;揞et;掴arenthesis;揜Ҁacfhilors๿ງຊຏຒດຝະ໼rtialD;戂y;䐟r;쀀𝔓i;䎦;䎠usMinus;䂱Āipຢອncareplanåڝf;愙Ȁ;eio຺ູ໠໤檻cedesȀ;EST່້໏໚扺qual;檯lantEqual;扼ilde;找me;怳Ādp໩໮uct;戏ortionĀ;aȥ໹l;戝Āci༁༆r;쀀𝒫;䎨ȀUfos༑༖༛༟OT耻"䀢r;쀀𝔔pf;愚cr;쀀𝒬؀BEacefhiorsu༾གྷཇའཱིྦྷྪྭ႖ႩႴႾarr;椐G耻®䂮ƀcnrཎནབute;䅔g;柫rĀ;tཛྷཝ憠l;椖ƀaeyཧཬཱron;䅘dil;䅖;䐠Ā;vླྀཹ愜erseĀEUྂྙĀlq྇ྎement;戋uilibrium;懋pEquilibrium;楯r»ཹo;䎡ghtЀACDFTUVa࿁࿫࿳ဢဨၛႇϘĀnr࿆࿒gleBracket;柩rowƀ;BL࿜࿝࿡憒ar;懥eftArrow;懄eiling;按oǵ࿹\0စbleBracket;柧nǔည\0နeeVector;楝ectorĀ;Bဝသ懂ar;楕loor;挋Āerိ၃eƀ;AVဵံြ抢rrow;憦ector;楛iangleƀ;BEၐၑၕ抳ar;槐qual;抵pƀDTVၣၮၸownVector;楏eeVector;楜ectorĀ;Bႂႃ憾ar;楔ectorĀ;B႑႒懀ar;楓Āpuႛ႞f;愝ndImplies;楰ightarrow;懛ĀchႹႼr;愛;憱leDelayed;槴ڀHOacfhimoqstuფჱჷჽᄙᄞᅑᅖᅡᅧᆵᆻᆿĀCcჩხHcy;䐩y;䐨FTcy;䐬cute;䅚ʀ;aeiyᄈᄉᄎᄓᄗ檼ron;䅠dil;䅞rc;䅜;䐡r;쀀𝔖ortȀDLRUᄪᄴᄾᅉownArrow»ОeftArrow»࢚ightArrow»࿝pArrow;憑gma;䎣allCircle;战pf;쀀𝕊ɲᅭ\0\0ᅰt;戚areȀ;ISUᅻᅼᆉᆯ斡ntersection;抓uĀbpᆏᆞsetĀ;Eᆗᆘ抏qual;抑ersetĀ;Eᆨᆩ抐qual;抒nion;抔cr;쀀𝒮ar;拆ȀbcmpᇈᇛሉላĀ;sᇍᇎ拐etĀ;Eᇍᇕqual;抆ĀchᇠህeedsȀ;ESTᇭᇮᇴᇿ扻qual;檰lantEqual;扽ilde;承Tháྌ;我ƀ;esሒሓሣ拑rsetĀ;Eሜም抃qual;抇et»ሓրHRSacfhiorsሾቄ቉ቕ቞ቱቶኟዂወዑORN耻Þ䃞ADE;愢ĀHc቎ቒcy;䐋y;䐦Ābuቚቜ;䀉;䎤ƀaeyብቪቯron;䅤dil;䅢;䐢r;쀀𝔗Āeiቻ኉Dzኀ\0ኇefore;戴a;䎘Ācn኎ኘkSpace;쀀  Space;怉ldeȀ;EFTካኬኲኼ戼qual;扃ullEqual;扅ilde;扈pf;쀀𝕋ipleDot;惛Āctዖዛr;쀀𝒯rok;䅦ૡዷጎጚጦ\0ጬጱ\0\0\0\0\0ጸጽ፷ᎅ\0᏿ᐄᐊᐐĀcrዻጁute耻Ú䃚rĀ;oጇገ憟cir;楉rǣጓ\0጖y;䐎ve;䅬Āiyጞጣrc耻Û䃛;䐣blac;䅰r;쀀𝔘rave耻Ù䃙acr;䅪Ādiፁ፩erĀBPፈ፝Āarፍፐr;䁟acĀekፗፙ;揟et;掵arenthesis;揝onĀ;P፰፱拃lus;抎Āgp፻፿on;䅲f;쀀𝕌ЀADETadps᎕ᎮᎸᏄϨᏒᏗᏳrrowƀ;BDᅐᎠᎤar;椒ownArrow;懅ownArrow;憕quilibrium;楮eeĀ;AᏋᏌ报rrow;憥ownáϳerĀLRᏞᏨeftArrow;憖ightArrow;憗iĀ;lᏹᏺ䏒on;䎥ing;䅮cr;쀀𝒰ilde;䅨ml耻Ü䃜ҀDbcdefosvᐧᐬᐰᐳᐾᒅᒊᒐᒖash;披ar;櫫y;䐒ashĀ;lᐻᐼ抩;櫦Āerᑃᑅ;拁ƀbtyᑌᑐᑺar;怖Ā;iᑏᑕcalȀBLSTᑡᑥᑪᑴar;戣ine;䁼eparator;杘ilde;所ThinSpace;怊r;쀀𝔙pf;쀀𝕍cr;쀀𝒱dash;抪ʀcefosᒧᒬᒱᒶᒼirc;䅴dge;拀r;쀀𝔚pf;쀀𝕎cr;쀀𝒲Ȁfiosᓋᓐᓒᓘr;쀀𝔛;䎞pf;쀀𝕏cr;쀀𝒳ҀAIUacfosuᓱᓵᓹᓽᔄᔏᔔᔚᔠcy;䐯cy;䐇cy;䐮cute耻Ý䃝Āiyᔉᔍrc;䅶;䐫r;쀀𝔜pf;쀀𝕐cr;쀀𝒴ml;䅸ЀHacdefosᔵᔹᔿᕋᕏᕝᕠᕤcy;䐖cute;䅹Āayᕄᕉron;䅽;䐗ot;䅻Dzᕔ\0ᕛoWidtè૙a;䎖r;愨pf;愤cr;쀀𝒵௡ᖃᖊᖐ\0ᖰᖶᖿ\0\0\0\0ᗆᗛᗫᙟ᙭\0ᚕ᚛ᚲᚹ\0ᚾcute耻á䃡reve;䄃̀;Ediuyᖜᖝᖡᖣᖨᖭ戾;쀀∾̳;房rc耻â䃢te肻´̆;䐰lig耻æ䃦Ā;r²ᖺ;쀀𝔞rave耻à䃠ĀepᗊᗖĀfpᗏᗔsym;愵èᗓha;䎱ĀapᗟcĀclᗤᗧr;䄁g;樿ɤᗰ\0\0ᘊʀ;adsvᗺᗻᗿᘁᘇ戧nd;橕;橜lope;橘;橚΀;elmrszᘘᘙᘛᘞᘿᙏᙙ戠;榤e»ᘙsdĀ;aᘥᘦ戡ѡᘰᘲᘴᘶᘸᘺᘼᘾ;榨;榩;榪;榫;榬;榭;榮;榯tĀ;vᙅᙆ戟bĀ;dᙌᙍ抾;榝Āptᙔᙗh;戢»¹arr;捼Āgpᙣᙧon;䄅f;쀀𝕒΀;Eaeiop዁ᙻᙽᚂᚄᚇᚊ;橰cir;橯;扊d;手s;䀧roxĀ;e዁ᚒñᚃing耻å䃥ƀctyᚡᚦᚨr;쀀𝒶;䀪mpĀ;e዁ᚯñʈilde耻ã䃣ml耻ä䃤Āciᛂᛈoninôɲnt;樑ࠀNabcdefiklnoprsu᛭ᛱᜰ᜼ᝃᝈ᝸᝽០៦ᠹᡐᜍ᤽᥈ᥰot;櫭Ācrᛶ᜞kȀcepsᜀᜅᜍᜓong;扌psilon;䏶rime;怵imĀ;e᜚᜛戽q;拍Ŷᜢᜦee;抽edĀ;gᜬᜭ挅e»ᜭrkĀ;t፜᜷brk;掶Āoyᜁᝁ;䐱quo;怞ʀcmprtᝓ᝛ᝡᝤᝨausĀ;eĊĉptyv;榰séᜌnoõēƀahwᝯ᝱ᝳ;䎲;愶een;扬r;쀀𝔟g΀costuvwឍឝឳេ៕៛៞ƀaiuបពរðݠrc;旯p»፱ƀdptឤឨឭot;樀lus;樁imes;樂ɱឹ\0\0ើcup;樆ar;昅riangleĀdu៍្own;施p;斳plus;樄eåᑄåᒭarow;植ƀako៭ᠦᠵĀcn៲ᠣkƀlst៺֫᠂ozenge;槫riangleȀ;dlr᠒᠓᠘᠝斴own;斾eft;旂ight;斸k;搣Ʊᠫ\0ᠳƲᠯ\0ᠱ;斒;斑4;斓ck;斈ĀeoᠾᡍĀ;qᡃᡆ쀀=⃥uiv;쀀≡⃥t;挐Ȁptwxᡙᡞᡧᡬf;쀀𝕓Ā;tᏋᡣom»Ꮜtie;拈؀DHUVbdhmptuvᢅᢖᢪᢻᣗᣛᣬ᣿ᤅᤊᤐᤡȀLRlrᢎᢐᢒᢔ;敗;敔;敖;敓ʀ;DUduᢡᢢᢤᢦᢨ敐;敦;敩;敤;敧ȀLRlrᢳᢵᢷᢹ;敝;敚;敜;教΀;HLRhlrᣊᣋᣍᣏᣑᣓᣕ救;敬;散;敠;敫;敢;敟ox;槉ȀLRlrᣤᣦᣨᣪ;敕;敒;攐;攌ʀ;DUduڽ᣷᣹᣻᣽;敥;敨;攬;攴inus;抟lus;択imes;抠ȀLRlrᤙᤛᤝ᤟;敛;敘;攘;攔΀;HLRhlrᤰᤱᤳᤵᤷ᤻᤹攂;敪;敡;敞;攼;攤;攜Āevģ᥂bar耻¦䂦Ȁceioᥑᥖᥚᥠr;쀀𝒷mi;恏mĀ;e᜚᜜lƀ;bhᥨᥩᥫ䁜;槅sub;柈Ŭᥴ᥾lĀ;e᥹᥺怢t»᥺pƀ;Eeįᦅᦇ;檮Ā;qۜۛೡᦧ\0᧨ᨑᨕᨲ\0ᨷᩐ\0\0᪴\0\0᫁\0\0ᬡᬮ᭍᭒\0᯽\0ᰌƀcpr᦭ᦲ᧝ute;䄇̀;abcdsᦿᧀᧄ᧊᧕᧙戩nd;橄rcup;橉Āau᧏᧒p;橋p;橇ot;橀;쀀∩︀Āeo᧢᧥t;恁îړȀaeiu᧰᧻ᨁᨅǰ᧵\0᧸s;橍on;䄍dil耻ç䃧rc;䄉psĀ;sᨌᨍ橌m;橐ot;䄋ƀdmnᨛᨠᨦil肻¸ƭptyv;榲t脀¢;eᨭᨮ䂢räƲr;쀀𝔠ƀceiᨽᩀᩍy;䑇ckĀ;mᩇᩈ朓ark»ᩈ;䏇r΀;Ecefms᩟᩠ᩢᩫ᪤᪪᪮旋;槃ƀ;elᩩᩪᩭ䋆q;扗eɡᩴ\0\0᪈rrowĀlr᩼᪁eft;憺ight;憻ʀRSacd᪒᪔᪖᪚᪟»ཇ;擈st;抛irc;抚ash;抝nint;樐id;櫯cir;槂ubsĀ;u᪻᪼晣it»᪼ˬ᫇᫔᫺\0ᬊonĀ;eᫍᫎ䀺Ā;qÇÆɭ᫙\0\0᫢aĀ;t᫞᫟䀬;䁀ƀ;fl᫨᫩᫫戁îᅠeĀmx᫱᫶ent»᫩eóɍǧ᫾\0ᬇĀ;dኻᬂot;橭nôɆƀfryᬐᬔᬗ;쀀𝕔oäɔ脀©;sŕᬝr;愗Āaoᬥᬩrr;憵ss;朗Ācuᬲᬷr;쀀𝒸Ābpᬼ᭄Ā;eᭁᭂ櫏;櫑Ā;eᭉᭊ櫐;櫒dot;拯΀delprvw᭠᭬᭷ᮂᮬᯔ᯹arrĀlr᭨᭪;椸;椵ɰ᭲\0\0᭵r;拞c;拟arrĀ;p᭿ᮀ憶;椽̀;bcdosᮏᮐᮖᮡᮥᮨ截rcap;橈Āauᮛᮞp;橆p;橊ot;抍r;橅;쀀∪︀Ȁalrv᮵ᮿᯞᯣrrĀ;mᮼᮽ憷;椼yƀevwᯇᯔᯘqɰᯎ\0\0ᯒreã᭳uã᭵ee;拎edge;拏en耻¤䂤earrowĀlrᯮ᯳eft»ᮀight»ᮽeäᯝĀciᰁᰇoninôǷnt;戱lcty;挭ঀAHabcdefhijlorstuwz᰸᰻᰿ᱝᱩᱵᲊᲞᲬᲷ᳻᳿ᴍᵻᶑᶫᶻ᷆᷍rò΁ar;楥Ȁglrs᱈ᱍ᱒᱔ger;怠eth;愸òᄳhĀ;vᱚᱛ怐»ऊūᱡᱧarow;椏aã̕Āayᱮᱳron;䄏;䐴ƀ;ao̲ᱼᲄĀgrʿᲁr;懊tseq;橷ƀglmᲑᲔᲘ耻°䂰ta;䎴ptyv;榱ĀirᲣᲨsht;楿;쀀𝔡arĀlrᲳᲵ»ࣜ»သʀaegsv᳂͸᳖᳜᳠mƀ;oș᳊᳔ndĀ;ș᳑uit;晦amma;䏝in;拲ƀ;io᳧᳨᳸䃷de脀÷;o᳧ᳰntimes;拇nø᳷cy;䑒cɯᴆ\0\0ᴊrn;挞op;挍ʀlptuwᴘᴝᴢᵉᵕlar;䀤f;쀀𝕕ʀ;emps̋ᴭᴷᴽᵂqĀ;d͒ᴳot;扑inus;戸lus;戔quare;抡blebarwedgåúnƀadhᄮᵝᵧownarrowóᲃarpoonĀlrᵲᵶefôᲴighôᲶŢᵿᶅkaro÷གɯᶊ\0\0ᶎrn;挟op;挌ƀcotᶘᶣᶦĀryᶝᶡ;쀀𝒹;䑕l;槶rok;䄑Ādrᶰᶴot;拱iĀ;fᶺ᠖斿Āah᷀᷃ròЩaòྦangle;榦Āci᷒ᷕy;䑟grarr;柿ऀDacdefglmnopqrstuxḁḉḙḸոḼṉṡṾấắẽỡἪἷὄ὎὚ĀDoḆᴴoôᲉĀcsḎḔute耻é䃩ter;橮ȀaioyḢḧḱḶron;䄛rĀ;cḭḮ扖耻ê䃪lon;払;䑍ot;䄗ĀDrṁṅot;扒;쀀𝔢ƀ;rsṐṑṗ檚ave耻è䃨Ā;dṜṝ檖ot;檘Ȁ;ilsṪṫṲṴ檙nters;揧;愓Ā;dṹṺ檕ot;檗ƀapsẅẉẗcr;䄓tyƀ;svẒẓẕ戅et»ẓpĀ1;ẝẤijạả;怄;怅怃ĀgsẪẬ;䅋p;怂ĀgpẴẸon;䄙f;쀀𝕖ƀalsỄỎỒrĀ;sỊị拕l;槣us;橱iƀ;lvỚớở䎵on»ớ;䏵ȀcsuvỪỳἋἣĀioữḱrc»Ḯɩỹ\0\0ỻíՈantĀglἂἆtr»ṝess»Ṻƀaeiἒ἖Ἒls;䀽st;扟vĀ;DȵἠD;橸parsl;槥ĀDaἯἳot;打rr;楱ƀcdiἾὁỸr;愯oô͒ĀahὉὋ;䎷耻ð䃰Āmrὓὗl耻ë䃫o;悬ƀcipὡὤὧl;䀡sôծĀeoὬὴctatioîՙnentialåչৡᾒ\0ᾞ\0ᾡᾧ\0\0ῆῌ\0ΐ\0ῦῪ \0 ⁚llingdotseñṄy;䑄male;晀ƀilrᾭᾳ῁lig;耀ffiɩᾹ\0\0᾽g;耀ffig;耀ffl;쀀𝔣lig;耀filig;쀀fjƀaltῙ῜ῡt;晭ig;耀flns;斱of;䆒ǰ΅\0ῳf;쀀𝕗ĀakֿῷĀ;vῼ´拔;櫙artint;樍Āao‌⁕Ācs‑⁒ႉ‸⁅⁈\0⁐β•‥‧‪‬\0‮耻½䂽;慓耻¼䂼;慕;慙;慛Ƴ‴\0‶;慔;慖ʴ‾⁁\0\0⁃耻¾䂾;慗;慜5;慘ƶ⁌\0⁎;慚;慝8;慞l;恄wn;挢cr;쀀𝒻ࢀEabcdefgijlnorstv₂₉₟₥₰₴⃰⃵⃺⃿℃ℒℸ̗ℾ⅒↞Ā;lٍ₇;檌ƀcmpₐₕ₝ute;䇵maĀ;dₜ᳚䎳;檆reve;䄟Āiy₪₮rc;䄝;䐳ot;䄡Ȁ;lqsؾق₽⃉ƀ;qsؾٌ⃄lanô٥Ȁ;cdl٥⃒⃥⃕c;檩otĀ;o⃜⃝檀Ā;l⃢⃣檂;檄Ā;e⃪⃭쀀⋛︀s;檔r;쀀𝔤Ā;gٳ؛mel;愷cy;䑓Ȁ;Eajٚℌℎℐ;檒;檥;檤ȀEaesℛℝ℩ℴ;扩pĀ;p℣ℤ檊rox»ℤĀ;q℮ℯ檈Ā;q℮ℛim;拧pf;쀀𝕘Āci⅃ⅆr;愊mƀ;el٫ⅎ⅐;檎;檐茀>;cdlqr׮ⅠⅪⅮⅳⅹĀciⅥⅧ;檧r;橺ot;拗Par;榕uest;橼ʀadelsↄⅪ←ٖ↛ǰ↉\0↎proø₞r;楸qĀlqؿ↖lesó₈ií٫Āen↣↭rtneqq;쀀≩︀Å↪ԀAabcefkosy⇄⇇⇱⇵⇺∘∝∯≨≽ròΠȀilmr⇐⇔⇗⇛rsðᒄf»․ilôکĀdr⇠⇤cy;䑊ƀ;cwࣴ⇫⇯ir;楈;憭ar;意irc;䄥ƀalr∁∎∓rtsĀ;u∉∊晥it»∊lip;怦con;抹r;쀀𝔥sĀew∣∩arow;椥arow;椦ʀamopr∺∾≃≞≣rr;懿tht;戻kĀlr≉≓eftarrow;憩ightarrow;憪f;쀀𝕙bar;怕ƀclt≯≴≸r;쀀𝒽asè⇴rok;䄧Ābp⊂⊇ull;恃hen»ᱛૡ⊣\0⊪\0⊸⋅⋎\0⋕⋳\0\0⋸⌢⍧⍢⍿\0⎆⎪⎴cute耻í䃭ƀ;iyݱ⊰⊵rc耻î䃮;䐸Ācx⊼⊿y;䐵cl耻¡䂡ĀfrΟ⋉;쀀𝔦rave耻ì䃬Ȁ;inoܾ⋝⋩⋮Āin⋢⋦nt;樌t;戭fin;槜ta;愩lig;䄳ƀaop⋾⌚⌝ƀcgt⌅⌈⌗r;䄫ƀelpܟ⌏⌓inåގarôܠh;䄱f;抷ed;䆵ʀ;cfotӴ⌬⌱⌽⍁are;愅inĀ;t⌸⌹戞ie;槝doô⌙ʀ;celpݗ⍌⍐⍛⍡al;抺Āgr⍕⍙eróᕣã⍍arhk;樗rod;樼Ȁcgpt⍯⍲⍶⍻y;䑑on;䄯f;쀀𝕚a;䎹uest耻¿䂿Āci⎊⎏r;쀀𝒾nʀ;EdsvӴ⎛⎝⎡ӳ;拹ot;拵Ā;v⎦⎧拴;拳Ā;iݷ⎮lde;䄩ǫ⎸\0⎼cy;䑖l耻ï䃯̀cfmosu⏌⏗⏜⏡⏧⏵Āiy⏑⏕rc;䄵;䐹r;쀀𝔧ath;䈷pf;쀀𝕛ǣ⏬\0⏱r;쀀𝒿rcy;䑘kcy;䑔Ѐacfghjos␋␖␢␧␭␱␵␻ppaĀ;v␓␔䎺;䏰Āey␛␠dil;䄷;䐺r;쀀𝔨reen;䄸cy;䑅cy;䑜pf;쀀𝕜cr;쀀𝓀஀ABEHabcdefghjlmnoprstuv⑰⒁⒆⒍⒑┎┽╚▀♎♞♥♹♽⚚⚲⛘❝❨➋⟀⠁⠒ƀart⑷⑺⑼rò৆òΕail;椛arr;椎Ā;gঔ⒋;檋ar;楢ॣ⒥\0⒪\0⒱\0\0\0\0\0⒵Ⓔ\0ⓆⓈⓍ\0⓹ute;䄺mptyv;榴raîࡌbda;䎻gƀ;dlࢎⓁⓃ;榑åࢎ;檅uo耻«䂫rЀ;bfhlpst࢙ⓞⓦⓩ⓫⓮⓱⓵Ā;f࢝ⓣs;椟s;椝ë≒p;憫l;椹im;楳l;憢ƀ;ae⓿─┄檫il;椙Ā;s┉┊檭;쀀⪭︀ƀabr┕┙┝rr;椌rk;杲Āak┢┬cĀek┨┪;䁻;䁛Āes┱┳;榋lĀdu┹┻;榏;榍Ȁaeuy╆╋╖╘ron;䄾Ādi═╔il;䄼ìࢰâ┩;䐻Ȁcqrs╣╦╭╽a;椶uoĀ;rนᝆĀdu╲╷har;楧shar;楋h;憲ʀ;fgqs▋▌উ◳◿扤tʀahlrt▘▤▷◂◨rrowĀ;t࢙□aé⓶arpoonĀdu▯▴own»њp»०eftarrows;懇ightƀahs◍◖◞rrowĀ;sࣴࢧarpoonó྘quigarro÷⇰hreetimes;拋ƀ;qs▋ও◺lanôবʀ;cdgsব☊☍☝☨c;檨otĀ;o☔☕橿Ā;r☚☛檁;檃Ā;e☢☥쀀⋚︀s;檓ʀadegs☳☹☽♉♋pproøⓆot;拖qĀgq♃♅ôউgtò⒌ôছiíলƀilr♕࣡♚sht;楼;쀀𝔩Ā;Eজ♣;檑š♩♶rĀdu▲♮Ā;l॥♳;楪lk;斄cy;䑙ʀ;achtੈ⚈⚋⚑⚖rò◁orneòᴈard;楫ri;旺Āio⚟⚤dot;䅀ustĀ;a⚬⚭掰che»⚭ȀEaes⚻⚽⛉⛔;扨pĀ;p⛃⛄檉rox»⛄Ā;q⛎⛏檇Ā;q⛎⚻im;拦Ѐa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di㩀㩑Ābg㩅㩉ar;機eĀ;qᗺ㩏;扙erp;愘r;쀀𝔴pf;쀀𝕨Ā;eᑹ㩦atèᑹcr;쀀𝓌ૣណ㪇\0㪋\0㪐㪛\0\0㪝㪨㪫㪯\0\0㫃㫎\0㫘ៜ៟tré៑r;쀀𝔵ĀAa㪔㪗ròσrò৶;䎾ĀAa㪡㪤ròθrò৫að✓is;拻ƀdptឤ㪵㪾Āfl㪺ឩ;쀀𝕩imåឲĀAa㫇㫊ròώròਁĀcq㫒ីr;쀀𝓍Āpt៖㫜ré។Ѐacefiosu㫰㫽㬈㬌㬑㬕㬛㬡cĀuy㫶㫻te耻ý䃽;䑏Āiy㬂㬆rc;䅷;䑋n耻¥䂥r;쀀𝔶cy;䑗pf;쀀𝕪cr;쀀𝓎Ācm㬦㬩y;䑎l耻ÿ䃿Ԁacdefhiosw㭂㭈㭔㭘㭤㭩㭭㭴㭺㮀cute;䅺Āay㭍㭒ron;䅾;䐷ot;䅼Āet㭝㭡træᕟa;䎶r;쀀𝔷cy;䐶grarr;懝pf;쀀𝕫cr;쀀𝓏Ājn㮅㮇;怍j;怌'.split("").map(s=>s.charCodeAt(0))),YK=new Map([[0,65533],[128,8364],[130,8218],[131,402],[132,8222],[133,8230],[134,8224],[135,8225],[136,710],[137,8240],[138,352],[139,8249],[140,338],[142,381],[145,8216],[146,8217],[147,8220],[148,8221],[149,8226],[150,8211],[151,8212],[152,732],[153,8482],[154,353],[155,8250],[156,339],[158,382],[159,376]]);function XK(s){var a;return s>=55296&&s<=57343||s>1114111?65533:(a=YK.get(s))!==null&&a!==void 0?a:s}var kn;(function(s){s[s.NUM=35]="NUM",s[s.SEMI=59]="SEMI",s[s.EQUALS=61]="EQUALS",s[s.ZERO=48]="ZERO",s[s.NINE=57]="NINE",s[s.LOWER_A=97]="LOWER_A",s[s.LOWER_F=102]="LOWER_F",s[s.LOWER_X=120]="LOWER_X",s[s.LOWER_Z=122]="LOWER_Z",s[s.UPPER_A=65]="UPPER_A",s[s.UPPER_F=70]="UPPER_F",s[s.UPPER_Z=90]="UPPER_Z"})(kn||(kn={}));const KK=32;var cp;(function(s){s[s.VALUE_LENGTH=49152]="VALUE_LENGTH",s[s.BRANCH_LENGTH=16256]="BRANCH_LENGTH",s[s.JUMP_TABLE=127]="JUMP_TABLE"})(cp||(cp={}));function Mv(s){return s>=kn.ZERO&&s<=kn.NINE}function ZK(s){return s>=kn.UPPER_A&&s<=kn.UPPER_F||s>=kn.LOWER_A&&s<=kn.LOWER_F}function QK(s){return s>=kn.UPPER_A&&s<=kn.UPPER_Z||s>=kn.LOWER_A&&s<=kn.LOWER_Z||Mv(s)}function JK(s){return s===kn.EQUALS||QK(s)}var xn;(function(s){s[s.EntityStart=0]="EntityStart",s[s.NumericStart=1]="NumericStart",s[s.NumericDecimal=2]="NumericDecimal",s[s.NumericHex=3]="NumericHex",s[s.NamedEntity=4]="NamedEntity"})(xn||(xn={}));var gi;(function(s){s[s.Legacy=0]="Legacy",s[s.Strict=1]="Strict",s[s.Attribute=2]="Attribute"})(gi||(gi={}));class sZ{constructor(a,t,n){this.decodeTree=a,this.emitCodePoint=t,this.errors=n,this.state=xn.EntityStart,this.consumed=1,this.result=0,this.treeIndex=0,this.excess=1,this.decodeMode=gi.Strict}startEntity(a){this.decodeMode=a,this.state=xn.EntityStart,this.result=0,this.treeIndex=0,this.excess=1,this.consumed=1}write(a,t){switch(this.state){case xn.EntityStart:return a.charCodeAt(t)===kn.NUM?(this.state=xn.NumericStart,this.consumed+=1,this.stateNumericStart(a,t+1)):(this.state=xn.NamedEntity,this.stateNamedEntity(a,t));case xn.NumericStart:return this.stateNumericStart(a,t);case xn.NumericDecimal:return this.stateNumericDecimal(a,t);case xn.NumericHex:return this.stateNumericHex(a,t);case xn.NamedEntity:return this.stateNamedEntity(a,t)}}stateNumericStart(a,t){return t>=a.length?-1:(a.charCodeAt(t)|KK)===kn.LOWER_X?(this.state=xn.NumericHex,this.consumed+=1,this.stateNumericHex(a,t+1)):(this.state=xn.NumericDecimal,this.stateNumericDecimal(a,t))}addToNumericResult(a,t,n,e){if(t!==n){const l=n-t;this.result=this.result*Math.pow(e,l)+Number.parseInt(a.substr(t,l),e),this.consumed+=l}}stateNumericHex(a,t){const n=t;for(;t<a.length;){const e=a.charCodeAt(t);if(Mv(e)||ZK(e))t+=1;else return this.addToNumericResult(a,n,t,16),this.emitNumericEntity(e,3)}return this.addToNumericResult(a,n,t,16),-1}stateNumericDecimal(a,t){const n=t;for(;t<a.length;){const e=a.charCodeAt(t);if(Mv(e))t+=1;else return this.addToNumericResult(a,n,t,10),this.emitNumericEntity(e,2)}return this.addToNumericResult(a,n,t,10),-1}emitNumericEntity(a,t){var n;if(this.consumed<=t)return(n=this.errors)===null||n===void 0||n.absenceOfDigitsInNumericCharacterReference(this.consumed),0;if(a===kn.SEMI)this.consumed+=1;else if(this.decodeMode===gi.Strict)return 0;return this.emitCodePoint(XK(this.result),this.consumed),this.errors&&(a!==kn.SEMI&&this.errors.missingSemicolonAfterCharacterReference(),this.errors.validateNumericCharacterReference(this.result)),this.consumed}stateNamedEntity(a,t){const{decodeTree:n}=this;let e=n[this.treeIndex],l=(e&cp.VALUE_LENGTH)>>14;for(;t<a.length;t++,this.excess++){const i=a.charCodeAt(t);if(this.treeIndex=aZ(n,e,this.treeIndex+Math.max(1,l),i),this.treeIndex<0)return this.result===0||this.decodeMode===gi.Attribute&&(l===0||JK(i))?0:this.emitNotTerminatedNamedEntity();if(e=n[this.treeIndex],l=(e&cp.VALUE_LENGTH)>>14,l!==0){if(i===kn.SEMI)return this.emitNamedEntityData(this.treeIndex,l,this.consumed+this.excess);this.decodeMode!==gi.Strict&&(this.result=this.treeIndex,this.consumed+=this.excess,this.excess=0)}}return-1}emitNotTerminatedNamedEntity(){var a;const{result:t,decodeTree:n}=this,e=(n[t]&cp.VALUE_LENGTH)>>14;return this.emitNamedEntityData(t,e,this.consumed),(a=this.errors)===null||a===void 0||a.missingSemicolonAfterCharacterReference(),this.consumed}emitNamedEntityData(a,t,n){const{decodeTree:e}=this;return this.emitCodePoint(t===1?e[a]&~cp.VALUE_LENGTH:e[a+1],n),t===3&&this.emitCodePoint(e[a+2],n),n}end(){var a;switch(this.state){case xn.NamedEntity:return this.result!==0&&(this.decodeMode!==gi.Attribute||this.result===this.treeIndex)?this.emitNotTerminatedNamedEntity():0;case xn.NumericDecimal:return this.emitNumericEntity(0,2);case xn.NumericHex:return this.emitNumericEntity(0,3);case xn.NumericStart:return(a=this.errors)===null||a===void 0||a.absenceOfDigitsInNumericCharacterReference(this.consumed),0;case xn.EntityStart:return 0}}}function aZ(s,a,t,n){const e=(a&cp.BRANCH_LENGTH)>>7,l=a&cp.JUMP_TABLE;if(e===0)return l!==0&&n===l?t:-1;if(l){const r=n-l;return r<0||r>=e?