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atom.xml
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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<title>Czruby's Blog</title>
<link href="https://www.czruby.eu.org/atom.xml" rel="self"/>
<link href="https://www.czruby.eu.org/"/>
<updated>2025-04-12T15:48:24.077Z</updated>
<id>https://www.czruby.eu.org/</id>
<author>
<name>Czruby</name>
</author>
<generator uri="https://hexo.io/">Hexo</generator>
<entry>
<title>e2课程作业1-SVG规范学习心得</title>
<link href="https://www.czruby.eu.org/posts/d2c92191.html"/>
<id>https://www.czruby.eu.org/posts/d2c92191.html</id>
<published>2025-04-12T13:29:21.000Z</published>
<updated>2025-04-12T15:48:24.077Z</updated>
<content type="html"><![CDATA[<h1 id="融媒体SVG交互设计技术规范"><a href="#融媒体SVG交互设计技术规范" class="headerlink" title="融媒体SVG交互设计技术规范"></a>融媒体SVG交互设计技术规范</h1><ul><li>缩略语<br> 下列缩略语适用于本文件。<ul><li>CSS:层叠样式表(Cascading Style Sheets)</li><li>HTML:超文本标记语言(Hyper Text Markup Language)</li><li>SVG:可伸缩矢量图形(Scalable Vector Graphics)</li><li>XML:可扩展标记语言(Extensible Markup Language)</li></ul></li><li>动画设计宜优先选择具备缓动函数的动画表达</li><li>应采用防误触设计,确保动画不出现预期外的重复执行、无法执行等状况。</li></ul><h1 id="学习心得"><a href="#学习心得" class="headerlink" title="学习心得"></a>学习心得</h1><ol><li>向微信公众号后台复制代码的时候,一定要使用代码编辑工具直接粘贴,否则预览界面会直接运行svg代码,影响展示效果。<ul><li>我在之前复制svg代码的时候选择的是在开发者工具里面使用替换段落的方法进行复制,可能会影响到svg动画的状态,影响最后的预览效果</li></ul></li><li>发布都使用定时发布,这样既可以直接预览,也可以在有问题的时候即使停止发布。<ul><li>目前我使用的时候都是定时发布</li></ul></li></ol>]]></content>
<summary type="html">e2课程作业</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="新媒体" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E6%96%B0%E5%AA%92%E4%BD%93/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="e2" scheme="https://www.czruby.eu.org/tags/e2/"/>
<category term="svg" scheme="https://www.czruby.eu.org/tags/svg/"/>
<category term="新媒体" scheme="https://www.czruby.eu.org/tags/%E6%96%B0%E5%AA%92%E4%BD%93/"/>
</entry>
<entry>
<title>软工计算一</title>
<link href="https://www.czruby.eu.org/posts/254c624.html"/>
<id>https://www.czruby.eu.org/posts/254c624.html</id>
<published>2024-06-13T02:27:23.000Z</published>
<updated>2024-06-17T11:18:12.043Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<div class="hbe hbe-content"> <div class="hbe hbe-input hbe-input-xray"> <input class="hbe hbe-input-field hbe-input-field-xray" type="password" id="hbePass"> <label class="hbe hbe-input-label hbe-input-label-xray" for="hbePass"> <span class="hbe hbe-input-label-content hbe-input-label-content-xray">您好, 这里需要密码.</span> </label> <svg class="hbe hbe-graphic hbe-graphic-xray" width="300%" height="100%" viewBox="0 0 1200 60" preserveAspectRatio="none"> <path d="M0,56.5c0,0,298.666,0,399.333,0C448.336,56.5,513.994,46,597,46c77.327,0,135,10.5,200.999,10.5c95.996,0,402.001,0,402.001,0"></path> <path d="M0,2.5c0,0,298.666,0,399.333,0C448.336,2.5,513.994,13,597,13c77.327,0,135-10.5,200.999-10.5c95.996,0,402.001,0,402.001,0"></path> </svg> </div> </div></div><script data-pjax src="/lib/hbe.js"></script><link href="/css/hbe.style.css" rel="stylesheet" type="text/css">]]></content>