-1:s[t+r]-1}let i=t,p=i+e-1;for(;i<=p;){const r=i+p>>>1,m=s[r];if(m<n)i=r+1;else if(m>n)p=r-1;else return s[r+e]}return-1}var sa;(function(s){s.HTML="http://www.w3.org/1999/xhtml",s.MATHML="http://www.w3.org/1998/Math/MathML",s.SVG="http://www.w3.org/2000/svg",s.XLINK="http://www.w3.org/1999/xlink",s.XML="http://www.w3.org/XML/1998/namespace",s.XMLNS="http://www.w3.org/2000/xmlns/"})(sa||(sa={}));var lr;(function(s){s.TYPE="type",s.ACTION="proxy.php?url=https%3A%2F%2Flincdocs.github.io%2Faction",s.ENCODING="encoding",s.PROMPT="prompt",s.NAME="name",s.COLOR="color",s.FACE="face",s.SIZE="size"})(lr||(lr={}));var be;(function(s){s.NO_QUIRKS="no-quirks",s.QUIRKS="quirks",s.LIMITED_QUIRKS="limited-quirks"})(be||(be={}));var Ts;(function(s){s.A="a",s.ADDRESS="address",s.ANNOTATION_XML="annotation-xml",s.APPLET="applet",s.AREA="area",s.ARTICLE="article",s.ASIDE="aside",s.B="b",s.BASE="base",s.BASEFONT="basefont",s.BGSOUND="bgsound",s.BIG="big",s.BLOCKQUOTE="blockquote",s.BODY="body",s.BR="br",s.BUTTON="button",s.CAPTION="caption",s.CENTER="center",s.CODE="code",s.COL="col",s.COLGROUP="colgroup",s.DD="dd",s.DESC="desc",s.DETAILS="details",s.DIALOG="dialog",s.DIR="dir",s.DIV="div",s.DL="dl",s.DT="dt",s.EM="em",s.EMBED="embed",s.FIELDSET="fieldset",s.FIGCAPTION="figcaption",s.FIGURE="figure",s.FONT="font",s.FOOTER="footer",s.FOREIGN_OBJECT="foreignObject",s.FORM="form",s.FRAME="frame",s.FRAMESET="frameset",s.H1="h1",s.H2="h2",s.H3="h3",s.H4="h4",s.H5="h5",s.H6="h6",s.HEAD="head",s.HEADER="header",s.HGROUP="hgroup",s.HR="hr",s.HTML="html",s.I="i",s.IMG="img",s.IMAGE="image",s.INPUT="input",s.IFRAME="iframe",s.KEYGEN="keygen",s.LABEL="label",s.LI="li",s.LINK="link",s.LISTING="listing",s.MAIN="main",s.MALIGNMARK="malignmark",s.MARQUEE="marquee",s.MATH="math",s.MENU="menu",s.META="meta",s.MGLYPH="mglyph",s.MI="mi",s.MO="mo",s.MN="mn",s.MS="ms",s.MTEXT="mtext",s.NAV="nav",s.NOBR="nobr",s.NOFRAMES="noframes",s.NOEMBED="noembed",s.NOSCRIPT="noscript",s.OBJECT="object",s.OL="ol",s.OPTGROUP="optgroup",s.OPTION="option",s.P="p",s.PARAM="param",s.PLAINTEXT="plaintext",s.PRE="pre",s.RB="rb",s.RP="rp",s.RT="rt",s.RTC="rtc",s.RUBY="ruby",s.S="s",s.SCRIPT="script",s.SEARCH="search",s.SECTION="section",s.SELECT="select",s.SOURCE="source",s.SMALL="small",s.SPAN="span",s.STRIKE="strike",s.STRONG="strong",s.STYLE="style",s.SUB="sub",s.SUMMARY="summary",s.SUP="sup",s.TABLE="table",s.TBODY="tbody",s.TEMPLATE="template",s.TEXTAREA="textarea",s.TFOOT="tfoot",s.TD="td",s.TH="th",s.THEAD="thead",s.TITLE="title",s.TR="tr",s.TRACK="track",s.TT="tt",s.U="u",s.UL="ul",s.SVG="svg",s.VAR="var",s.WBR="wbr",s.XMP="xmp"})(Ts||(Ts={}));var k;(function(s){s[s.UNKNOWN=0]="UNKNOWN",s[s.A=1]="A",s[s.ADDRESS=2]="ADDRESS",s[s.ANNOTATION_XML=3]="ANNOTATION_XML",s[s.APPLET=4]="APPLET",s[s.AREA=5]="AREA",s[s.ARTICLE=6]="ARTICLE",s[s.ASIDE=7]="ASIDE",s[s.B=8]="B",s[s.BASE=9]="BASE",s[s.BASEFONT=10]="BASEFONT",s[s.BGSOUND=11]="BGSOUND",s[s.BIG=12]="BIG",s[s.BLOCKQUOTE=13]="BLOCKQUOTE",s[s.BODY=14]="BODY",s[s.BR=15]="BR",s[s.BUTTON=16]="BUTTON",s[s.CAPTION=17]="CAPTION",s[s.CENTER=18]="CENTER",s[s.CODE=19]="CODE",s[s.COL=20]="COL",s[s.COLGROUP=21]="COLGROUP",s[s.DD=22]="DD",s[s.DESC=23]="DESC",s[s.DETAILS=24]="DETAILS",s[s.DIALOG=25]="DIALOG",s[s.DIR=26]="DIR",s[s.DIV=27]="DIV",s[s.DL=28]="DL",s[s.DT=29]="DT",s[s.EM=30]="EM",s[s.EMBED=31]="EMBED",s[s.FIELDSET=32]="FIELDSET",s[s.FIGCAPTION=33]="FIGCAPTION",s[s.FIGURE=34]="FIGURE",s[s.FONT=35]="FONT",s[s.FOOTER=36]="FOOTER",s[s.FOREIGN_OBJECT=37]="FOREIGN_OBJECT",s[s.FORM=38]="FORM",s[s.FRAME=39]="FRAME",s[s.FRAMESET=40]="FRAMESET",s[s.H1=41]="H1",s[s.H2=42]="H2",s[s.H3=43]="H3",s[s.H4=44]="H4",s[s.H5=45]="H5",s[s.H6=46]="H6",s[s.HEAD=47]="HEAD",s[s.HEADER=48]="HEADER",s[s.HGROUP=49]="HGROUP",s[s.HR=50]="HR",s[s.HTML=51]="HTML",s[s.I=52]="I",s[s.IMG=53]="IMG",s[s.IMAGE=54]="IMAGE",s[s.INPUT=55]="INPUT",s[s.IFRAME=56]="IFRAME",s[s.KEYGEN=57]="KEYGEN",s[s.LABEL=58]="LABEL",s[s.LI=59]="LI",s[s.LINK=60]="LINK",s[s.LISTING=61]="LISTING",s[s.MAIN=62]="MAIN",s[s.MALIGNMARK=63]="MALIGNMARK",s[s.MARQUEE=64]="MARQUEE",s[s.MATH=65]="MATH",s[s.MENU=66]="MENU",s[s.META=67]="META",s[s.MGLYPH=68]="MGLYPH",s[s.MI=69]="MI",s[s.MO=70]="MO",s[s.MN=71]="MN",s[s.MS=72]="MS",s[s.MTEXT=73]="MTEXT",s[s.NAV=74]="NAV",s[s.NOBR=75]="NOBR",s[s.NOFRAMES=76]="NOFRAMES",s[s.NOEMBED=77]="NOEMBED",s[s.NOSCRIPT=78]="NOSCRIPT",s[s.OBJECT=79]="OBJECT",s[s.OL=80]="OL",s[s.OPTGROUP=81]="OPTGROUP",s[s.OPTION=82]="OPTION",s[s.P=83]="P",s[s.PARAM=84]="PARAM",s[s.PLAINTEXT=85]="PLAINTEXT",s[s.PRE=86]="PRE",s[s.RB=87]="RB",s[s.RP=88]="RP",s[s.RT=89]="RT",s[s.RTC=90]="RTC",s[s.RUBY=91]="RUBY",s[s.S=92]="S",s[s.SCRIPT=93]="SCRIPT",s[s.SEARCH=94]="SEARCH",s[s.SECTION=95]="SECTION",s[s.SELECT=96]="SELECT",s[s.SOURCE=97]="SOURCE",s[s.SMALL=98]="SMALL",s[s.SPAN=99]="SPAN",s[s.STRIKE=100]="STRIKE",s[s.STRONG=101]="STRONG",s[s.STYLE=102]="STYLE",s[s.SUB=103]="SUB",s[s.SUMMARY=104]="SUMMARY",s[s.SUP=105]="SUP",s[s.TABLE=106]="TABLE",s[s.TBODY=107]="TBODY",s[s.TEMPLATE=108]="TEMPLATE",s[s.TEXTAREA=109]="TEXTAREA",s[s.TFOOT=110]="TFOOT",s[s.TD=111]="TD",s[s.TH=112]="TH",s[s.THEAD=113]="THEAD",s[s.TITLE=114]="TITLE",s[s.TR=115]="TR",s[s.TRACK=116]="TRACK",s[s.TT=117]="TT",s[s.U=118]="U",s[s.UL=119]="UL",s[s.SVG=120]="SVG",s[s.VAR=121]="VAR",s[s.WBR=122]="WBR",s[s.XMP=123]="XMP"})(k||(k={}));const tZ=new Map([[Ts.A,k.A],[Ts.ADDRESS,k.ADDRESS],[Ts.ANNOTATION_XML,k.ANNOTATION_XML],[Ts.APPLET,k.APPLET],[Ts.AREA,k.AREA],[Ts.ARTICLE,k.ARTICLE],[Ts.ASIDE,k.ASIDE],[Ts.B,k.B],[Ts.BASE,k.BASE],[Ts.BASEFONT,k.BASEFONT],[Ts.BGSOUND,k.BGSOUND],[Ts.BIG,k.BIG],[Ts.BLOCKQUOTE,k.BLOCKQUOTE],[Ts.BODY,k.BODY],[Ts.BR,k.BR],[Ts.BUTTON,k.BUTTON],[Ts.CAPTION,k.CAPTION],[Ts.CENTER,k.CENTER],[Ts.CODE,k.CODE],[Ts.COL,k.COL],[Ts.COLGROUP,k.COLGROUP],[Ts.DD,k.DD],[Ts.DESC,k.DESC],[Ts.DETAILS,k.DETAILS],[Ts.DIALOG,k.DIALOG],[Ts.DIR,k.DIR],[Ts.DIV,k.DIV],[Ts.DL,k.DL],[Ts.DT,k.DT],[Ts.EM,k.EM],[Ts.EMBED,k.EMBED],[Ts.FIELDSET,k.FIELDSET],[Ts.FIGCAPTION,k.FIGCAPTION],[Ts.FIGURE,k.FIGURE],[Ts.FONT,k.FONT],[Ts.FOOTER,k.FOOTER],[Ts.FOREIGN_OBJECT,k.FOREIGN_OBJECT],[Ts.FORM,k.FORM],[Ts.FRAME,k.FRAME],[Ts.FRAMESET,k.FRAMESET],[Ts.H1,k.H1],[Ts.H2,k.H2],[Ts.H3,k.H3],[Ts.H4,k.H4],[Ts.H5,k.H5],[Ts.H6,k.H6],[Ts.HEAD,k.HEAD],[Ts.HEADER,k.HEADER],[Ts.HGROUP,k.HGROUP],[Ts.HR,k.HR],[Ts.HTML,k.HTML],[Ts.I,k.I],[Ts.IMG,k.IMG],[Ts.IMAGE,k.IMAGE],[Ts.INPUT,k.INPUT],[Ts.IFRAME,k.IFRAME],[Ts.KEYGEN,k.KEYGEN],[Ts.LABEL,k.LABEL],[Ts.LI,k.LI],[Ts.LINK,k.LINK],[Ts.LISTING,k.LISTING],[Ts.MAIN,k.MAIN],[Ts.MALIGNMARK,k.MALIGNMARK],[Ts.MARQUEE,k.MARQUEE],[Ts.MATH,k.MATH],[Ts.MENU,k.MENU],[Ts.META,k.META],[Ts.MGLYPH,k.MGLYPH],[Ts.MI,k.MI],[Ts.MO,k.MO],[Ts.MN,k.MN],[Ts.MS,k.MS],[Ts.MTEXT,k.MTEXT],[Ts.NAV,k.NAV],[Ts.NOBR,k.NOBR],[Ts.NOFRAMES,k.NOFRAMES],[Ts.NOEMBED,k.NOEMBED],[Ts.NOSCRIPT,k.NOSCRIPT],[Ts.OBJECT,k.OBJECT],[Ts.OL,k.OL],[Ts.OPTGROUP,k.OPTGROUP],[Ts.OPTION,k.OPTION],[Ts.P,k.P],[Ts.PARAM,k.PARAM],[Ts.PLAINTEXT,k.PLAINTEXT],[Ts.PRE,k.PRE],[Ts.RB,k.RB],[Ts.RP,k.RP],[Ts.RT,k.RT],[Ts.RTC,k.RTC],[Ts.RUBY,k.RUBY],[Ts.S,k.S],[Ts.SCRIPT,k.SCRIPT],[Ts.SEARCH,k.SEARCH],[Ts.SECTION,k.SECTION],[Ts.SELECT,k.SELECT],[Ts.SOURCE,k.SOURCE],[Ts.SMALL,k.SMALL],[Ts.SPAN,k.SPAN],[Ts.STRIKE,k.STRIKE],[Ts.STRONG,k.STRONG],[Ts.STYLE,k.STYLE],[Ts.SUB,k.SUB],[Ts.SUMMARY,k.SUMMARY],[Ts.SUP,k.SUP],[Ts.TABLE,k.TABLE],[Ts.TBODY,k.TBODY],[Ts.TEMPLATE,k.TEMPLATE],[Ts.TEXTAREA,k.TEXTAREA],[Ts.TFOOT,k.TFOOT],[Ts.TD,k.TD],[Ts.TH,k.TH],[Ts.THEAD,k.THEAD],[Ts.TITLE,k.TITLE],[Ts.TR,k.TR],[Ts.TRACK,k.TRACK],[Ts.TT,k.TT],[Ts.U,k.U],[Ts.UL,k.UL],[Ts.SVG,k.SVG],[Ts.VAR,k.VAR],[Ts.WBR,k.WBR],[Ts.XMP,k.XMP]]);function jd(s){var a;return(a=tZ.get(s))!==null&&a!==void 0?a:k.UNKNOWN}const ea=k,nZ={[sa.HTML]:new Set([ea.ADDRESS,ea.APPLET,ea.AREA,ea.ARTICLE,ea.ASIDE,ea.BASE,ea.BASEFONT,ea.BGSOUND,ea.BLOCKQUOTE,ea.BODY,ea.BR,ea.BUTTON,ea.CAPTION,ea.CENTER,ea.COL,ea.COLGROUP,ea.DD,ea.DETAILS,ea.DIR,ea.DIV,ea.DL,ea.DT,ea.EMBED,ea.FIELDSET,ea.FIGCAPTION,ea.FIGURE,ea.FOOTER,ea.FORM,ea.FRAME,ea.FRAMESET,ea.H1,ea.H2,ea.H3,ea.H4,ea.H5,ea.H6,ea.HEAD,ea.HEADER,ea.HGROUP,ea.HR,ea.HTML,ea.IFRAME,ea.IMG,ea.INPUT,ea.LI,ea.LINK,ea.LISTING,ea.MAIN,ea.MARQUEE,ea.MENU,ea.META,ea.NAV,ea.NOEMBED,ea.NOFRAMES,ea.NOSCRIPT,ea.OBJECT,ea.OL,ea.P,ea.PARAM,ea.PLAINTEXT,ea.PRE,ea.SCRIPT,ea.SECTION,ea.SELECT,ea.SOURCE,ea.STYLE,ea.SUMMARY,ea.TABLE,ea.TBODY,ea.TD,ea.TEMPLATE,ea.TEXTAREA,ea.TFOOT,ea.TH,ea.THEAD,ea.TITLE,ea.TR,ea.TRACK,ea.UL,ea.WBR,ea.XMP]),[sa.MATHML]:new Set([ea.MI,ea.MO,ea.MN,ea.MS,ea.MTEXT,ea.ANNOTATION_XML]),[sa.SVG]:new Set([ea.TITLE,ea.FOREIGN_OBJECT,ea.DESC]),[sa.XLINK]:new Set,[sa.XML]:new Set,[sa.XMLNS]:new Set},Tv=new Set([ea.H1,ea.H2,ea.H3,ea.H4,ea.H5,ea.H6]),eZ=new Set([Ts.STYLE,Ts.SCRIPT,Ts.XMP,Ts.IFRAME,Ts.NOEMBED,Ts.NOFRAMES,Ts.PLAINTEXT]);function lZ(s,a){return eZ.has(s)||a&&s===Ts.NOSCRIPT}var ls;(function(s){s[s.DATA=0]="DATA",s[s.RCDATA=1]="RCDATA",s[s.RAWTEXT=2]="RAWTEXT",s[s.SCRIPT_DATA=3]="SCRIPT_DATA",s[s.PLAINTEXT=4]="PLAINTEXT",s[s.TAG_OPEN=5]="TAG_OPEN",s[s.END_TAG_OPEN=6]="END_TAG_OPEN",s[s.TAG_NAME=7]="TAG_NAME",s[s.RCDATA_LESS_THAN_SIGN=8]="RCDATA_LESS_THAN_SIGN",s[s.RCDATA_END_TAG_OPEN=9]="RCDATA_END_TAG_OPEN",s[s.RCDATA_END_TAG_NAME=10]="RCDATA_END_TAG_NAME",s[s.RAWTEXT_LESS_THAN_SIGN=11]="RAWTEXT_LESS_THAN_SIGN",s[s.RAWTEXT_END_TAG_OPEN=12]="RAWTEXT_END_TAG_OPEN",s[s.RAWTEXT_END_TAG_NAME=13]="RAWTEXT_END_TAG_NAME",s[s.SCRIPT_DATA_LESS_THAN_SIGN=14]="SCRIPT_DATA_LESS_THAN_SIGN",s[s.SCRIPT_DATA_END_TAG_OPEN=15]="SCRIPT_DATA_END_TAG_OPEN",s[s.SCRIPT_DATA_END_TAG_NAME=16]="SCRIPT_DATA_END_TAG_NAME",s[s.SCRIPT_DATA_ESCAPE_START=17]="SCRIPT_DATA_ESCAPE_START",s[s.SCRIPT_DATA_ESCAPE_START_DASH=18]="SCRIPT_DATA_ESCAPE_START_DASH",s[s.SCRIPT_DATA_ESCAPED=19]="SCRIPT_DATA_ESCAPED",s[s.SCRIPT_DATA_ESCAPED_DASH=20]="SCRIPT_DATA_ESCAPED_DASH",s[s.SCRIPT_DATA_ESCAPED_DASH_DASH=21]="SCRIPT_DATA_ESCAPED_DASH_DASH",s[s.SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN=22]="SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN",s[s.SCRIPT_DATA_ESCAPED_END_TAG_OPEN=23]="SCRIPT_DATA_ESCAPED_END_TAG_OPEN",s[s.SCRIPT_DATA_ESCAPED_END_TAG_NAME=24]="SCRIPT_DATA_ESCAPED_END_TAG_NAME",s[s.SCRIPT_DATA_DOUBLE_ESCAPE_START=25]="SCRIPT_DATA_DOUBLE_ESCAPE_START",s[s.SCRIPT_DATA_DOUBLE_ESCAPED=26]="SCRIPT_DATA_DOUBLE_ESCAPED",s[s.SCRIPT_DATA_DOUBLE_ESCAPED_DASH=27]="SCRIPT_DATA_DOUBLE_ESCAPED_DASH",s[s.SCRIPT_DATA_DOUBLE_ESCAPED_DASH_DASH=28]="SCRIPT_DATA_DOUBLE_ESCAPED_DASH_DASH",s[s.SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN=29]="SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN",s[s.SCRIPT_DATA_DOUBLE_ESCAPE_END=30]="SCRIPT_DATA_DOUBLE_ESCAPE_END",s[s.BEFORE_ATTRIBUTE_NAME=31]="BEFORE_ATTRIBUTE_NAME",s[s.ATTRIBUTE_NAME=32]="ATTRIBUTE_NAME",s[s.AFTER_ATTRIBUTE_NAME=33]="AFTER_ATTRIBUTE_NAME",s[s.BEFORE_ATTRIBUTE_VALUE=34]="BEFORE_ATTRIBUTE_VALUE",s[s.ATTRIBUTE_VALUE_DOUBLE_QUOTED=35]="ATTRIBUTE_VALUE_DOUBLE_QUOTED",s[s.ATTRIBUTE_VALUE_SINGLE_QUOTED=36]="ATTRIBUTE_VALUE_SINGLE_QUOTED",s[s.ATTRIBUTE_VALUE_UNQUOTED=37]="ATTRIBUTE_VALUE_UNQUOTED",s[s.AFTER_ATTRIBUTE_VALUE_QUOTED=38]="AFTER_ATTRIBUTE_VALUE_QUOTED",s[s.SELF_CLOSING_START_TAG=39]="SELF_CLOSING_START_TAG",s[s.BOGUS_COMMENT=40]="BOGUS_COMMENT",s[s.MARKUP_DECLARATION_OPEN=41]="MARKUP_DECLARATION_OPEN",s[s.COMMENT_START=42]="COMMENT_START",s[s.COMMENT_START_DASH=43]="COMMENT_START_DASH",s[s.COMMENT=44]="COMMENT",s[s.COMMENT_LESS_THAN_SIGN=45]="COMMENT_LESS_THAN_SIGN",s[s.COMMENT_LESS_THAN_SIGN_BANG=46]="COMMENT_LESS_THAN_SIGN_BANG",s[s.COMMENT_LESS_THAN_SIGN_BANG_DASH=47]="COMMENT_LESS_THAN_SIGN_BANG_DASH",s[s.COMMENT_LESS_THAN_SIGN_BANG_DASH_DASH=48]="COMMENT_LESS_THAN_SIGN_BANG_DASH_DASH",s[s.COMMENT_END_DASH=49]="COMMENT_END_DASH",s[s.COMMENT_END=50]="COMMENT_END",s[s.COMMENT_END_BANG=51]="COMMENT_END_BANG",s[s.DOCTYPE=52]="DOCTYPE",s[s.BEFORE_DOCTYPE_NAME=53]="BEFORE_DOCTYPE_NAME",s[s.DOCTYPE_NAME=54]="DOCTYPE_NAME",s[s.AFTER_DOCTYPE_NAME=55]="AFTER_DOCTYPE_NAME",s[s.AFTER_DOCTYPE_PUBLIC_KEYWORD=56]="AFTER_DOCTYPE_PUBLIC_KEYWORD",s[s.BEFORE_DOCTYPE_PUBLIC_IDENTIFIER=57]="BEFORE_DOCTYPE_PUBLIC_IDENTIFIER",s[s.DOCTYPE_PUBLIC_IDENTIFIER_DOUBLE_QUOTED=58]="DOCTYPE_PUBLIC_IDENTIFIER_DOUBLE_QUOTED",s[s.DOCTYPE_PUBLIC_IDENTIFIER_SINGLE_QUOTED=59]="DOCTYPE_PUBLIC_IDENTIFIER_SINGLE_QUOTED",s[s.AFTER_DOCTYPE_PUBLIC_IDENTIFIER=60]="AFTER_DOCTYPE_PUBLIC_IDENTIFIER",s[s.BETWEEN_DOCTYPE_PUBLIC_AND_SYSTEM_IDENTIFIERS=61]="BETWEEN_DOCTYPE_PUBLIC_AND_SYSTEM_IDENTIFIERS",s[s.AFTER_DOCTYPE_SYSTEM_KEYWORD=62]="AFTER_DOCTYPE_SYSTEM_KEYWORD",s[s.BEFORE_DOCTYPE_SYSTEM_IDENTIFIER=63]="BEFORE_DOCTYPE_SYSTEM_IDENTIFIER",s[s.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED=64]="DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED",s[s.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED=65]="DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED",s[s.AFTER_DOCTYPE_SYSTEM_IDENTIFIER=66]="AFTER_DOCTYPE_SYSTEM_IDENTIFIER",s[s.BOGUS_DOCTYPE=67]="BOGUS_DOCTYPE",s[s.CDATA_SECTION=68]="CDATA_SECTION",s[s.CDATA_SECTION_BRACKET=69]="CDATA_SECTION_BRACKET",s[s.CDATA_SECTION_END=70]="CDATA_SECTION_END",s[s.CHARACTER_REFERENCE=71]="CHARACTER_REFERENCE",s[s.AMBIGUOUS_AMPERSAND=72]="AMBIGUOUS_AMPERSAND"})(ls||(ls={}));const ce={DATA:ls.DATA,RCDATA:ls.RCDATA,RAWTEXT:ls.RAWTEXT,SCRIPT_DATA:ls.SCRIPT_DATA,PLAINTEXT:ls.PLAINTEXT,CDATA_SECTION:ls.CDATA_SECTION};function iZ(s){return s>=ts.DIGIT_0&&s<=ts.DIGIT_9}function Fo(s){return s>=ts.LATIN_CAPITAL_A&&s<=ts.LATIN_CAPITAL_Z}function pZ(s){return s>=ts.LATIN_SMALL_A&&s<=ts.LATIN_SMALL_Z}function Yi(s){return pZ(s)||Fo(s)}function s9(s){return Yi(s)||iZ(s)}function y0(s){return s+32}function yM(s){return s===ts.SPACE||s===ts.LINE_FEED||s===ts.TABULATION||s===ts.FORM_FEED}function a9(s){return yM(s)||s===ts.SOLIDUS||s===ts.GREATER_THAN_SIGN}function rZ(s){return s===ts.NULL?Us.nullCharacterReference:s>1114111?Us.characterReferenceOutsideUnicodeRange:uM(s)?Us.surrogateCharacterReference:gM(s)?Us.noncharacterCharacterReference:dM(s)||s===ts.CARRIAGE_RETURN?Us.controlCharacterReference:null}let mZ=class{constructor(a,t){this.options=a,this.handler=t,this.paused=!1,this.inLoop=!1,this.inForeignNode=!1,this.lastStartTagName="",this.active=!1,this.state=ls.DATA,this.returnState=ls.DATA,this.entityStartPos=0,this.consumedAfterSnapshot=-1,this.currentCharacterToken=null,this.currentToken=null,this.currentAttr={name:"",value:""},this.preprocessor=new GK(t),this.currentLocation=this.getCurrentLocation(-1),this.entityDecoder=new sZ(WK,(n,e)=>{this.preprocessor.pos=this.entityStartPos+e-1,this._flushCodePointConsumedAsCharacterReference(n)},t.onParseError?{missingSemicolonAfterCharacterReference:()=>{this._err(Us.missingSemicolonAfterCharacterReference,1)},absenceOfDigitsInNumericCharacterReference:n=>{this._err(Us.absenceOfDigitsInNumericCharacterReference,this.entityStartPos-this.preprocessor.pos+n)},validateNumericCharacterReference:n=>{const e=rZ(n);e&&this._err(e,1)}}:void 0)}_err(a,t=0){var n,e;(e=(n=this.handler).onParseError)===null||e===void 0||e.call(n,this.preprocessor.getError(a,t))}getCurrentLocation(a){return this.options.sourceCodeLocationInfo?{startLine:this.preprocessor.line,startCol:this.preprocessor.col-a,startOffset:this.preprocessor.offset-a,endLine:-1,endCol:-1,endOffset:-1}:null}_runParsingLoop(){if(!this.inLoop){for(this.inLoop=!0;this.active&&!this.paused;){this.consumedAfterSnapshot=0;const a=this._consume();this._ensureHibernation()||this._callState(a)}this.inLoop=!1}}pause(){this.paused=!0}resume(a){if(!this.paused)throw new Error("Parser was already resumed");this.paused=!1,!this.inLoop&&(this._runParsingLoop(),this.paused||a?.())}write(a,t,n){this.active=!0,this.preprocessor.write(a,t),this._runParsingLoop(),this.paused||n?.()}insertHtmlAtCurrentPos(a){this.active=!0,this.preprocessor.insertHtmlAtCurrentPos(a),this._runParsingLoop()}_ensureHibernation(){return this.preprocessor.endOfChunkHit?(this.preprocessor.retreat(this.consumedAfterSnapshot),this.consumedAfterSnapshot=0,this.active=!1,!0):!1}_consume(){return this.consumedAfterSnapshot++,this.preprocessor.advance()}_advanceBy(a){this.consumedAfterSnapshot+=a;for(let t=0;t<a;t++)this.preprocessor.advance()}_consumeSequenceIfMatch(a,t){return this.preprocessor.startsWith(a,t)?(this._advanceBy(a.length-1),!0):!1}_createStartTagToken(){this.currentToken={type:nt.START_TAG,tagName:"",tagID:k.UNKNOWN,selfClosing:!1,ackSelfClosing:!1,attrs:[],location:this.getCurrentLocation(1)}}_createEndTagToken(){this.currentToken={type:nt.END_TAG,tagName:"",tagID:k.UNKNOWN,selfClosing:!1,ackSelfClosing:!1,attrs:[],location:this.getCurrentLocation(2)}}_createCommentToken(a){this.currentToken={type:nt.COMMENT,data:"",location:this.getCurrentLocation(a)}}_createDoctypeToken(a){this.currentToken={type:nt.DOCTYPE,name:a,forceQuirks:!1,publicId:null,systemId:null,location:this.currentLocation}}_createCharacterToken(a,t){this.currentCharacterToken={type:a,chars:t,location:this.currentLocation}}_createAttr(a){this.currentAttr={name:a,value:""},this.currentLocation=this.getCurrentLocation(0)}_leaveAttrName(){var a,t;const n=this.currentToken;if(fM(n,this.currentAttr.name)===null){if(n.attrs.push(this.currentAttr),n.location&&this.currentLocation){const e=(a=(t=n.location).attrs)!==null&&a!==void 0?a:t.attrs=Object.create(null);e[this.currentAttr.name]=this.currentLocation,this._leaveAttrValue()}}else this._err(Us.duplicateAttribute)}_leaveAttrValue(){this.currentLocation&&(this.currentLocation.endLine=this.preprocessor.line,this.currentLocation.endCol=this.preprocessor.col,this.currentLocation.endOffset=this.preprocessor.offset)}prepareToken(a){this._emitCurrentCharacterToken(a.location),this.currentToken=null,a.location&&(a.location.endLine=this.preprocessor.line,a.location.endCol=this.preprocessor.col+1,a.location.endOffset=this.preprocessor.offset+1),this.currentLocation=this.getCurrentLocation(-1)}emitCurrentTagToken(){const a=this.currentToken;this.prepareToken(a),a.tagID=jd(a.tagName),a.type===nt.START_TAG?(this.lastStartTagName=a.tagName,this.handler.onStartTag(a)):(a.attrs.length>0&&this._err(Us.endTagWithAttributes),a.selfClosing&&this._err(Us.endTagWithTrailingSolidus),this.handler.onEndTag(a)),this.preprocessor.dropParsedChunk()}emitCurrentComment(a){this.prepareToken(a),this.handler.onComment(a),this.preprocessor.dropParsedChunk()}emitCurrentDoctype(a){this.prepareToken(a),this.handler.onDoctype(a),this.preprocessor.dropParsedChunk()}_emitCurrentCharacterToken(a){if(this.currentCharacterToken){switch(a&&this.currentCharacterToken.location&&(this.currentCharacterToken.location.endLine=a.startLine,this.currentCharacterToken.location.endCol=a.startCol,this.currentCharacterToken.location.endOffset=a.startOffset),this.currentCharacterToken.type){case nt.CHARACTER:{this.handler.onCharacter(this.currentCharacterToken);break}case nt.NULL_CHARACTER:{this.handler.onNullCharacter(this.currentCharacterToken);break}case nt.WHITESPACE_CHARACTER:{this.handler.onWhitespaceCharacter(this.currentCharacterToken);break}}this.currentCharacterToken=null}}_emitEOFToken(){const a=this.getCurrentLocation(0);a&&(a.endLine=a.startLine,a.endCol=a.startCol,a.endOffset=a.startOffset),this._emitCurrentCharacterToken(a),this.handler.onEof({type:nt.EOF,location:a}),this.active=!1}_appendCharToCurrentCharacterToken(a,t){if(this.currentCharacterToken)if(this.currentCharacterToken.type===a){this.currentCharacterToken.chars+=t;return}else this.currentLocation=this.getCurrentLocation(0),this._emitCurrentCharacterToken(this.currentLocation),this.preprocessor.dropParsedChunk();this._createCharacterToken(a,t)}_emitCodePoint(a){const t=yM(a)?nt.WHITESPACE_CHARACTER:a===ts.NULL?nt.NULL_CHARACTER:nt.CHARACTER;this._appendCharToCurrentCharacterToken(t,String.fromCodePoint(a))}_emitChars(a){this._appendCharToCurrentCharacterToken(nt.CHARACTER,a)}_startCharacterReference(){this.returnState=this.state,this.state=ls.CHARACTER_REFERENCE,this.entityStartPos=this.preprocessor.pos,this.entityDecoder.startEntity(this._isCharacterReferenceInAttribute()?gi.Attribute:gi.Legacy)}_isCharacterReferenceInAttribute(){return this.returnState===ls.ATTRIBUTE_VALUE_DOUBLE_QUOTED||this.returnState===ls.ATTRIBUTE_VALUE_SINGLE_QUOTED||this.returnState===ls.ATTRIBUTE_VALUE_UNQUOTED}_flushCodePointConsumedAsCharacterReference(a){this._isCharacterReferenceInAttribute()?this.currentAttr.value+=String.fromCodePoint(a):this._emitCodePoint(a)}_callState(a){switch(this.state){case ls.DATA:{this._stateData(a);break}case ls.RCDATA:{this._stateRcdata(a);break}case ls.RAWTEXT:{this._stateRawtext(a);break}case ls.SCRIPT_DATA:{this._stateScriptData(a);break}case ls.PLAINTEXT:{this._statePlaintext(a);break}case ls.TAG_OPEN:{this._stateTagOpen(a);break}case ls.END_TAG_OPEN:{this._stateEndTagOpen(a);break}case ls.TAG_NAME:{this._stateTagName(a);break}case ls.RCDATA_LESS_THAN_SIGN:{this._stateRcdataLessThanSign(a);break}case ls.RCDATA_END_TAG_OPEN:{this._stateRcdataEndTagOpen(a);break}case ls.RCDATA_END_TAG_NAME:{this._stateRcdataEndTagName(a);break}case ls.RAWTEXT_LESS_THAN_SIGN:{this._stateRawtextLessThanSign(a);break}case ls.RAWTEXT_END_TAG_OPEN:{this._stateRawtextEndTagOpen(a);break}case ls.RAWTEXT_END_TAG_NAME:{this._stateRawtextEndTagName(a);break}case ls.SCRIPT_DATA_LESS_THAN_SIGN:{this._stateScriptDataLessThanSign(a);break}case ls.SCRIPT_DATA_END_TAG_OPEN:{this._stateScriptDataEndTagOpen(a);break}case ls.SCRIPT_DATA_END_TAG_NAME:{this._stateScriptDataEndTagName(a);break}case ls.SCRIPT_DATA_ESCAPE_START:{this._stateScriptDataEscapeStart(a);break}case ls.SCRIPT_DATA_ESCAPE_START_DASH:{this._stateScriptDataEscapeStartDash(a);break}case ls.SCRIPT_DATA_ESCAPED:{this._stateScriptDataEscaped(a);break}case ls.SCRIPT_DATA_ESCAPED_DASH:{this._stateScriptDataEscapedDash(a);break}case ls.SCRIPT_DATA_ESCAPED_DASH_DASH:{this._stateScriptDataEscapedDashDash(a);break}case ls.SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN:{this._stateScriptDataEscapedLessThanSign(a);break}case ls.SCRIPT_DATA_ESCAPED_END_TAG_OPEN:{this._stateScriptDataEscapedEndTagOpen(a);break}case ls.SCRIPT_DATA_ESCAPED_END_TAG_NAME:{this._stateScriptDataEscapedEndTagName(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPE_START:{this._stateScriptDataDoubleEscapeStart(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPED:{this._stateScriptDataDoubleEscaped(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPED_DASH:{this._stateScriptDataDoubleEscapedDash(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPED_DASH_DASH:{this._stateScriptDataDoubleEscapedDashDash(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN:{this._stateScriptDataDoubleEscapedLessThanSign(a);break}case ls.SCRIPT_DATA_DOUBLE_ESCAPE_END:{this._stateScriptDataDoubleEscapeEnd(a);break}case ls.BEFORE_ATTRIBUTE_NAME:{this._stateBeforeAttributeName(a);break}case ls.ATTRIBUTE_NAME:{this._stateAttributeName(a);break}case ls.AFTER_ATTRIBUTE_NAME:{this._stateAfterAttributeName(a);break}case ls.BEFORE_ATTRIBUTE_VALUE:{this._stateBeforeAttributeValue(a);break}case ls.ATTRIBUTE_VALUE_DOUBLE_QUOTED:{this._stateAttributeValueDoubleQuoted(a);break}case ls.ATTRIBUTE_VALUE_SINGLE_QUOTED:{this._stateAttributeValueSingleQuoted(a);break}case ls.ATTRIBUTE_VALUE_UNQUOTED:{this._stateAttributeValueUnquoted(a);break}case ls.AFTER_ATTRIBUTE_VALUE_QUOTED:{this._stateAfterAttributeValueQuoted(a);break}case ls.SELF_CLOSING_START_TAG:{this._stateSelfClosingStartTag(a);break}case ls.BOGUS_COMMENT:{this._stateBogusComment(a);break}case ls.MARKUP_DECLARATION_OPEN:{this._stateMarkupDeclarationOpen(a);break}case ls.COMMENT_START:{this._stateCommentStart(a);break}case ls.COMMENT_START_DASH:{this._stateCommentStartDash(a);break}case ls.COMMENT:{this._stateComment(a);break}case ls.COMMENT_LESS_THAN_SIGN:{this._stateCommentLessThanSign(a);break}case ls.COMMENT_LESS_THAN_SIGN_BANG:{this._stateCommentLessThanSignBang(a);break}case ls.COMMENT_LESS_THAN_SIGN_BANG_DASH:{this._stateCommentLessThanSignBangDash(a);break}case ls.COMMENT_LESS_THAN_SIGN_BANG_DASH_DASH:{this._stateCommentLessThanSignBangDashDash(a);break}case ls.COMMENT_END_DASH:{this._stateCommentEndDash(a);break}case ls.COMMENT_END:{this._stateCommentEnd(a);break}case ls.COMMENT_END_BANG:{this._stateCommentEndBang(a);break}case ls.DOCTYPE:{this._stateDoctype(a);break}case ls.BEFORE_DOCTYPE_NAME:{this._stateBeforeDoctypeName(a);break}case ls.DOCTYPE_NAME:{this._stateDoctypeName(a);break}case ls.AFTER_DOCTYPE_NAME:{this._stateAfterDoctypeName(a);break}case ls.AFTER_DOCTYPE_PUBLIC_KEYWORD:{this._stateAfterDoctypePublicKeyword(a);break}case ls.BEFORE_DOCTYPE_PUBLIC_IDENTIFIER:{this._stateBeforeDoctypePublicIdentifier(a);break}case ls.DOCTYPE_PUBLIC_IDENTIFIER_DOUBLE_QUOTED:{this._stateDoctypePublicIdentifierDoubleQuoted(a);break}case ls.DOCTYPE_PUBLIC_IDENTIFIER_SINGLE_QUOTED:{this._stateDoctypePublicIdentifierSingleQuoted(a);break}case ls.AFTER_DOCTYPE_PUBLIC_IDENTIFIER:{this._stateAfterDoctypePublicIdentifier(a);break}case ls.BETWEEN_DOCTYPE_PUBLIC_AND_SYSTEM_IDENTIFIERS:{this._stateBetweenDoctypePublicAndSystemIdentifiers(a);break}case ls.AFTER_DOCTYPE_SYSTEM_KEYWORD:{this._stateAfterDoctypeSystemKeyword(a);break}case ls.BEFORE_DOCTYPE_SYSTEM_IDENTIFIER:{this._stateBeforeDoctypeSystemIdentifier(a);break}case ls.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED:{this._stateDoctypeSystemIdentifierDoubleQuoted(a);break}case ls.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED:{this._stateDoctypeSystemIdentifierSingleQuoted(a);break}case ls.AFTER_DOCTYPE_SYSTEM_IDENTIFIER:{this._stateAfterDoctypeSystemIdentifier(a);break}case ls.BOGUS_DOCTYPE:{this._stateBogusDoctype(a);break}case ls.CDATA_SECTION:{this._stateCdataSection(a);break}case ls.CDATA_SECTION_BRACKET:{this._stateCdataSectionBracket(a);break}case ls.CDATA_SECTION_END:{this._stateCdataSectionEnd(a);break}case ls.CHARACTER_REFERENCE:{this._stateCharacterReference();break}case ls.AMBIGUOUS_AMPERSAND:{this._stateAmbiguousAmpersand(a);break}default:throw new Error("Unknown state")}}_stateData(a){switch(a){case ts.LESS_THAN_SIGN:{this.state=ls.TAG_OPEN;break}case ts.AMPERSAND:{this._startCharacterReference();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitCodePoint(a);break}case ts.EOF:{this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateRcdata(a){switch(a){case ts.AMPERSAND:{this._startCharacterReference();break}case ts.LESS_THAN_SIGN:{this.state=ls.RCDATA_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateRawtext(a){switch(a){case ts.LESS_THAN_SIGN:{this.state=ls.RAWTEXT_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateScriptData(a){switch(a){case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._emitEOFToken();break}default:this._emitCodePoint(a)}}_statePlaintext(a){switch(a){case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateTagOpen(a){if(Yi(a))this._createStartTagToken(),this.state=ls.TAG_NAME,this._stateTagName(a);else switch(a){case ts.EXCLAMATION_MARK:{this.state=ls.MARKUP_DECLARATION_OPEN;break}case ts.SOLIDUS:{this.state=ls.END_TAG_OPEN;break}case ts.QUESTION_MARK:{this._err(Us.unexpectedQuestionMarkInsteadOfTagName),this._createCommentToken(1),this.state=ls.BOGUS_COMMENT,this._stateBogusComment(a);break}case ts.EOF:{this._err(Us.eofBeforeTagName),this._emitChars("<"),this._emitEOFToken();break}default:this._err(Us.invalidFirstCharacterOfTagName),this._emitChars("<"),this.state=ls.DATA,this._stateData(a)}}_stateEndTagOpen(a){if(Yi(a))this._createEndTagToken(),this.state=ls.TAG_NAME,this._stateTagName(a);else switch(a){case ts.GREATER_THAN_SIGN:{this._err(Us.missingEndTagName),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofBeforeTagName),this._emitChars("</"),this._emitEOFToken();break}default:this._err(Us.invalidFirstCharacterOfTagName),this._createCommentToken(2),this.state=ls.BOGUS_COMMENT,this._stateBogusComment(a)}}_stateTagName(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.BEFORE_ATTRIBUTE_NAME;break}case ts.SOLIDUS:{this.state=ls.SELF_CLOSING_START_TAG;break}case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentTagToken();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.tagName+=Ft;break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:t.tagName+=String.fromCodePoint(Fo(a)?y0(a):a)}}_stateRcdataLessThanSign(a){a===ts.SOLIDUS?this.state=ls.RCDATA_END_TAG_OPEN:(this._emitChars("<"),this.state=ls.RCDATA,this._stateRcdata(a))}_stateRcdataEndTagOpen(a){Yi(a)?