<summary type="html">软工一期末复习资料</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="软工一" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E8%BD%AF%E5%B7%A5%E4%B8%80/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
<category term="软工一" scheme="https://www.czruby.eu.org/tags/%E8%BD%AF%E5%B7%A5%E4%B8%80/"/>
</entry>
<entry>
<title>微积分二复习</title>
<link href="https://www.czruby.eu.org/posts/e32f1441.html"/>
<id>https://www.czruby.eu.org/posts/e32f1441.html</id>
<published>2024-04-20T03:39:42.000Z</published>
<updated>2024-06-13T02:27:02.302Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<div class="hbe hbe-content"> <div class="hbe hbe-input hbe-input-xray"> <input class="hbe hbe-input-field hbe-input-field-xray" type="password" id="hbePass"> <label class="hbe hbe-input-label hbe-input-label-xray" for="hbePass"> <span class="hbe hbe-input-label-content hbe-input-label-content-xray">您好, 这里需要密码.</span> </label> <svg class="hbe hbe-graphic hbe-graphic-xray" width="300%" height="100%" viewBox="0 0 1200 60" preserveAspectRatio="none"> <path d="M0,56.5c0,0,298.666,0,399.333,0C448.336,56.5,513.994,46,597,46c77.327,0,135,10.5,200.999,10.5c95.996,0,402.001,0,402.001,0"></path> <path d="M0,2.5c0,0,298.666,0,399.333,0C448.336,2.5,513.994,13,597,13c77.327,0,135-10.5,200.999-10.5c95.996,0,402.001,0,402.001,0"></path> </svg> </div> </div></div><script data-pjax src="/lib/hbe.js"></script><link href="/css/hbe.style.css" rel="stylesheet" type="text/css">]]></content>
<summary type="html">本文为微积分2知识点梳理</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/tags/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
</entry>
<entry>
<title>需求课笔记</title>
<link href="https://www.czruby.eu.org/posts/4b727690.html"/>
<id>https://www.czruby.eu.org/posts/4b727690.html</id>
<published>2024-03-13T10:57:16.000Z</published>
<updated>2024-06-18T07:23:50.457Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<div class="hbe hbe-content"> <div class="hbe hbe-input hbe-input-xray"> <input class="hbe hbe-input-field hbe-input-field-xray" type="password" id="hbePass"> <label class="hbe hbe-input-label hbe-input-label-xray" for="hbePass"> <span class="hbe hbe-input-label-content hbe-input-label-content-xray">您好, 这里需要密码.</span> </label> <svg class="hbe hbe-graphic hbe-graphic-xray" width="300%" height="100%" viewBox="0 0 1200 60" preserveAspectRatio="none"> <path d="M0,56.5c0,0,298.666,0,399.333,0C448.336,56.5,513.994,46,597,46c77.327,0,135,10.5,200.999,10.5c95.996,0,402.001,0,402.001,0"></path> <path d="M0,2.5c0,0,298.666,0,399.333,0C448.336,2.5,513.994,13,597,13c77.327,0,135-10.5,200.999-10.5c95.996,0,402.001,0,402.001,0"></path> </svg> </div> </div></div><script data-pjax src="/lib/hbe.js"></script><link href="/css/hbe.style.css" rel="stylesheet" type="text/css">]]></content>
<summary type="html">有东西被加密了, 请输入密码查看.</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="需求" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E9%9C%80%E6%B1%82/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
<category term="需求" scheme="https://www.czruby.eu.org/tags/%E9%9C%80%E6%B1%82/"/>
</entry>
<entry>
<title>需求与商业模式创新</title>