(this.state=ls.RCDATA_END_TAG_NAME,this._stateRcdataEndTagName(a)):(this._emitChars("</"),this.state=ls.RCDATA,this._stateRcdata(a))}handleSpecialEndTag(a){if(!this.preprocessor.startsWith(this.lastStartTagName,!1))return!this._ensureHibernation();this._createEndTagToken();const t=this.currentToken;switch(t.tagName=this.lastStartTagName,this.preprocessor.peek(this.lastStartTagName.length)){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:return this._advanceBy(this.lastStartTagName.length),this.state=ls.BEFORE_ATTRIBUTE_NAME,!1;case ts.SOLIDUS:return this._advanceBy(this.lastStartTagName.length),this.state=ls.SELF_CLOSING_START_TAG,!1;case ts.GREATER_THAN_SIGN:return this._advanceBy(this.lastStartTagName.length),this.emitCurrentTagToken(),this.state=ls.DATA,!1;default:return!this._ensureHibernation()}}_stateRcdataEndTagName(a){this.handleSpecialEndTag(a)&&(this._emitChars("</"),this.state=ls.RCDATA,this._stateRcdata(a))}_stateRawtextLessThanSign(a){a===ts.SOLIDUS?this.state=ls.RAWTEXT_END_TAG_OPEN:(this._emitChars("<"),this.state=ls.RAWTEXT,this._stateRawtext(a))}_stateRawtextEndTagOpen(a){Yi(a)?(this.state=ls.RAWTEXT_END_TAG_NAME,this._stateRawtextEndTagName(a)):(this._emitChars("</"),this.state=ls.RAWTEXT,this._stateRawtext(a))}_stateRawtextEndTagName(a){this.handleSpecialEndTag(a)&&(this._emitChars("</"),this.state=ls.RAWTEXT,this._stateRawtext(a))}_stateScriptDataLessThanSign(a){switch(a){case ts.SOLIDUS:{this.state=ls.SCRIPT_DATA_END_TAG_OPEN;break}case ts.EXCLAMATION_MARK:{this.state=ls.SCRIPT_DATA_ESCAPE_START,this._emitChars("<!");break}default:this._emitChars("<"),this.state=ls.SCRIPT_DATA,this._stateScriptData(a)}}_stateScriptDataEndTagOpen(a){Yi(a)?(this.state=ls.SCRIPT_DATA_END_TAG_NAME,this._stateScriptDataEndTagName(a)):(this._emitChars("</"),this.state=ls.SCRIPT_DATA,this._stateScriptData(a))}_stateScriptDataEndTagName(a){this.handleSpecialEndTag(a)&&(this._emitChars("</"),this.state=ls.SCRIPT_DATA,this._stateScriptData(a))}_stateScriptDataEscapeStart(a){a===ts.HYPHEN_MINUS?(this.state=ls.SCRIPT_DATA_ESCAPE_START_DASH,this._emitChars("-")):(this.state=ls.SCRIPT_DATA,this._stateScriptData(a))}_stateScriptDataEscapeStartDash(a){a===ts.HYPHEN_MINUS?(this.state=ls.SCRIPT_DATA_ESCAPED_DASH_DASH,this._emitChars("-")):(this.state=ls.SCRIPT_DATA,this._stateScriptData(a))}_stateScriptDataEscaped(a){switch(a){case ts.HYPHEN_MINUS:{this.state=ls.SCRIPT_DATA_ESCAPED_DASH,this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateScriptDataEscapedDash(a){switch(a){case ts.HYPHEN_MINUS:{this.state=ls.SCRIPT_DATA_ESCAPED_DASH_DASH,this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.state=ls.SCRIPT_DATA_ESCAPED,this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this.state=ls.SCRIPT_DATA_ESCAPED,this._emitCodePoint(a)}}_stateScriptDataEscapedDashDash(a){switch(a){case ts.HYPHEN_MINUS:{this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_ESCAPED_LESS_THAN_SIGN;break}case ts.GREATER_THAN_SIGN:{this.state=ls.SCRIPT_DATA,this._emitChars(">");break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.state=ls.SCRIPT_DATA_ESCAPED,this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this.state=ls.SCRIPT_DATA_ESCAPED,this._emitCodePoint(a)}}_stateScriptDataEscapedLessThanSign(a){a===ts.SOLIDUS?this.state=ls.SCRIPT_DATA_ESCAPED_END_TAG_OPEN:Yi(a)?(this._emitChars("<"),this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPE_START,this._stateScriptDataDoubleEscapeStart(a)):(this._emitChars("<"),this.state=ls.SCRIPT_DATA_ESCAPED,this._stateScriptDataEscaped(a))}_stateScriptDataEscapedEndTagOpen(a){Yi(a)?(this.state=ls.SCRIPT_DATA_ESCAPED_END_TAG_NAME,this._stateScriptDataEscapedEndTagName(a)):(this._emitChars("</"),this.state=ls.SCRIPT_DATA_ESCAPED,this._stateScriptDataEscaped(a))}_stateScriptDataEscapedEndTagName(a){this.handleSpecialEndTag(a)&&(this._emitChars("</"),this.state=ls.SCRIPT_DATA_ESCAPED,this._stateScriptDataEscaped(a))}_stateScriptDataDoubleEscapeStart(a){if(this.preprocessor.startsWith(pe.SCRIPT,!1)&&a9(this.preprocessor.peek(pe.SCRIPT.length))){this._emitCodePoint(a);for(let t=0;t<pe.SCRIPT.length;t++)this._emitCodePoint(this._consume());this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED}else this._ensureHibernation()||(this.state=ls.SCRIPT_DATA_ESCAPED,this._stateScriptDataEscaped(a))}_stateScriptDataDoubleEscaped(a){switch(a){case ts.HYPHEN_MINUS:{this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED_DASH,this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN,this._emitChars("<");break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateScriptDataDoubleEscapedDash(a){switch(a){case ts.HYPHEN_MINUS:{this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED_DASH_DASH,this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN,this._emitChars("<");break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._emitCodePoint(a)}}_stateScriptDataDoubleEscapedDashDash(a){switch(a){case ts.HYPHEN_MINUS:{this._emitChars("-");break}case ts.LESS_THAN_SIGN:{this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED_LESS_THAN_SIGN,this._emitChars("<");break}case ts.GREATER_THAN_SIGN:{this.state=ls.SCRIPT_DATA,this._emitChars(">");break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._emitChars(Ft);break}case ts.EOF:{this._err(Us.eofInScriptHtmlCommentLikeText),this._emitEOFToken();break}default:this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._emitCodePoint(a)}}_stateScriptDataDoubleEscapedLessThanSign(a){a===ts.SOLIDUS?(this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPE_END,this._emitChars("/")):(this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._stateScriptDataDoubleEscaped(a))}_stateScriptDataDoubleEscapeEnd(a){if(this.preprocessor.startsWith(pe.SCRIPT,!1)&&a9(this.preprocessor.peek(pe.SCRIPT.length))){this._emitCodePoint(a);for(let t=0;t<pe.SCRIPT.length;t++)this._emitCodePoint(this._consume());this.state=ls.SCRIPT_DATA_ESCAPED}else this._ensureHibernation()||(this.state=ls.SCRIPT_DATA_DOUBLE_ESCAPED,this._stateScriptDataDoubleEscaped(a))}_stateBeforeAttributeName(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.SOLIDUS:case ts.GREATER_THAN_SIGN:case ts.EOF:{this.state=ls.AFTER_ATTRIBUTE_NAME,this._stateAfterAttributeName(a);break}case ts.EQUALS_SIGN:{this._err(Us.unexpectedEqualsSignBeforeAttributeName),this._createAttr("="),this.state=ls.ATTRIBUTE_NAME;break}default:this._createAttr(""),this.state=ls.ATTRIBUTE_NAME,this._stateAttributeName(a)}}_stateAttributeName(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:case ts.SOLIDUS:case ts.GREATER_THAN_SIGN:case ts.EOF:{this._leaveAttrName(),this.state=ls.AFTER_ATTRIBUTE_NAME,this._stateAfterAttributeName(a);break}case ts.EQUALS_SIGN:{this._leaveAttrName(),this.state=ls.BEFORE_ATTRIBUTE_VALUE;break}case ts.QUOTATION_MARK:case ts.APOSTROPHE:case ts.LESS_THAN_SIGN:{this._err(Us.unexpectedCharacterInAttributeName),this.currentAttr.name+=String.fromCodePoint(a);break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.currentAttr.name+=Ft;break}default:this.currentAttr.name+=String.fromCodePoint(Fo(a)?y0(a):a)}}_stateAfterAttributeName(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.SOLIDUS:{this.state=ls.SELF_CLOSING_START_TAG;break}case ts.EQUALS_SIGN:{this.state=ls.BEFORE_ATTRIBUTE_VALUE;break}case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentTagToken();break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this._createAttr(""),this.state=ls.ATTRIBUTE_NAME,this._stateAttributeName(a)}}_stateBeforeAttributeValue(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.QUOTATION_MARK:{this.state=ls.ATTRIBUTE_VALUE_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{this.state=ls.ATTRIBUTE_VALUE_SINGLE_QUOTED;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingAttributeValue),this.state=ls.DATA,this.emitCurrentTagToken();break}default:this.state=ls.ATTRIBUTE_VALUE_UNQUOTED,this._stateAttributeValueUnquoted(a)}}_stateAttributeValueDoubleQuoted(a){switch(a){case ts.QUOTATION_MARK:{this.state=ls.AFTER_ATTRIBUTE_VALUE_QUOTED;break}case ts.AMPERSAND:{this._startCharacterReference();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.currentAttr.value+=Ft;break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this.currentAttr.value+=String.fromCodePoint(a)}}_stateAttributeValueSingleQuoted(a){switch(a){case ts.APOSTROPHE:{this.state=ls.AFTER_ATTRIBUTE_VALUE_QUOTED;break}case ts.AMPERSAND:{this._startCharacterReference();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.currentAttr.value+=Ft;break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this.currentAttr.value+=String.fromCodePoint(a)}}_stateAttributeValueUnquoted(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this._leaveAttrValue(),this.state=ls.BEFORE_ATTRIBUTE_NAME;break}case ts.AMPERSAND:{this._startCharacterReference();break}case ts.GREATER_THAN_SIGN:{this._leaveAttrValue(),this.state=ls.DATA,this.emitCurrentTagToken();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),this.currentAttr.value+=Ft;break}case ts.QUOTATION_MARK:case ts.APOSTROPHE:case ts.LESS_THAN_SIGN:case ts.EQUALS_SIGN:case ts.GRAVE_ACCENT:{this._err(Us.unexpectedCharacterInUnquotedAttributeValue),this.currentAttr.value+=String.fromCodePoint(a);break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this.currentAttr.value+=String.fromCodePoint(a)}}_stateAfterAttributeValueQuoted(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this._leaveAttrValue(),this.state=ls.BEFORE_ATTRIBUTE_NAME;break}case ts.SOLIDUS:{this._leaveAttrValue(),this.state=ls.SELF_CLOSING_START_TAG;break}case ts.GREATER_THAN_SIGN:{this._leaveAttrValue(),this.state=ls.DATA,this.emitCurrentTagToken();break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this._err(Us.missingWhitespaceBetweenAttributes),this.state=ls.BEFORE_ATTRIBUTE_NAME,this._stateBeforeAttributeName(a)}}_stateSelfClosingStartTag(a){switch(a){case ts.GREATER_THAN_SIGN:{const t=this.currentToken;t.selfClosing=!0,this.state=ls.DATA,this.emitCurrentTagToken();break}case ts.EOF:{this._err(Us.eofInTag),this._emitEOFToken();break}default:this._err(Us.unexpectedSolidusInTag),this.state=ls.BEFORE_ATTRIBUTE_NAME,this._stateBeforeAttributeName(a)}}_stateBogusComment(a){const t=this.currentToken;switch(a){case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentComment(t);break}case ts.EOF:{this.emitCurrentComment(t),this._emitEOFToken();break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.data+=Ft;break}default:t.data+=String.fromCodePoint(a)}}_stateMarkupDeclarationOpen(a){this._consumeSequenceIfMatch(pe.DASH_DASH,!0)?(this._createCommentToken(pe.DASH_DASH.length+1),this.state=ls.COMMENT_START):this._consumeSequenceIfMatch(pe.DOCTYPE,!1)?(this.currentLocation=this.getCurrentLocation(pe.DOCTYPE.length+1),this.state=ls.DOCTYPE):this._consumeSequenceIfMatch(pe.CDATA_START,!0)?this.inForeignNode?this.state=ls.CDATA_SECTION:(this._err(Us.cdataInHtmlContent),this._createCommentToken(pe.CDATA_START.length+1),this.currentToken.data="[CDATA[",this.state=ls.BOGUS_COMMENT):this._ensureHibernation()||(this._err(Us.incorrectlyOpenedComment),this._createCommentToken(2),this.state=ls.BOGUS_COMMENT,this._stateBogusComment(a))}_stateCommentStart(a){switch(a){case ts.HYPHEN_MINUS:{this.state=ls.COMMENT_START_DASH;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptClosingOfEmptyComment),this.state=ls.DATA;const t=this.currentToken;this.emitCurrentComment(t);break}default:this.state=ls.COMMENT,this._stateComment(a)}}_stateCommentStartDash(a){const t=this.currentToken;switch(a){case ts.HYPHEN_MINUS:{this.state=ls.COMMENT_END;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptClosingOfEmptyComment),this.state=ls.DATA,this.emitCurrentComment(t);break}case ts.EOF:{this._err(Us.eofInComment),this.emitCurrentComment(t),this._emitEOFToken();break}default:t.data+="-",this.state=ls.COMMENT,this._stateComment(a)}}_stateComment(a){const t=this.currentToken;switch(a){case ts.HYPHEN_MINUS:{this.state=ls.COMMENT_END_DASH;break}case ts.LESS_THAN_SIGN:{t.data+="<",this.state=ls.COMMENT_LESS_THAN_SIGN;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.data+=Ft;break}case ts.EOF:{this._err(Us.eofInComment),this.emitCurrentComment(t),this._emitEOFToken();break}default:t.data+=String.fromCodePoint(a)}}_stateCommentLessThanSign(a){const t=this.currentToken;switch(a){case ts.EXCLAMATION_MARK:{t.data+="!",this.state=ls.COMMENT_LESS_THAN_SIGN_BANG;break}case ts.LESS_THAN_SIGN:{t.data+="<";break}default:this.state=ls.COMMENT,this._stateComment(a)}}_stateCommentLessThanSignBang(a){a===ts.HYPHEN_MINUS?this.state=ls.COMMENT_LESS_THAN_SIGN_BANG_DASH:(this.state=ls.COMMENT,this._stateComment(a))}_stateCommentLessThanSignBangDash(a){a===ts.HYPHEN_MINUS?this.state=ls.COMMENT_LESS_THAN_SIGN_BANG_DASH_DASH:(this.state=ls.COMMENT_END_DASH,this._stateCommentEndDash(a))}_stateCommentLessThanSignBangDashDash(a){a!==ts.GREATER_THAN_SIGN&&a!==ts.EOF&&this._err(Us.nestedComment),this.state=ls.COMMENT_END,this._stateCommentEnd(a)}_stateCommentEndDash(a){const t=this.currentToken;switch(a){case ts.HYPHEN_MINUS:{this.state=ls.COMMENT_END;break}case ts.EOF:{this._err(Us.eofInComment),this.emitCurrentComment(t),this._emitEOFToken();break}default:t.data+="-",this.state=ls.COMMENT,this._stateComment(a)}}_stateCommentEnd(a){const t=this.currentToken;switch(a){case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentComment(t);break}case ts.EXCLAMATION_MARK:{this.state=ls.COMMENT_END_BANG;break}case ts.HYPHEN_MINUS:{t.data+="-";break}case ts.EOF:{this._err(Us.eofInComment),this.emitCurrentComment(t),this._emitEOFToken();break}default:t.data+="--",this.state=ls.COMMENT,this._stateComment(a)}}_stateCommentEndBang(a){const t=this.currentToken;switch(a){case ts.HYPHEN_MINUS:{t.data+="--!",this.state=ls.COMMENT_END_DASH;break}case ts.GREATER_THAN_SIGN:{this._err(Us.incorrectlyClosedComment),this.state=ls.DATA,this.emitCurrentComment(t);break}case ts.EOF:{this._err(Us.eofInComment),this.emitCurrentComment(t),this._emitEOFToken();break}default:t.data+="--!",this.state=ls.COMMENT,this._stateComment(a)}}_stateDoctype(a){switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.BEFORE_DOCTYPE_NAME;break}case ts.GREATER_THAN_SIGN:{this.state=ls.BEFORE_DOCTYPE_NAME,this._stateBeforeDoctypeName(a);break}case ts.EOF:{this._err(Us.eofInDoctype),this._createDoctypeToken(null);const t=this.currentToken;t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingWhitespaceBeforeDoctypeName),this.state=ls.BEFORE_DOCTYPE_NAME,this._stateBeforeDoctypeName(a)}}_stateBeforeDoctypeName(a){if(Fo(a))this._createDoctypeToken(String.fromCharCode(y0(a))),this.state=ls.DOCTYPE_NAME;else switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.NULL:{this._err(Us.unexpectedNullCharacter),this._createDoctypeToken(Ft),this.state=ls.DOCTYPE_NAME;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingDoctypeName),this._createDoctypeToken(null);const t=this.currentToken;t.forceQuirks=!0,this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),this._createDoctypeToken(null);const t=this.currentToken;t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._createDoctypeToken(String.fromCodePoint(a)),this.state=ls.DOCTYPE_NAME}}_stateDoctypeName(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.AFTER_DOCTYPE_NAME;break}case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.name+=Ft;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:t.name+=String.fromCodePoint(Fo(a)?y0(a):a)}}_stateAfterDoctypeName(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._consumeSequenceIfMatch(pe.PUBLIC,!1)?this.state=ls.AFTER_DOCTYPE_PUBLIC_KEYWORD:this._consumeSequenceIfMatch(pe.SYSTEM,!1)?this.state=ls.AFTER_DOCTYPE_SYSTEM_KEYWORD:this._ensureHibernation()||(this._err(Us.invalidCharacterSequenceAfterDoctypeName),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a))}}_stateAfterDoctypePublicKeyword(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.BEFORE_DOCTYPE_PUBLIC_IDENTIFIER;break}case ts.QUOTATION_MARK:{this._err(Us.missingWhitespaceAfterDoctypePublicKeyword),t.publicId="",this.state=ls.DOCTYPE_PUBLIC_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{this._err(Us.missingWhitespaceAfterDoctypePublicKeyword),t.publicId="",this.state=ls.DOCTYPE_PUBLIC_IDENTIFIER_SINGLE_QUOTED;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingDoctypePublicIdentifier),t.forceQuirks=!0,this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypePublicIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateBeforeDoctypePublicIdentifier(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.QUOTATION_MARK:{t.publicId="",this.state=ls.DOCTYPE_PUBLIC_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{t.publicId="",this.state=ls.DOCTYPE_PUBLIC_IDENTIFIER_SINGLE_QUOTED;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingDoctypePublicIdentifier),t.forceQuirks=!0,this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypePublicIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateDoctypePublicIdentifierDoubleQuoted(a){const t=this.currentToken;switch(a){case ts.QUOTATION_MARK:{this.state=ls.AFTER_DOCTYPE_PUBLIC_IDENTIFIER;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.publicId+=Ft;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptDoctypePublicIdentifier),t.forceQuirks=!0,this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:t.publicId+=String.fromCodePoint(a)}}_stateDoctypePublicIdentifierSingleQuoted(a){const t=this.currentToken;switch(a){case ts.APOSTROPHE:{this.state=ls.AFTER_DOCTYPE_PUBLIC_IDENTIFIER;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.publicId+=Ft;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptDoctypePublicIdentifier),t.forceQuirks=!0,this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:t.publicId+=String.fromCodePoint(a)}}_stateAfterDoctypePublicIdentifier(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.BETWEEN_DOCTYPE_PUBLIC_AND_SYSTEM_IDENTIFIERS;break}case ts.GREATER_THAN_SIGN:{this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.QUOTATION_MARK:{this._err(Us.missingWhitespaceBetweenDoctypePublicAndSystemIdentifiers),t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{this._err(Us.missingWhitespaceBetweenDoctypePublicAndSystemIdentifiers),t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateBetweenDoctypePublicAndSystemIdentifiers(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.GREATER_THAN_SIGN:{this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.QUOTATION_MARK:{t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateAfterDoctypeSystemKeyword(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:{this.state=ls.BEFORE_DOCTYPE_SYSTEM_IDENTIFIER;break}case ts.QUOTATION_MARK:{this._err(Us.missingWhitespaceAfterDoctypeSystemKeyword),t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{this._err(Us.missingWhitespaceAfterDoctypeSystemKeyword),t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateBeforeDoctypeSystemIdentifier(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.QUOTATION_MARK:{t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_DOUBLE_QUOTED;break}case ts.APOSTROPHE:{t.systemId="",this.state=ls.DOCTYPE_SYSTEM_IDENTIFIER_SINGLE_QUOTED;break}case ts.GREATER_THAN_SIGN:{this._err(Us.missingDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.DATA,this.emitCurrentDoctype(t);break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.missingQuoteBeforeDoctypeSystemIdentifier),t.forceQuirks=!0,this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateDoctypeSystemIdentifierDoubleQuoted(a){const t=this.currentToken;switch(a){case ts.QUOTATION_MARK:{this.state=ls.AFTER_DOCTYPE_SYSTEM_IDENTIFIER;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.systemId+=Ft;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptDoctypeSystemIdentifier),t.forceQuirks=!0,this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:t.systemId+=String.fromCodePoint(a)}}_stateDoctypeSystemIdentifierSingleQuoted(a){const t=this.currentToken;switch(a){case ts.APOSTROPHE:{this.state=ls.AFTER_DOCTYPE_SYSTEM_IDENTIFIER;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter),t.systemId+=Ft;break}case ts.GREATER_THAN_SIGN:{this._err(Us.abruptDoctypeSystemIdentifier),t.forceQuirks=!0,this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:t.systemId+=String.fromCodePoint(a)}}_stateAfterDoctypeSystemIdentifier(a){const t=this.currentToken;switch(a){case ts.SPACE:case ts.LINE_FEED:case ts.TABULATION:case ts.FORM_FEED:break;case ts.GREATER_THAN_SIGN:{this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.EOF:{this._err(Us.eofInDoctype),t.forceQuirks=!0,this.emitCurrentDoctype(t),this._emitEOFToken();break}default:this._err(Us.unexpectedCharacterAfterDoctypeSystemIdentifier),this.state=ls.BOGUS_DOCTYPE,this._stateBogusDoctype(a)}}_stateBogusDoctype(a){const t=this.currentToken;switch(a){case ts.GREATER_THAN_SIGN:{this.emitCurrentDoctype(t),this.state=ls.DATA;break}case ts.NULL:{this._err(Us.unexpectedNullCharacter);break}case ts.EOF:{this.emitCurrentDoctype(t),this._emitEOFToken();break}}}_stateCdataSection(a){switch(a){case ts.RIGHT_SQUARE_BRACKET:{this.state=ls.CDATA_SECTION_BRACKET;break}case ts.EOF:{this._err(Us.eofInCdata),this._emitEOFToken();break}default:this._emitCodePoint(a)}}_stateCdataSectionBracket(a){a===ts.RIGHT_SQUARE_BRACKET?this.state=ls.CDATA_SECTION_END:(this._emitChars("]"),this.state=ls.CDATA_SECTION,this._stateCdataSection(a))}_stateCdataSectionEnd(a){switch(a){case ts.GREATER_THAN_SIGN:{this.state=ls.DATA;break}case ts.RIGHT_SQUARE_BRACKET:{this._emitChars("]");break}default:this._emitChars("]]"),this.state=ls.CDATA_SECTION,this._stateCdataSection(a)}}_stateCharacterReference(){let a=this.entityDecoder.write(this.preprocessor.html,this.preprocessor.pos);if(a<0)if(this.preprocessor.lastChunkWritten)a=this.entityDecoder.end();else{this.active=!1,this.preprocessor.pos=this.preprocessor.html.length-1,this.consumedAfterSnapshot=0,this.preprocessor.endOfChunkHit=!0;return}a===0?(this.preprocessor.pos=this.entityStartPos,this._flushCodePointConsumedAsCharacterReference(ts.AMPERSAND),this.state=!this._isCharacterReferenceInAttribute()&&s9(this.preprocessor.peek(1))?ls.AMBIGUOUS_AMPERSAND:this.returnState):this.state=this.returnState}_stateAmbiguousAmpersand(a){s9(a)?this._flushCodePointConsumedAsCharacterReference(a):(a===ts.SEMICOLON&&this._err(Us.unknownNamedCharacterReference),this.state=this.returnState,this._callState(a))}};const vM=new Set([k.DD,k.DT,k.LI,k.OPTGROUP,k.OPTION,k.P,k.RB,k.RP,k.RT,k.RTC]),t9=new Set([...vM,k.CAPTION,k.COLGROUP,k.TBODY,k.TD,k.TFOOT,k.TH,k.THEAD,k.TR]),Iu=new Set([k.APPLET,k.CAPTION,k.HTML,k.MARQUEE,k.OBJECT,k.TABLE,k.TD,k.TEMPLATE,k.TH]),oZ=new Set([...Iu,k.OL,k.UL]),cZ=new Set([...Iu,k.BUTTON]),n9=new Set([k.ANNOTATION_XML,k.MI,k.MN,k.MO,k.MS,k.MTEXT]),e9=new Set([k.DESC,k.FOREIGN_OBJECT,k.TITLE]),hZ=new Set([k.TR,k.TEMPLATE,k.HTML]),uZ=new Set([k.TBODY,k.TFOOT,k.THEAD,k.TEMPLATE,k.HTML]),dZ=new Set([k.TABLE,k.TEMPLATE,k.HTML]),gZ=new Set([k.TD,k.TH]);class fZ{get currentTmplContentOrNode(){return this._isInTemplate()?this.treeAdapter.getTemplateContent(this.current):this.current}constructor(a,t,n){this.treeAdapter=t,this.handler=n,this.items=[],this.tagIDs=[],this.stackTop=-1,this.tmplCount=0,this.currentTagId=k.UNKNOWN,this.current=a}_indexOf(a){return this.items.lastIndexOf(a,this.stackTop)}_isInTemplate(){return this.currentTagId===k.TEMPLATE&&this.treeAdapter.getNamespaceURI(this.current)===sa.HTML}_updateCurrentElement(){this.current=this.items[this.stackTop],this.currentTagId=this.tagIDs[this.stackTop]}push(a,t){this.stackTop++,this.items[this.stackTop]=a,this.current=a,this.tagIDs[this.stackTop]=t,this.currentTagId=t,this._isInTemplate()&&this.tmplCount++,this.handler.onItemPush(a,t,!0)}pop(){const a=this.current;this.tmplCount>0&&this._isInTemplate()&&this.tmplCount--,this.stackTop--,this._updateCurrentElement(),this.handler.onItemPop(a,!0)}replace(a,t){const n=this._indexOf(a);this.items[n]=t,n===this.stackTop&&(this.current=t)}insertAfter(a,t,n){const e=this._indexOf(a)+1;this.items.splice(e,0,t),this.tagIDs.splice(e,0,n),this.stackTop++,e===this.stackTop&&this._updateCurrentElement(),this.current&&this.currentTagId!==void 0&&this.handler.onItemPush(this.current,this.currentTagId,e===this.stackTop)}popUntilTagNamePopped(a){let t=this.stackTop+1;do t=this.tagIDs.lastIndexOf(a,t-1);while(t>0&&this.treeAdapter.getNamespaceURI(this.items[t])!==sa.HTML);this.shortenToLength(Math.max(t,0))}shortenToLength(a){for(;this.stackTop>=a;){const t=this.current;this.tmplCount>0&&this._isInTemplate()&&(this.tmplCount-=1),this.stackTop--,this._updateCurrentElement(),this.handler.onItemPop(t,this.stackTop<a)}}popUntilElementPopped(a){const t=this._indexOf(a);this.shortenToLength(Math.max(t,0))}popUntilPopped(a,t){const n=this._indexOfTagNames(a,t);this.shortenToLength(Math.max(n,0))}popUntilNumberedHeaderPopped(){this.popUntilPopped(Tv,sa.HTML)}popUntilTableCellPopped(){this.popUntilPopped(gZ,sa.HTML)}popAllUpToHtmlElement(){this.tmplCount=0,this.shortenToLength(1)}_indexOfTagNames(a,t){for(let n=this.stackTop;n>=0;n--)if(a.has(this.tagIDs[n])&&this.treeAdapter.getNamespaceURI(this.items[n])===t)return n;return-1}clearBackTo(a,t){const n=this._indexOfTagNames(a,t);this.shortenToLength(n+1)}clearBackToTableContext(){this.clearBackTo(dZ,sa.HTML)}clearBackToTableBodyContext(){this.clearBackTo(uZ,sa.HTML)}clearBackToTableRowContext(){this.clearBackTo(hZ,sa.HTML)}remove(a){const t=this._indexOf(a);t>=0&&(t===this.stackTop?this.pop():(this.items.splice(t,1),this.tagIDs.splice(t,1),this.stackTop--,this._updateCurrentElement(),this.handler.onItemPop(a,!1)))}tryPeekProperlyNestedBodyElement(){return this.stackTop>=1&&this.tagIDs[1]===k.BODY?this.items[1]:null}contains(a){return this._indexOf(a)>-1}getCommonAncestor(a){const t=this._indexOf(a)-1;return t>=0?this.items[t]:null}isRootHtmlElementCurrent(){return this.stackTop===0&&this.tagIDs[0]===k.HTML}hasInDynamicScope(a,t){for(let n=this.stackTop;n>=0;n--){const e=this.tagIDs[n];switch(this.treeAdapter.getNamespaceURI(this.items[n])){case sa.HTML:{if(e===a)return!0;if(t.has(e))return!1;break}case sa.SVG:{if(e9.has(e))return!1;break}case sa.MATHML:{if(n9.has(e))return!1;break}}}return!0}hasInScope(a){return this.hasInDynamicScope(a,Iu)}hasInListItemScope(a){return this.hasInDynamicScope(a,oZ)}hasInButtonScope(a){return this.hasInDynamicScope(a,cZ)}hasNumberedHeaderInScope(){for(let a=this.stackTop;a>=0;a--){const t=this.tagIDs[a];switch(this.treeAdapter.getNamespaceURI(this.items[a])){case sa.HTML:{if(Tv.has(t))return!0;if(Iu.has(t))return!1;break}case sa.SVG:{if(e9.has(t))return!1;break}case sa.MATHML:{if(n9.has(t))return!1;break}}}return!0}hasInTableScope(a){for(let t=this.stackTop;t>=0;t--)if(this.treeAdapter.getNamespaceURI(this.items[t])===sa.HTML)switch(this.tagIDs[t]){case a:return!0;case k.TABLE:case k.HTML:return!1}return!0}hasTableBodyContextInTableScope(){for(let a=this.stackTop;a>=0;a--)if(this.treeAdapter.getNamespaceURI(this.items[a])===sa.HTML)switch(this.tagIDs[a]){case k.TBODY:case k.THEAD:case k.TFOOT:return!0;case k.TABLE:case k.HTML:return!1}return!0}hasInSelectScope(a){for(let t=this.stackTop;t>=0;t--)if(this.treeAdapter.getNamespaceURI(this.items[t])===sa.HTML)switch(this.tagIDs[t]){case a:return!0;case k.OPTION:case k.OPTGROUP:break;default:return!1}return!0}generateImpliedEndTags(){for(;this.currentTagId!