<link href="https://www.czruby.eu.org/posts/b7bd168.html"/>
<id>https://www.czruby.eu.org/posts/b7bd168.html</id>
<published>2024-03-11T12:55:19.000Z</published>
<updated>2024-04-26T14:29:11.592Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<summary type="html">需求与商业模式创新课的知识点和相关内容</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="需求" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E9%9C%80%E6%B1%82/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
<category term="需求" scheme="https://www.czruby.eu.org/tags/%E9%9C%80%E6%B1%82/"/>
</entry>
<entry>
<title>微积分2小黄书题目收集</title>
<link href="https://www.czruby.eu.org/posts/a564d22c.html"/>
<id>https://www.czruby.eu.org/posts/a564d22c.html</id>
<published>2024-03-11T10:49:13.000Z</published>
<updated>2024-04-26T14:27:29.336Z</updated>
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<summary type="html">本文为学习微积分2中小黄书题目收集</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/tags/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
</entry>
<entry>
<title>记录bmap的一个开发中可能遇到的问题</title>
<link href="https://www.czruby.eu.org/posts/bd3111ab.html"/>
<id>https://www.czruby.eu.org/posts/bd3111ab.html</id>
<published>2024-03-03T03:19:43.000Z</published>
<updated>2024-03-03T03:38:37.132Z</updated>
<content type="html"><![CDATA[<h1 id="问题描述">问题描述</h1><p>在使用Vue3框架结合echarts实现百度地图并标点时,出现了在本地运行正常,但是一打包部署便出现页面白屏的问题</p><h1 id="Debug过程">Debug过程</h1><ul><li>首先考虑到是echarts脚本未加载出来<ul><li>经过调试发现echarts表格dom以建立,标题也显示,就是地图空白</li></ul></li><li>考虑bmap脚本加载失败<ul><li>检查发现控制台bmap脚本成功加载</li></ul></li><li>查找互联网同类型错误<ul><li>发现出现情况均与本次遇到情况不符</li></ul></li><li>最后将代码回滚到可以正常打包运行的版本,一步步调试终于发现了问题所在</li></ul><h1 id="问题解决">问题解决</h1><h2 id="原代码">原代码</h2><figure class="highlight ts"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> <span class="string">'echarts/extension-src/bmap/bmap'</span></span><br></pre></td></tr></table></figure><h2 id="改正后代码">改正后代码</h2><figure class="highlight ts"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> <span class="string">'echarts/extension/bmap/bmap'</span></span><br></pre></td></tr></table></figure><h2 id="复盘解释">复盘解释</h2><ul><li>猜想1:原代码是通过copilot生成的,可能是早期写法</li><li>猜想2: extension-src包下内容是ts写的,可能在打包后有更严格的代码规范检查,导致运行出错,extension是js代码,要求不高</li></ul>]]></content>
<summary type="html">本文记录在2024年寒假参加社会实践活动中制作行迹图时遇到的一个bug以及解决方法</summary>
<category term="编程学习" scheme="https://www.czruby.eu.org/categories/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/"/>
<category term="debug" scheme="https://www.czruby.eu.org/categories/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/debug/"/>
<category term="编程学习" scheme="https://www.czruby.eu.org/tags/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/"/>
<category term="bmap" scheme="https://www.czruby.eu.org/tags/bmap/"/>
<category term="debug" scheme="https://www.czruby.eu.org/tags/debug/"/>
</entry>
<entry>
<title>耍一耍逆向</title>
<link href="https://www.czruby.eu.org/posts/12e37b44.html"/>
<id>https://www.czruby.eu.org/posts/12e37b44.html</id>
<published>2024-01-09T09:02:28.000Z</published>
<updated>2024-01-11T15:33:32.000Z</updated>