==void 0&&vM.has(this.currentTagId);)this.pop()}generateImpliedEndTagsThoroughly(){for(;this.currentTagId!==void 0&&t9.has(this.currentTagId);)this.pop()}generateImpliedEndTagsWithExclusion(a){for(;this.currentTagId!==void 0&&this.currentTagId!==a&&t9.has(this.currentTagId);)this.pop()}}const pf=3;var Dl;(function(s){s[s.Marker=0]="Marker",s[s.Element=1]="Element"})(Dl||(Dl={}));const l9={type:Dl.Marker};class yZ{constructor(a){this.treeAdapter=a,this.entries=[],this.bookmark=null}_getNoahArkConditionCandidates(a,t){const n=[],e=t.length,l=this.treeAdapter.getTagName(a),i=this.treeAdapter.getNamespaceURI(a);for(let p=0;p<this.entries.length;p++){const r=this.entries[p];if(r.type===Dl.Marker)break;const{element:m}=r;if(this.treeAdapter.getTagName(m)===l&&this.treeAdapter.getNamespaceURI(m)===i){const o=this.treeAdapter.getAttrList(m);o.length===e&&n.push({idx:p,attrs:o})}}return n}_ensureNoahArkCondition(a){if(this.entries.length<pf)return;const t=this.treeAdapter.getAttrList(a),n=this._getNoahArkConditionCandidates(a,t);if(n.length<pf)return;const e=new Map(t.map(i=>[i.name,i.value]));let l=0;for(let i=0;i<n.length;i++){const p=n[i];p.attrs.every(r=>e.get(r.name)===r.value)&&(l+=1,l>=pf&&this.entries.splice(p.idx,1))}}insertMarker(){this.entries.unshift(l9)}pushElement(a,t){this._ensureNoahArkCondition(a),this.entries.unshift({type:Dl.Element,element:a,token:t})}insertElementAfterBookmark(a,t){const n=this.entries.indexOf(this.bookmark);this.entries.splice(n,0,{type:Dl.Element,element:a,token:t})}removeEntry(a){const t=this.entries.indexOf(a);t!==-1&&this.entries.splice(t,1)}clearToLastMarker(){const a=this.entries.indexOf(l9);a===-1?this.entries.length=0:this.entries.splice(0,a+1)}getElementEntryInScopeWithTagName(a){const t=this.entries.find(n=>n.type===Dl.Marker||this.treeAdapter.getTagName(n.element)===a);return t&&t.type===Dl.Element?t:null}getElementEntry(a){return this.entries.find(t=>t.type===Dl.Element&&t.element===a)}}const oi={createDocument(){return{nodeName:"#document",mode:be.NO_QUIRKS,childNodes:[]}},createDocumentFragment(){return{nodeName:"#document-fragment",childNodes:[]}},createElement(s,a,t){return{nodeName:s,tagName:s,attrs:t,namespaceURI:a,childNodes:[],parentNode:null}},createCommentNode(s){return{nodeName:"#comment",data:s,parentNode:null}},createTextNode(s){return{nodeName:"#text",value:s,parentNode:null}},appendChild(s,a){s.childNodes.push(a),a.parentNode=s},insertBefore(s,a,t){const n=s.childNodes.indexOf(t);s.childNodes.splice(n,0,a),a.parentNode=s},setTemplateContent(s,a){s.content=a},getTemplateContent(s){return s.content},setDocumentType(s,a,t,n){const e=s.childNodes.find(l=>l.nodeName==="#documentType");if(e)e.name=a,e.publicId=t,e.systemId=n;else{const l={nodeName:"#documentType",name:a,publicId:t,systemId:n,parentNode:null};oi.appendChild(s,l)}},setDocumentMode(s,a){s.mode=a},getDocumentMode(s){return s.mode},detachNode(s){if(s.parentNode){const a=s.parentNode.childNodes.indexOf(s);s.parentNode.childNodes.splice(a,1),s.parentNode=null}},insertText(s,a){if(s.childNodes.length>0){const t=s.childNodes[s.childNodes.length-1];if(oi.isTextNode(t)){t.value+=a;return}}oi.appendChild(s,oi.createTextNode(a))},insertTextBefore(s,a,t){const n=s.childNodes[s.childNodes.indexOf(t)-1];n&&oi.isTextNode(n)?n.value+=a:oi.insertBefore(s,oi.createTextNode(a),t)},adoptAttributes(s,a){const t=new Set(s.attrs.map(n=>n.name));for(let n=0;n<a.length;n++)t.has(a[n].name)||s.attrs.push(a[n])},getFirstChild(s){return s.childNodes[0]},getChildNodes(s){return s.childNodes},getParentNode(s){return s.parentNode},getAttrList(s){return s.attrs},getTagName(s){return s.tagName},getNamespaceURI(s){return s.namespaceURI},getTextNodeContent(s){return s.value},getCommentNodeContent(s){return s.data},getDocumentTypeNodeName(s){return s.name},getDocumentTypeNodePublicId(s){return s.publicId},getDocumentTypeNodeSystemId(s){return s.systemId},isTextNode(s){return s.nodeName==="#text"},isCommentNode(s){return s.nodeName==="#comment"},isDocumentTypeNode(s){return s.nodeName==="#documentType"},isElementNode(s){return Object.prototype.hasOwnProperty.call(s,"tagName")},setNodeSourceCodeLocation(s,a){s.sourceCodeLocation=a},getNodeSourceCodeLocation(s){return s.sourceCodeLocation},updateNodeSourceCodeLocation(s,a){s.sourceCodeLocation={...s.sourceCodeLocation,...a}}},EM="html",vZ="about:legacy-compat",EZ="http://www.ibm.com/data/dtd/v11/ibmxhtml1-transitional.dtd",xM=["+//silmaril//dtd html pro v0r11 19970101//","-//as//dtd html 3.0 aswedit + extensions//","-//advasoft ltd//dtd html 3.0 aswedit + 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s.attributes},getTagName(s){return s.name},getNamespaceURI(s){return s.namespace},getTextNodeContent(s){return s.data},getCommentNodeContent(s){return s.data},getDocumentTypeNodeName(s){var a;return(a=s["x-name"])!==null&&a!==void 0?a:""},getDocumentTypeNodePublicId(s){var a;return(a=s["x-publicId"])!==null&&a!==void 0?a:""},getDocumentTypeNodeSystemId(s){var a;return(a=s["x-systemId"])!==null&&a!==void 0?a:""},isDocumentTypeNode(s){return vv(s)&&s.name==="!doctype"},setNodeSourceCodeLocation(s,a){a&&(s.startIndex=a.startOffset,s.endIndex=a.endOffset),s.sourceCodeLocation=a},getNodeSourceCodeLocation(s){return s.sourceCodeLocation},updateNodeSourceCodeLocation(s,a){a.endOffset!=null&&(s.endIndex=a.endOffset),s.sourceCodeLocation={...s.sourceCodeLocation,...a}}};function SJ(s,a,t,n){const e={scriptingEnabled:typeof a.scriptingEnabled=="boolean"?a.scriptingEnabled:!0,treeAdapter:sp,sourceCodeLocationInfo:a.sourceCodeLocationInfo};return t?zJ(s,e):DJ(n,s,e)}const FJ={treeAdapter:sp};function OJ(s){const a="length"in s?s:[s];for(let n=0;n<a.length;n+=1){const e=a[n];kr(e)&&Array.prototype.splice.call(a,n,1,...e.children)}let t="";for(let n=0;n<a.length;n+=1){const e=a[n];t+=AJ(e,FJ)}return t}var xa;(function(s){s[s.Tab=9]="Tab",s[s.NewLine=10]="NewLine",s[s.FormFeed=12]="FormFeed",s[s.CarriageReturn=13]="CarriageReturn",s[s.Space=32]="Space",s[s.ExclamationMark=33]="ExclamationMark",s[s.Number=35]="Number",s[s.Amp=38]="Amp",s[s.SingleQuote=39]="SingleQuote",s[s.DoubleQuote=34]="DoubleQuote",s[s.Dash=45]="Dash",s[s.Slash=47]="Slash",s[s.Zero=48]="Zero",s[s.Nine=57]="Nine",s[s.Semi=59]="Semi",s[s.Lt=60]="Lt",s[s.Eq=61]="Eq",s[s.Gt=62]="Gt",s[s.Questionmark=63]="Questionmark",s[s.UpperA=65]="UpperA",s[s.LowerA=97]="LowerA",s[s.UpperF=70]="UpperF",s[s.LowerF=102]="LowerF",s[s.UpperZ=90]="UpperZ",s[s.LowerZ=122]="LowerZ",s[s.LowerX=120]="LowerX",s[s.OpeningSquareBracket=91]="OpeningSquareBracket"})(xa||(xa={}));var Qs;(function(s){s[s.Text=1]="Text",s[s.BeforeTagName=2]="BeforeTagName",s[s.InTagName=3]="InTagName",s[s.InSelfClosingTag=4]="InSelfClosingTag",s[s.BeforeClosingTagName=5]="BeforeClosingTagName",s[s.InClosingTagName=6]="InClosingTagName",s[s.AfterClosingTagName=7]="AfterClosingTagName",s[s.BeforeAttributeName=8]="BeforeAttributeName",s[s.InAttributeName=9]="InAttributeName",s[s.AfterAttributeName=10]="AfterAttributeName",s[s.BeforeAttributeValue=11]="BeforeAttributeValue",s[s.InAttributeValueDq=12]="InAttributeValueDq",s[s.InAttributeValueSq=13]="InAttributeValueSq",s[s.InAttributeValueNq=14]="InAttributeValueNq",s[s.BeforeDeclaration=15]="BeforeDeclaration",s[s.InDeclaration=16]="InDeclaration",s[s.InProcessingInstruction=17]="InProcessingInstruction",s[s.BeforeComment=18]="BeforeComment",s[s.CDATASequence=19]="CDATASequence",s[s.InSpecialComment=20]="InSpecialComment",s[s.InCommentLike=21]="InCommentLike",s[s.BeforeSpecialS=22]="BeforeSpecialS",s[s.SpecialStartSequence=23]="SpecialStartSequence",s[s.InSpecialTag=24]="InSpecialTag",s[s.BeforeEntity=25]="BeforeEntity",s[s.BeforeNumericEntity=26]="BeforeNumericEntity",s[s.InNamedEntity=27]="InNamedEntity",s[s.InNumericEntity=28]="InNumericEntity",s[s.InHexEntity=29]="InHexEntity"})(Qs||(Qs={}));function ri(s){return s===xa.Space||s===xa.NewLine||s===xa.Tab||s===xa.FormFeed||s===xa.CarriageReturn}function v0(s){return s===xa.Slash||s===xa.Gt||ri(s)}function h9(s){return s>=xa.Zero&&s<=xa.Nine}function IJ(s){return s>=xa.LowerA&&s<=xa.LowerZ||s>=xa.UpperA&&s<=xa.UpperZ}function NJ(s){return s>=xa.UpperA&&s<=xa.UpperF||s>=xa.LowerA&&s<=xa.LowerF}var Sl;(function(s){s[s.NoValue=0]="NoValue",s[s.Unquoted=1]="Unquoted",s[s.Single=2]="Single",s[s.Double=3]="Double"})(Sl||(Sl={}));const re={Cdata:new Uint8Array([67,68,65,84,65,91]),CdataEnd:new Uint8Array([93,93,62]),CommentEnd:new Uint8Array([45,45,62]),ScriptEnd:new Uint8Array([60,47,115,99,114,105,112,116]),StyleEnd:new Uint8Array([60,47,115,116,121,108,101]),TitleEnd:new Uint8Array([60,47,116,105,116,108,101])};class PJ{constructor({xmlMode:a=!1,decodeEntities:t=!0},n){this.cbs=n,this.state=Qs.Text,this.buffer="",this.sectionStart=0,this.index=0,this.baseState=Qs.Text,this.isSpecial=!1,this.running=!0,this.offset=0,this.currentSequence=void 0,this.sequenceIndex=0,this.trieIndex=0,this.trieCurrent=0,this.entityResult=0,this.entityExcess=0,this.xmlMode=a,this.decodeEntities=t,this.entityTrie=a?CB:TB}reset(){this.state=Qs.Text,this.buffer="",this.sectionStart=0,this.index=0,this.baseState=Qs.Text,this.currentSequence=void 0,this.running=!0,this.offset=0}write(a){this.offset+=this.buffer.length,this.buffer=a,this.parse()}end(){this.running&&this.finish()}pause(){this.running=!1}resume(){this.running=!0,this.index<this.buffer.length+this.offset&&this.parse()}getIndex(){return this.index}getSectionStart(){return this.sectionStart}stateText(a){a===xa.Lt||!this.decodeEntities&&this.fastForwardTo(xa.Lt)?(this.index>this.sectionStart&&this.cbs.ontext(this.sectionStart,this.index),this.state=Qs.BeforeTagName,this.sectionStart=this.index):this.decodeEntities&&a===xa.Amp&&(this.state=Qs.BeforeEntity)}stateSpecialStartSequence(a){const t=this.sequenceIndex===this.currentSequence.length;if(!(t?v0(a):(a|32)===this.currentSequence[this.sequenceIndex]))this.isSpecial=!1;else if(!t){this.sequenceIndex++;return}this.sequenceIndex=0,this.state=Qs.InTagName,this.stateInTagName(a)}stateInSpecialTag(a){if(this.sequenceIndex===this.currentSequence.length){if(a===xa.Gt||ri(a)){const t=this.index-this.currentSequence.length;if(this.sectionStart<t){const n=this.index;this.index=t,this.cbs.ontext(this.sectionStart,t),this.index=n}this.isSpecial=!1,this.sectionStart=t+2,this.stateInClosingTagName(a);return}this.sequenceIndex=0}(a|32)===this.currentSequence[this.sequenceIndex]?this.sequenceIndex+=1:this.sequenceIndex===0?this.currentSequence===re.TitleEnd?this.decodeEntities&&a===xa.Amp&&(this.state=Qs.BeforeEntity):this.fastForwardTo(xa.Lt)&&(this.sequenceIndex=1):this.sequenceIndex=+(a===xa.Lt)}stateCDATASequence(a){a===re.Cdata[this.sequenceIndex]?++this.sequenceIndex===re.Cdata.length&&(this.state=Qs.InCommentLike,this.currentSequence=re.CdataEnd,this.sequenceIndex=0,this.sectionStart=this.index+1):(this.sequenceIndex=0,this.state=Qs.InDeclaration,this.stateInDeclaration(a))}fastForwardTo(a){for(;++this.index<this.buffer.length+this.offset;)if(this.buffer.charCodeAt(this.index-this.offset)===a)return!0;return this.index=this.buffer.length+this.offset-1,!1}stateInCommentLike(a){a===this.currentSequence[this.sequenceIndex]?++this.sequenceIndex===this.currentSequence.length&&(this.currentSequence===re.CdataEnd?this.cbs.oncdata(this.sectionStart,this.index,2):this.cbs.oncomment(this.sectionStart,this.index,2),this.sequenceIndex=0,this.sectionStart=this.index+1,this.state=Qs.Text):this.sequenceIndex===0?this.fastForwardTo(this.currentSequence[0])&&(this.sequenceIndex=1):a!==this.currentSequence[this.sequenceIndex-1]&&(this.sequenceIndex=0)}isTagStartChar(a){return this.xmlMode?!v0(a):IJ(a)}startSpecial(a,t){this.isSpecial=!0,this.currentSequence=a,this.sequenceIndex=t,this.state=Qs.SpecialStartSequence}stateBeforeTagName(a){if(a===xa.ExclamationMark)this.state=Qs.BeforeDeclaration,this.sectionStart=this.index+1;else if(a===xa.Questionmark)this.state=Qs.InProcessingInstruction,this.sectionStart=this.index+1;else if(this.isTagStartChar(a)){const t=a|32;this.sectionStart=this.index,!this.xmlMode&&t===re.TitleEnd[2]?this.startSpecial(re.TitleEnd,3):this.state=!this.xmlMode&&t===re.ScriptEnd[2]?Qs.BeforeSpecialS:Qs.InTagName}else a===xa.Slash?this.state=Qs.BeforeClosingTagName:(this.state=Qs.Text,this.stateText(a))}stateInTagName(a){v0(a)&&(this.cbs.onopentagname(this.sectionStart,this.index),this.sectionStart=-1,this.state=Qs.BeforeAttributeName,this.stateBeforeAttributeName(a))}stateBeforeClosingTagName(a){ri(a)||(a===xa.Gt?this.state=Qs.Text:(this.state=this.isTagStartChar(a)?Qs.InClosingTagName:Qs.InSpecialComment,this.sectionStart=this.index))}stateInClosingTagName(a){(a===xa.Gt||ri(a))&&(this.cbs.onclosetag(this.sectionStart,this.index),this.sectionStart=-1,this.state=Qs.AfterClosingTagName,this.stateAfterClosingTagName(a))}stateAfterClosingTagName(a){(a===xa.Gt||this.fastForwardTo(xa.Gt))&&(this.state=Qs.Text,this.baseState=Qs.Text,this.sectionStart=this.index+1)}stateBeforeAttributeName(a){a===xa.Gt?(this.cbs.onopentagend(this.index),this.isSpecial?(this.state=Qs.InSpecialTag,this.sequenceIndex=0):this.state=Qs.Text,this.baseState=this.state,this.sectionStart=this.index+1):a===xa.Slash?this.state=Qs.InSelfClosingTag:ri(a)||(this.state=Qs.InAttributeName,this.sectionStart=this.index)}stateInSelfClosingTag(a){a===xa.Gt?(this.cbs.onselfclosingtag(this.index),this.state=Qs.Text,this.baseState=Qs.Text,this.sectionStart=this.index+1,this.isSpecial=!1):ri(a)||(this.state=Qs.BeforeAttributeName,this.stateBeforeAttributeName(a))}stateInAttributeName(a){(a===xa.Eq||v0(a))&&(this.cbs.onattribname(this.sectionStart,this.index),this.sectionStart=-1,this.state=Qs.AfterAttributeName,this.stateAfterAttributeName(a))}stateAfterAttributeName(a){a===xa.Eq?this.state=Qs.BeforeAttributeValue:a===xa.Slash||a===xa.Gt?(this.cbs.onattribend(Sl.NoValue,this.index),this.state=Qs.BeforeAttributeName,this.stateBeforeAttributeName(a)):ri(a)||(this.cbs.onattribend(Sl.NoValue,this.index),this.state=Qs.InAttributeName,this.sectionStart=this.index)}stateBeforeAttributeValue(a){a===xa.DoubleQuote?(this.state=Qs.InAttributeValueDq,this.sectionStart=this.index+1):a===xa.SingleQuote?(this.state=Qs.InAttributeValueSq,this.sectionStart=this.index+1):ri(a)||(this.sectionStart=this.index,this.state=Qs.InAttributeValueNq,this.stateInAttributeValueNoQuotes(a))}handleInAttributeValue(a,t){a===t||!this.decodeEntities&&this.fastForwardTo(t)?(this.cbs.onattribdata(this.sectionStart,this.index),this.sectionStart=-1,this.cbs.onattribend(t===xa.DoubleQuote?Sl.Double:Sl.Single,this.index),this.state=Qs.BeforeAttributeName):this.decodeEntities&&a===xa.Amp&&(this.baseState=this.state,this.state=Qs.BeforeEntity)}stateInAttributeValueDoubleQuotes(a){this.handleInAttributeValue(a,xa.DoubleQuote)}stateInAttributeValueSingleQuotes(a){this.handleInAttributeValue(a,xa.SingleQuote)}stateInAttributeValueNoQuotes(a){ri(a)||a===xa.Gt?(this.cbs.onattribdata(this.sectionStart,this.index),this.sectionStart=-1,this.cbs.onattribend(Sl.Unquoted,this.index),this.state=Qs.BeforeAttributeName,this.stateBeforeAttributeName(a)):this.decodeEntities&&a===xa.Amp&&(this.baseState=this.state,this.state=Qs.BeforeEntity)}stateBeforeDeclaration(a){a===xa.OpeningSquareBracket?(this.state=Qs.CDATASequence,this.sequenceIndex=0):this.state=a===xa.Dash?Qs.BeforeComment:Qs.InDeclaration}stateInDeclaration(a){(a===xa.Gt||this.fastForwardTo(xa.Gt))&&(this.cbs.ondeclaration(this.sectionStart,this.index),this.state=Qs.Text,this.sectionStart=this.index+1)}stateInProcessingInstruction(a){(a===xa.Gt||this.fastForwardTo(xa.Gt))&&(this.cbs.onprocessinginstruction(this.sectionStart,this.index),this.state=Qs.Text,this.sectionStart=this.index+1)}stateBeforeComment(a){a===xa.Dash?(this.state=Qs.InCommentLike,this.currentSequence=re.CommentEnd,this.sequenceIndex=2,this.sectionStart=this.index+1):this.state=Qs.InDeclaration}stateInSpecialComment(a){(a===xa.Gt||this.fastForwardTo(xa.Gt))&&(this.cbs.oncomment(this.sectionStart,this.index,0),this.state=Qs.Text,this.sectionStart=this.index+1)}stateBeforeSpecialS(a){const t=a|32;t===re.ScriptEnd[3]?this.startSpecial(re.ScriptEnd,4):t===re.StyleEnd[3]?this.startSpecial(re.StyleEnd,4):(this.state=Qs.InTagName,this.stateInTagName(a))}stateBeforeEntity(a){this.entityExcess=1,this.entityResult=0,a===xa.Number?this.state=Qs.BeforeNumericEntity:a===xa.Amp||(this.trieIndex=0,this.trieCurrent=this.entityTrie[0],this.state=Qs.InNamedEntity,this.stateInNamedEntity(a))}stateInNamedEntity(a){if(this.entityExcess+=1,this.trieIndex=LB(this.entityTrie,this.trieCurrent,this.trieIndex+1,a),this.trieIndex<0){this.emitNamedEntity(),this.index--;return}this.trieCurrent=this.entityTrie[this.trieIndex];const t=this.trieCurrent&rl.VALUE_LENGTH;if(t){const n=(t>>14)-1;if(!this.allowLegacyEntity()&&a!==xa.Semi)this.trieIndex+=n;else{const e=this.index-this.entityExcess+1;e>this.sectionStart&&this.emitPartial(this.sectionStart,e),this.entityResult=this.trieIndex,this.trieIndex+=n,this.entityExcess=0,this.sectionStart=this.index+1,n===0&&this.emitNamedEntity()}}}emitNamedEntity(){if(this.state=this.baseState,this.entityResult===0)return;switch((this.entityTrie[this.entityResult]&rl.VALUE_LENGTH)>>14){case 1:{this.emitCodePoint(this.entityTrie[this.entityResult]&~rl.VALUE_LENGTH);break}case 2:{this.emitCodePoint(this.entityTrie[this.entityResult+1]);break}case 3:this.emitCodePoint(this.entityTrie[this.entityResult+1]),this.emitCodePoint(this.entityTrie[this.entityResult+2])}}stateBeforeNumericEntity(a){(a|32)===xa.LowerX?(this.entityExcess++,this.state=Qs.InHexEntity):(this.state=Qs.InNumericEntity,this.stateInNumericEntity(a))}emitNumericEntity(a){const t=this.index-this.entityExcess-1;t+2+ +(this.state===Qs.InHexEntity)!==this.index&&(t>this.sectionStart&&this.emitPartial(this.sectionStart,t),this.sectionStart=this.index+Number(a),this.emitCodePoint(zB(this.entityResult))),this.state=this.baseState}stateInNumericEntity(a){a===xa.Semi?this.emitNumericEntity(!0):h9(a)?(this.entityResult=this.entityResult*10+(a-xa.Zero),this.entityExcess++):(this.allowLegacyEntity()?this.emitNumericEntity(!1):this.state=this.baseState,this.index--)}stateInHexEntity(a){a===xa.Semi?this.emitNumericEntity(!0):h9(a)?(this.entityResult=this.entityResult*16+(a-xa.Zero),this.entityExcess++):NJ(a)?(this.entityResult=this.entityResult*16+((a|32)-xa.LowerA+10),this.entityExcess++):(this.allowLegacyEntity()?this.emitNumericEntity(!1):this.state=this.baseState,this.index--)}allowLegacyEntity(){return!this.xmlMode&&(this.baseState===Qs.Text||this.baseState===Qs.InSpecialTag)}cleanup(){this.running&&this.sectionStart!==this.index&&(this.state===Qs.Text||this.state===Qs.InSpecialTag&&this.sequenceIndex===0?(this.cbs.ontext(this.sectionStart,this.index),this.sectionStart=this.index):(this.state===Qs.InAttributeValueDq||this.state===Qs.InAttributeValueSq||this.state===Qs.InAttributeValueNq)&&(this.cbs.onattribdata(this.sectionStart,this.index),this.sectionStart=this.index))}shouldContinue(){return this.index<this.buffer.length+this.offset&&this.running}parse(){for(;this.shouldContinue();){const a=this.buffer.charCodeAt(this.index-this.offset);switch(this.state){case Qs.Text:{this.stateText(a);break}case Qs.SpecialStartSequence:{this.stateSpecialStartSequence(a);break}case Qs.InSpecialTag:{this.stateInSpecialTag(a);break}case Qs.CDATASequence:{this.stateCDATASequence(a);break}case Qs.InAttributeValueDq:{this.stateInAttributeValueDoubleQuotes(a);break}case Qs.InAttributeName:{this.stateInAttributeName(a);break}case Qs.InCommentLike:{this.stateInCommentLike(a);break}case Qs.InSpecialComment:{this.stateInSpecialComment(a);break}case Qs.BeforeAttributeName:{this.stateBeforeAttributeName(a);break}case Qs.InTagName:{this.stateInTagName(a);break}case Qs.InClosingTagName:{this.stateInClosingTagName(a);break}case Qs.BeforeTagName:{this.stateBeforeTagName(a);break}case Qs.AfterAttributeName:{this.stateAfterAttributeName(a);break}case Qs.InAttributeValueSq:{this.stateInAttributeValueSingleQuotes(a);break}case Qs.BeforeAttributeValue:{this.stateBeforeAttributeValue(a);break}case Qs.BeforeClosingTagName:{this.stateBeforeClosingTagName(a);break}case Qs.AfterClosingTagName:{this.stateAfterClosingTagName(a);break}case Qs.BeforeSpecialS:{this.stateBeforeSpecialS(a);break}case Qs.InAttributeValueNq:{this.stateInAttributeValueNoQuotes(a);break}case Qs.InSelfClosingTag:{this.stateInSelfClosingTag(a);break}case Qs.InDeclaration:{this.stateInDeclaration(a);break}case Qs.BeforeDeclaration:{this.stateBeforeDeclaration(a);break}case Qs.BeforeComment:{this.stateBeforeComment(a);break}case Qs.InProcessingInstruction:{this.stateInProcessingInstruction(a);break}case Qs.InNamedEntity:{this.stateInNamedEntity(a);break}case Qs.BeforeEntity:{this.stateBeforeEntity(a);break}case Qs.InHexEntity:{this.stateInHexEntity(a);break}case Qs.InNumericEntity:{this.stateInNumericEntity(a);break}default:this.stateBeforeNumericEntity(a)}this.index++}this.cleanup()}finish(){this.state===Qs.InNamedEntity&&this.emitNamedEntity(),this.sectionStart<this.index&&this.handleTrailingData(),this.cbs.onend()}handleTrailingData(){const 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Set(["input","option","optgroup","select","button","datalist","textarea"]),Et=new Set(["p"]),u9=new Set(["thead","tbody"]),d9=new Set(["dd","dt"]),g9=new Set(["rt","rp"]),RJ=new Map([["tr",new Set(["tr","th","td"])],["th",new Set(["th"])],["td",new Set(["thead","th","td"])],["body",new Set(["head","link","script"])],["li",new Set(["li"])],["p",Et],["h1",Et],["h2",Et],["h3",Et],["h4",Et],["h5",Et],["h6",Et],["select",Rr],["input",Rr],["output",Rr],["button",Rr],["datalist",Rr],["textarea",Rr],["option",new Set(["option"])],["optgroup",new Set(["optgroup","option"])],["dd",d9],["dt",d9],["address",Et],["article",Et],["aside",Et],["blockquote",Et],["details",Et],["div",Et],["dl",Et],["fieldset",Et],["figcaption",Et],["figure",Et],["footer",Et],["form",Et],["header",Et],["hr",Et],["main",Et],["nav",Et],["ol",Et],["pre",Et],["section",Et],["table",Et],["ul",Et],["rt",g9],["rp",g9],["tbody",u9],["tfoot",u9]]),$J=new Set(["area","base","basefont","br","col","command","embed","frame","hr","img","input","isindex","keygen","link","meta","param","source","track","wbr"]),f9=new Set(["math","svg"]),y9=new Set(["mi","mo","mn","ms","mtext","annotation-xml","foreignobject","desc","title"]),VJ=/\s|\//;let HJ=class{constructor(a,t={}){var n,e,l,i,p;this.options=t,this.startIndex=0,this.endIndex=0,this.openTagStart=0,this.tagname="",this.attribname="",this.attribvalue="",this.attribs=null,this.stack=[],this.foreignContext=[],this.buffers=[],this.bufferOffset=0,this.writeIndex=0,this.ended=!1,this.cbs=a??{},this.lowerCaseTagNames=(n=t.lowerCaseTags)!==null&&n!==void 0?n:!t.xmlMode,this.lowerCaseAttributeNames=(e=t.lowerCaseAttributeNames)!==null&&e!==void 0?e:!t.xmlMode,this.tokenizer=new((l=t.Tokenizer)!==null&&l!==void 0?l:PJ)(this.options,this),(p=(i=this.cbs).onparserinit)===null||p===void 0||p.call(i,this)}ontext(a,t){var n,e;const l=this.getSlice(a,t);this.endIndex=t-1,(e=(n=this.cbs).ontext)===null||e===void 0||e.call(n,l),this.startIndex=t}ontextentity(a){var t,n;const e=this.tokenizer.getSectionStart();this.endIndex=e-1,(n=(t=this.cbs).ontext)===null||n===void 0||n.call(t,Ev(a)),this.startIndex=e}isVoidElement(a){return!this.options.xmlMode&&$J.has(a)}onopentagname(a,t){this.endIndex=t;let n=this.getSlice(a,t);this.lowerCaseTagNames&&(n=n.toLowerCase()),this.emitOpenTag(n)}emitOpenTag(a){var t,n,e,l;this.openTagStart=this.startIndex,this.tagname=a;const i=!this.options.xmlMode&&RJ.get(a);if(i)for(;this.stack.length>0&&i.has(this.stack[this.stack.length-1]);){const p=this.stack.pop();(n=(t=this.cbs).onclosetag)===null||n===void 0||n.call(t,p,!0)}this.isVoidElement(a)||(this.stack.push(a),f9.has(a)?this.foreignContext.push(!0):y9.has(a)&&this.foreignContext.push(!1)),(l=(e=this.cbs).onopentagname)===null||l===void 0||l.call(e,a),this.cbs.onopentag&&(this.attribs={})}endOpenTag(a){var t,n;this.startIndex=this.openTagStart,this.attribs&&((n=(t=this.cbs).onopentag)===null||n===void 0||n.call(t,this.tagname,this.attribs,a),this.attribs=null),this.cbs.onclosetag&&this.isVoidElement(this.tagname)&&this.cbs.onclosetag(this.tagname,!0),this.tagname=""}onopentagend(a){this.endIndex=a,this.endOpenTag(!1),this.startIndex=a+1}onclosetag(a,t){var n,e,l,i,p,r;this.endIndex=t;let m=this.getSlice(a,t);if(this.lowerCaseTagNames&&(m=m.toLowerCase()),(f9.has(m)||y9.has(m))&&this.foreignContext.pop(),this.isVoidElement(m))!this.options.xmlMode&&m==="br"&&((e=(n=this.cbs).onopentagname)===null||e===void 0||e.call(n,"br"),(i=(l=this.cbs).onopentag)===null||i===void 0||i.call(l,"br",{},!0),(r=(p=this.cbs).onclosetag)===null||r===void 0||r.call(p,"br",!1));else{const o=this.stack.lastIndexOf(m);if(o!==-1)if(this.cbs.onclosetag){let c=this.stack.length-o;for(;c--;)this.cbs.onclosetag(this.stack.pop(),c!==0)}else 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y=this.getSlice(a,t-n);this.options.xmlMode||this.options.recognizeCDATA?((l=(e=this.cbs).oncdatastart)===null||l===void 0||l.call(e),(p=(i=this.cbs).ontext)===null||p===void 0||p.call(i,y),(m=(r=this.cbs).oncdataend)===null||m===void 0||m.call(r)):((c=(o=this.cbs).oncomment)===null||c===void 0||c.call(o,`[CDATA[${y}]]`),(u=(h=this.cbs).oncommentend)===null||u===void 0||u.call(h)),this.startIndex=t+1}onend(){var a,t;if(this.cbs.onclosetag){this.endIndex=this.startIndex;for(let n=this.stack.length;n>0;this.cbs.onclosetag(this.stack[--n],!0));}(t=(a=this.cbs).onend)===null||t===void 0||t.call(a)}reset(){var a,t,n,e;(t=(a=this.cbs).onreset)===null||t===void 0||t.call(a),this.tokenizer.reset(),this.tagname="",this.attribname="",this.attribs=null,this.stack.length=0,this.startIndex=0,this.endIndex=0,(e=(n=this.cbs).onparserinit)===null||e===void 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l=e.index(t);l>=0&&(e=e.slice(0,l)),n=a(e)}return{queue:t,nesting:!0,...n}},"table,pre,p>img:only-child":({$node:s,getContent:a})=>({...a(s)})},GJ={selector:"h1,h2,h3,h4,h5,h6,ul,ol,li,table,pre,p>img:only-child",selectorRules:jJ},v9="markmap: ",WJ=/^h[1-6]$/,YJ=/^[uo]l$/,XJ=/^li$/;function KJ(s){return WJ.test(s)?+s[1]:YJ.test(s)?8:XJ.test(s)?9:7}function ZJ(s,a){const t={...GJ,...a},n=qM(s),e=n("body");let l=0;const i={id:l,tag:"",html:"",level:0,parent:0,childrenLevel:0,children:[]},p=[];let r=0;return u(e.children()),i;function m(y){var x;const{parent:b}=y,_={id:++l,tag:y.tagName,level:y.level,html:y.html,childrenLevel:0,children:y.nesting?[]:void 0,parent:b.id};return(x=y.comments)!=null&&x.length&&(_.comments=y.comments),Object.keys(y.data||{}).length&&(_.data=y.data),b.children&&((b.childrenLevel===0||b.childrenLevel>_.level)&&(b.children=[],b.childrenLevel=_.level),b.childrenLevel===_.level&&b.children.push(_)),_}function o(y){let x;for(;(x=p.at(-1))&&x.level>=y;)p.pop();return 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`:case`\r
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`,this.pos);if(e!==-1){for(;e!==-1;){const l=this.continueScalar(e+1);if(l===-1)break;e=n.indexOf(`
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body",Es);if(Es=j[--ns],Es.text==="#")j.splice(ns+1,1);else if(/^[1-9]$/.test(Es.text)){var _s;(_s=j).splice.apply(_s,[ns,2].concat(K[+Es.text-1]))}else throw new r("Not a valid argument number",Es)}}}return this.pushTokens(j),j},d.expandAfterFuture=function(){return this.expandOnce(),this.future()},d.expandNextToken=function(){for(;;){var f=this.expandOnce();if(f instanceof i)if(f.text==="\\relax"||f.treatAsRelax)this.stack.pop();else return this.stack.pop()}throw new Error},d.expandMacro=function(f){return this.macros.has(f)?this.expandTokens([new i(f)]):void 0},d.expandTokens=function(f){var w=[],z=this.stack.length;for(this.pushTokens(f);this.stack.length>z;){var F=this.expandOnce(!0);F instanceof i&&(F.treatAsRelax&&(F.noexpand=!1,F.treatAsRelax=!1),w.push(this.stack.pop()))}return w},d.expandMacroAsText=function(f){var w=this.expandMacro(f);return w&&w.map(function(z){return z.text}).join("")},d._getExpansion=function(f){var w=this.macros.get(f);if(w==null)return w;var z=typeof w=="function"?w(this):w;if(typeof z=="string"){var F=0;if(z.indexOf("#")!==-1)for(var j=z.replace(/##/g,"");j.indexOf("#"+(F+1))!==-1;)++F;for(var K=new V3(z,this.settings),ns=[],Es=K.lex();Es.text!=="EOF";)ns.push(Es),Es=K.lex();ns.reverse();var _s={tokens:ns,numArgs:F};return _s}return z},d.isDefined=function(f){return this.macros.has(f)||Lp.hasOwnProperty(f)||Rs.math.hasOwnProperty(f)||Rs.text.hasOwnProperty(f)||G3.hasOwnProperty(f)},d.isExpandable=function(f){var w=this.macros.get(f);return w!=null?typeof w=="string"||typeof w=="function"||!w.unexpandable:Lp.hasOwnProperty(f)},O})(),W3={"́":{text:"\\'",math:"\\acute"},"̀":{text:"\\`",math:"\\grave"},"̈":{text:'\\"',math:"\\ddot"},"̃":{text:"\\~",math:"\\tilde"},"̄":{text:"\\=",math:"\\bar"},"̆":{text:"\\u",math:"\\breve"},"̌":{text:"\\v",math:"\\check"},"̂":{text:"\\^",math:"\\hat"},"̇":{text:"\\.",math:"\\dot"},"̊":{text:"\\r",math:"\\mathring"},"̋":{text:"\\H"}},Y3={á:"á",à:"à",ä:"ä",ǟ:"ǟ",ã:"ã",ā:"ā",ă:"ă",ắ:"ắ",ằ:"ằ",ẵ:"ẵ",ǎ:"ǎ",â:"â",ấ:"ấ",ầ:"ầ",ẫ:"ẫ",ȧ:"ȧ",ǡ:"ǡ",å:"å",ǻ:"ǻ",ḃ:"ḃ",ć:"ć",č:"č",ĉ:"ĉ",ċ:"ċ",ď:"ď",ḋ:"ḋ",é:"é",è:"è",ë:"ë",ẽ:"ẽ",ē:"ē",ḗ:"ḗ",ḕ:"ḕ",ĕ:"ĕ",ě:"ě",ê:"ê",ế:"ế",ề:"ề",ễ:"ễ",ė:"ė",ḟ:"ḟ",ǵ:"ǵ",ḡ:"ḡ",ğ:"ğ",ǧ:"ǧ",ĝ:"ĝ",ġ:"ġ",ḧ:"ḧ",ȟ:"ȟ",ĥ:"ĥ",ḣ:"ḣ",í:"í",ì:"ì",ï:"ï",ḯ:"ḯ",ĩ:"ĩ",ī:"ī",ĭ:"ĭ",ǐ:"ǐ",î:"î",ǰ:"ǰ",ĵ:"ĵ",ḱ:"ḱ",ǩ:"ǩ",ĺ:"ĺ",ľ:"ľ",ḿ:"ḿ",ṁ:"ṁ",ń:"ń",ǹ:"ǹ",ñ:"ñ",ň:"ň",ṅ:"ṅ",ó:"ó",ò:"ò",ö:"ö",ȫ:"ȫ",õ:"õ",ṍ:"ṍ",ṏ:"ṏ",ȭ:"ȭ",ō:"ō",ṓ:"ṓ",ṑ:"ṑ",ŏ:"ŏ",ǒ:"ǒ",ô:"ô",ố:"ố",ồ:"ồ",ỗ:"ỗ",ȯ:"ȯ",ȱ:"ȱ",ő:"ő",ṕ:"ṕ",ṗ:"ṗ",ŕ:"ŕ",ř:"ř",ṙ:"ṙ",ś:"ś",ṥ:"ṥ",š:"š",ṧ:"ṧ",ŝ:"ŝ",ṡ:"ṡ",ẗ:"ẗ",ť:"ť",ṫ:"ṫ",ú:"ú",ù:"ù",ü:"ü",ǘ:"ǘ",ǜ:"ǜ",ǖ:"ǖ",ǚ:"ǚ",ũ:"ũ",ṹ:"ṹ",ū:"ū",ṻ:"ṻ",ŭ:"ŭ",ǔ:"ǔ",û:"û",ů:"ů",ű:"ű",ṽ:"ṽ",ẃ:"ẃ",ẁ:"ẁ",ẅ:"ẅ",ŵ:"ŵ",ẇ:"ẇ",ẘ:"ẘ",ẍ:"ẍ",ẋ:"ẋ",ý:"ý",ỳ:"ỳ",ÿ:"ÿ",ỹ:"ỹ",ȳ:"ȳ",ŷ:"ŷ",ẏ:"ẏ",ẙ:"ẙ",ź:"ź",ž:"ž",ẑ:"ẑ",ż:"ż",Á:"Á",À:"À",Ä:"Ä",Ǟ:"Ǟ",Ã:"Ã",Ā:"Ā",Ă:"Ă",Ắ:"Ắ",Ằ:"Ằ",Ẵ:"Ẵ",Ǎ:"Ǎ",Â:"Â",Ấ:"Ấ",Ầ:"Ầ",Ẫ:"Ẫ",Ȧ:"Ȧ",Ǡ:"Ǡ",Å:"Å",Ǻ:"Ǻ",Ḃ:"Ḃ",Ć:"Ć",Č:"Č",Ĉ:"Ĉ",Ċ:"Ċ",Ď:"Ď",Ḋ:"Ḋ",É:"É",È:"È",Ë:"Ë",Ẽ:"Ẽ",Ē:"Ē",Ḗ:"Ḗ",Ḕ:"Ḕ",Ĕ:"Ĕ",Ě:"Ě",Ê:"Ê",Ế:"Ế",Ề:"Ề",Ễ:"Ễ",Ė:"Ė",Ḟ:"Ḟ",Ǵ:"Ǵ",Ḡ:"Ḡ",Ğ:"Ğ",Ǧ:"Ǧ",Ĝ:"Ĝ",Ġ:"Ġ",Ḧ:"Ḧ",Ȟ:"Ȟ",Ĥ:"Ĥ",Ḣ:"Ḣ",Í:"Í",Ì:"Ì",Ï:"Ï",Ḯ:"Ḯ",Ĩ:"Ĩ",Ī:"Ī",Ĭ:"Ĭ",Ǐ:"Ǐ",Î:"Î",İ:"İ",Ĵ:"Ĵ",Ḱ:"Ḱ",Ǩ:"Ǩ",Ĺ:"Ĺ",Ľ:"Ľ",Ḿ:"Ḿ",Ṁ:"Ṁ",Ń:"Ń",Ǹ:"Ǹ",Ñ:"Ñ",Ň:"Ň",Ṅ:"Ṅ",Ó:"Ó",Ò:"Ò",Ö:"Ö",Ȫ:"Ȫ",Õ:"Õ",Ṍ:"Ṍ",Ṏ:"Ṏ",Ȭ:"Ȭ",Ō:"Ō",Ṓ:"Ṓ",Ṑ:"Ṑ",Ŏ:"Ŏ",Ǒ:"Ǒ",Ô:"Ô",Ố:"Ố",Ồ:"Ồ",Ỗ:"Ỗ",Ȯ:"Ȯ",Ȱ:"Ȱ",Ő:"Ő",Ṕ:"Ṕ",Ṗ:"Ṗ",Ŕ:"Ŕ",Ř:"Ř",Ṙ:"Ṙ",Ś:"Ś",Ṥ:"Ṥ",Š:"Š",Ṧ:"Ṧ",Ŝ:"Ŝ",Ṡ:"Ṡ",Ť:"Ť",Ṫ:"Ṫ",Ú:"Ú",Ù:"Ù",Ü:"Ü",Ǘ:"Ǘ",Ǜ:"Ǜ",Ǖ:"Ǖ",Ǚ:"Ǚ",Ũ:"Ũ",Ṹ:"Ṹ",Ū:"Ū",Ṻ:"Ṻ",Ŭ:"Ŭ",Ǔ:"Ǔ",Û:"Û",Ů:"Ů",Ű:"Ű",Ṽ:"Ṽ",Ẃ:"Ẃ",Ẁ:"Ẁ",Ẅ:"Ẅ",Ŵ:"Ŵ",Ẇ:"Ẇ",Ẍ:"Ẍ",Ẋ:"Ẋ",Ý:"Ý",Ỳ:"Ỳ",Ÿ:"Ÿ",Ỹ:"Ỹ",Ȳ:"Ȳ",Ŷ:"Ŷ",Ẏ:"Ẏ",Ź:"Ź",Ž:"Ž",Ẑ:"Ẑ",Ż:"Ż",ά:"ά",ὰ:"ὰ",ᾱ:"ᾱ",ᾰ:"ᾰ",έ:"έ",ὲ:"ὲ",ή:"ή",ὴ:"ὴ",ί:"ί",ὶ:"ὶ",ϊ:"ϊ",ΐ:"ΐ",ῒ:"ῒ",ῑ:"ῑ",ῐ:"ῐ",ό:"ό",ὸ:"ὸ",ύ:"ύ",ὺ:"ὺ",ϋ:"ϋ",ΰ:"ΰ",ῢ:"ῢ",ῡ:"ῡ",ῠ:"ῠ",ώ:"ώ",ὼ:"ὼ",Ύ:"Ύ",Ὺ:"Ὺ",Ϋ:"Ϋ",Ῡ:"Ῡ",Ῠ:"Ῠ",Ώ:"Ώ",Ὼ:"Ὼ"},jh=(function(){function O(g,f){this.mode=void 0,this.gullet=void 0,this.settings=void 0,this.leftrightDepth=void 0,this.nextToken=void 0,this.mode="math",this.gullet=new BD(g,f,this.mode),this.settings=f,this.leftrightDepth=0}var d=O.prototype;return d.expect=function(f,w){if(w===void 0&&(w=!0),this.fetch().text!==f)throw new r("Expected '"+f+"', got '"+this.fetch().text+"'",this.fetch());w&&this.consume()},d.consume=function(){this.nextToken=null},d.fetch=function(){return this.nextToken==null&&(this.nextToken=this.gullet.expandNextToken()),this.nextToken},d.switchMode=function(f){this.mode=f,this.gullet.switchMode(f)},d.parse=function(){this.settings.globalGroup||this.gullet.beginGroup(),this.settings.colorIsTextColor&&this.gullet.macros.set("\\color","\\textcolor");var f=this.parseExpression(!1);return this.expect("EOF"),this.settings.globalGroup||this.gullet.endGroup(),f},d.parseExpression=function(f,w){for(var z=[];;){this.mode==="math"&&this.consumeSpaces();var F=this.fetch();if(O.endOfExpression.indexOf(F.text)!==-1||w&&F.text===w||f&&Lp[F.text]&&Lp[F.text].infix)break;var j=this.parseAtom(w);if(j){if(j.type==="internal")continue}else break;z.push(j)}return this.mode==="text"&&this.formLigatures(z),this.handleInfixNodes(z)},d.handleInfixNodes=function(f){for(var w=-1,z,F=0;F<f.length;F++)if(f[F].type==="infix"){if(w!==-1)throw new r("only one infix operator per group",f[F].token);w=F,z=f[F].replaceWith}if(w!==-1&&z){var j,K,ns=f.slice(0,w),Es=f.slice(w+1);ns.length===1&&ns[0].type==="ordgroup"?j=ns[0]:j={type:"ordgroup",mode:this.mode,body:ns},Es.length===1&&Es[0].type==="ordgroup"?K=Es[0]:K={type:"ordgroup",mode:this.mode,body:Es};var _s;return z==="\\\\abovefrac"?_s=this.callFunction(z,[j,f[w],K],[]):_s=this.callFunction(z,[j,K],[]),[_s]}else return f},d.handleSupSubscript=function(f){var w=this.fetch(),z=w.text;this.consume();var F=this.parseGroup(f,!1,O.SUPSUB_GREEDINESS,void 0,void 0,!0);if(!F)throw new r("Expected group after '"+z+"'",w);return F},d.formatUnsupportedCmd=function(f){for(var w=[],z=0;z<f.length;z++)w.push({type:"textord",mode:"text",text:f[z]});var F={type:"text",mode:this.mode,body:w},j={type:"color",mode:this.mode,color:this.settings.errorColor,body:[F]};return j},d.parseAtom=function(f){var w=this.parseGroup("atom",!1,null,f);if(this.mode==="text")return w;for(var z,F;;){this.consumeSpaces();var j=this.fetch();if(j.text==="\\limits"||j.text==="\\nolimits"){if(w&&w.type==="op"){var K=j.text==="\\limits";w.limits=K,w.alwaysHandleSupSub=!0}else if(w&&w.type==="operatorname"&&w.alwaysHandleSupSub){var ns=j.text==="\\limits";w.limits=ns}else throw new r("Limit controls must follow a math operator",j);this.consume()}else if(j.text==="^"){if(z)throw new r("Double superscript",j);z=this.handleSupSubscript("superscript")}else if(j.text==="_"){if(F)throw new r("Double subscript",j);F=this.handleSupSubscript("subscript")}else if(j.text==="'"){if(z)throw new r("Double superscript",j);var Es={type:"textord",mode:this.mode,text:"\\prime"},_s=[Es];for(this.consume();this.fetch().text==="'";)_s.push(Es),this.consume();this.fetch().text==="^"&&_s.push(this.handleSupSubscript("superscript")),z={type:"ordgroup",mode:this.mode,body:_s}}else break}return z||F?{type:"supsub",mode:this.mode,base:w,sup:z,sub:F}:w},d.parseFunction=function(f,w,z){var F=this.fetch(),j=F.text,K=Lp[j];if(!K)return null;if(this.consume(),z!=null&&K.greediness<=z)throw new r("Got function '"+j+"' with no arguments"+(w?" as "+w:""),F);if(this.mode==="text"&&!K.allowedInText)throw new r("Can't use function '"+j+"' in text mode",F);if(this.mode==="math"&&K.allowedInMath===!1)throw new r("Can't use function '"+j+"' in math mode",F);var ns=this.parseArguments(j,K),Es=ns.args,_s=ns.optArgs;return this.callFunction(j,Es,_s,F,f)},d.callFunction=function(f,w,z,F,j){var K={funcName:f,parser:this,token:F,breakOnTokenText:j},ns=Lp[f];if(ns&&ns.handler)return ns.handler(K,w,z);throw new r("No function handler for "+f)},d.parseArguments=function(f,w){var z=w.numArgs+w.numOptionalArgs;if(z===0)return{args:[],optArgs:[]};for(var F=w.greediness,j=[],K=[],ns=0;ns<z;ns++){var Es=w.argTypes&&w.argTypes[ns],_s=ns<w.numOptionalArgs,Hs=ns>0&&!_s||ns===0&&!_s&&this.mode==="math",na=this.parseGroupOfType("argument to '"+f+"'",Es,_s,F,Hs);if(!na){if(_s){K.push(null);continue}throw new r("Expected group after '"+f+"'",this.fetch())}(_s?K:j).push(na)}return{args:j,optArgs:K}},d.parseGroupOfType=function(f,w,z,F,j){switch(w){case"color":return j&&this.consumeSpaces(),this.parseColorGroup(z);case"size":return j&&this.consumeSpaces(),this.parseSizeGroup(z);case"url":return this.parseUrlGroup(z,j);case"math":case"text":return this.parseGroup(f,z,F,void 0,w,j);case"hbox":{var K=this.parseGroup(f,z,F,void 0,"text",j);if(!K)return K;var ns={type:"styling",mode:K.mode,body:[K],style:"text"};return ns}case"raw":{if(j&&this.consumeSpaces(),z&&this.fetch().text==="{")return null;var Es=this.parseStringGroup("raw",z,!0);if(Es)return{type:"raw",mode:"text",string:Es.text};throw new r("Expected raw group",this.fetch())}case"original":case null:case void 0:return this.parseGroup(f,z,F,void 0,void 0,j);default:throw new r("Unknown group type as "+f,this.fetch())}},d.consumeSpaces=function(){for(;this.fetch().text===" ";)this.consume()},d.parseStringGroup=function(f,w,z){var F=w?"[":"{",j=w?"]":"}",K=this.fetch();if(K.text!==F){if(w)return null;if(z&&K.text!=="EOF"&&/[^{}[\]]/.test(K.text))return this.consume(),K}var ns=this.mode;this.mode="text",this.expect(F);for(var Es="",_s=this.fetch(),Hs=0,na=_s,ra;(ra=this.fetch()).text!==j||z&&Hs>0;){switch(ra.text){case"EOF":throw new r("Unexpected end of input in "+f,_s.range(na,Es));case F:Hs++;break;case j:Hs--;break}na=ra,Es+=na.text,this.consume()}return this.expect(j),this.mode=ns,_s.range(na,Es)},d.parseRegexGroup=function(f,w){var z=this.mode;this.mode="text";for(var F=this.fetch(),j=F,K="",ns;(ns=this.fetch()).text!=="EOF"&&f.test(K+ns.text);)j=ns,K+=j.text,this.consume();if(K==="")throw new r("Invalid "+w+": '"+F.text+"'",F);return this.mode=z,F.range(j,K)},d.parseColorGroup=function(f){var w=this.parseStringGroup("color",f);if(!w)return null;var z=/^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(w.text);if(!z)throw new r("Invalid color: '"+w.text+"'",w);var F=z[0];return/^[0-9a-f]{6}$/i.test(F)&&(F="#"+F),{type:"color-token",mode:this.mode,color:F}},d.parseSizeGroup=function(f){var w,z=!1;if(!f&&this.fetch().text!=="{"?w=this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/,"size"):w=this.parseStringGroup("size",f),!w)return null;!f&&w.text.length===0&&(w.text="0pt",z=!0);var F=/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(w.text);if(!F)throw new r("Invalid size: '"+w.text+"'",w);var j={number:+(F[1]+F[2]),unit:F[3]};if(!$E(j))throw new r("Invalid unit: 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**Test** *1*
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    - Latent, i
    - seed,
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    - steps,, 20
    - CFG,, 8.0
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    - denoise,, 1.00
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    - 负面条件, i
    - 潜空间, i
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