<content type="html"><![CDATA[<h1 id="CPU-寄存器以及数据传输指令">CPU 寄存器以及数据传输指令</h1><h2 id="寄存器">寄存器</h2><p>EAX、ECX、EDX、EBX、ESP、EBP、ESI、EDI 和 EIP</p><ul><li>EAX (accumulator,累加器)</li><li>EBX (Base index,基址寄存器)</li><li>ECX (counter,计数器)</li><li>EDX (data):EDX 通常用于存储乘积的部分位数以及除法的余数,同时也 能存储起始内存地址。</li><li>EBP (base pointer):EBP 指向一个内存地址,主要作为一个函数中的参 数以及变量的基址。</li><li>EDI (destination index):EDI 常用于字符串指令,指向目标字符串。</li><li>ESI (source index):ESI 常用于字符串指令,指向初始字符串。</li><li>EIP:存储即将执行的下一条指令的地址。</li><li>ESP:存储栈顶地址。</li></ul><p>还有16 位和 8位寄存器,作为 32位和 16 位寄存器的一部分。AX 就是最<br>后 16 位,AH 是 AX 的前面 8 位,AL 是 AX 的后八位。</p><p><img src="https://img.czruby.eu.org/imgs/%E8%80%8D%E4%B8%80%E8%80%8D%E9%80%86%E5%90%91/image-20240109170801211.png" alt="image-20240109170801211"></p><p>最基本的是 BYTE占用1个字节(8 bit)内 存,WORD 占用 2 个字节(16 bit)内存,DWORD 占用 4 个字节(32bit)内存、 QWORD 占用 8 个字节(64 bit)内存。</p><h2 id="数据传输指令">数据传输指令</h2><ul><li><p>数据传输指令 MOV:将其始操作单元(src)的内容复制到目标单元(dest)。</p><p>MOV dest src</p><p>(“OFFSET”这个词说明传递的是内存地址而不是内存地址上的值。)</p></li><li><p>XCHG A, B指令将A与B的值互换</p></li><li><p>push 压栈</p><ul><li>IDA发现的函数在调用前一般需要传递参数,大部分是通过PUSH指令(仅指32位)。PUSH指令将这些值存放到栈上,这些值被称为函数参数。</li></ul></li><li><p>pop 出栈</p></li><li><p>PUSHA :按如下顺序将所有寄存器的值保存到栈上。<br>EAX、ECX、EDX、EBX、original ESP、EBP、ESI 和 EDI</p></li><li><p>LEA 即 LOAD EFFECTIVE ADDRESS。“LEA A, B”指令将 B 的地址传递给 A。该指令不会获取B存储的内容,只会传递地址或者后一个操作数的运算结果(外加中括号)。这种方法普遍运用于获取变量参数的地址</p><ul><li>所以,LEA EAX,[EBP - 0C]将 EBP-0C 计算的结果也就是这个局部变量的地址传给 EAX。而 MOV EAX, [EBP - 0C]除了计算 EBP - 0C 表达式结果得到变量地址外,之后还会读取该地址上存储的值并将其传给EAX。</li></ul></li><li><p>MOVSX 和 MOVZX 都会把一个字节填充到一个寄存器当中。MOVZX 将高位用0填充,MOVSX则会考虑符号。如果这个字节小于等于 0x7f,那么就用0填充高 位,如果大于等于0x80,那么高位就用0xff填充</p></li></ul><h2 id="整数运算与逻辑运算指令">整数运算与逻辑运算指令</h2><p><strong>整数运算</strong></p><ul><li>ADD A,B指令将B 的值与A 相加,并将结果保存到 A</li><li>SUB A,B 这个指令和ADD 指令一致,只不过它是对两个操作数相减,最后<br>结果保存到A。SUB允许的操作数组合和ADD是一样的。</li><li>INC A 和 DEC A指令对一个寄存器或者内存值±1。这是加减操作的一个<br>特例。一般这两个指令用来对计数器±1。</li><li>IMUL 是带符号整数乘法指令,这里介绍两种使用方式。IMUL A,B 和 IMUL A,B,C。第一种对A和B相乘,结果返回给A。第二种方式对B和C相乘,将结果返回给A。<ul><li>在这两种方式下,A只能是一个寄存器,B可以是寄存器或者内存值(第一种方式下可以是常数),而C只能是个常数。</li></ul></li><li>IDIV A指令中,仅仅指定了除法的除数。被除数没有指定,因为存放的地<br>方是固定的。<ul><li>在 32 位运算中,EDX 和 EAX 组成一个 64 位数,EDX 在高位,EAX 在低位。这个64位数除以A后,商返回给EAX,余数返回给EDX。</li></ul></li></ul><p><strong>逻辑运算</strong></p><ul><li><p>AND A,B 对 A 和 B 进行与运算,将最终结果保存到 A。对于 OR 和 XOR 运算也是一样的。指令对2个操作数进行运算,结果保存到A。每一种运算都有一个真值表。</p><ul><li>A和B可以是寄存器或者内存值,但同一条指令中A和B都是内存值是不允许的。</li></ul></li><li><p>NOT A将A 所有的位取反然后保存到A。Python 中按位取反是“~”符号。 对于0101,结果会对每一位取反。所有的0反转为1,1反转为0。</p></li><li><p>NEG A将A转变成-A。NEG运算和按位取反不一样,按位取反多减了1。(相当于按位取反后加1)</p></li><li><p>SHL A,B;SHR A,B 中 A 是一个寄存器或内存值,B 是一个常数或者一个 8<br>位寄存器。这两个指令将操作数按位向左或向右偏移,缺少的位用0来填补。</p></li><li><p>还有2个类似的指令ROL和ROR,指令会将每个比特偏移一定的位置,但是<br>一端超出的字节会重新返回给另一端。指令只对每个比特进行了偏移但不改变内容</p><p><img src="https://img.czruby.eu.org/imgs/%E8%80%8D%E4%B8%80%E8%80%8D%E9%80%86%E5%90%91/image-20240109180318661.png" alt="image-20240109180318661"></p></li></ul><h2 id="流程控制指令">流程控制指令</h2><p><strong>无条件跳转指令</strong></p><ul><li>JMP A指令当中,A是程序作者期望的无条件转向的一个内存地址。<ul><li>跳转距离的计算方式:终点指令起始地址 - 起点指令起始地址 – 5</li><li>5是跳转指令的字节数,结果是0x300也就是长跳转指令操作码E9后面的dword。</li></ul></li><li>JMP SHORT 指令是一个两字节的短程跳转指令,只能向前或者往回跳转。<ul><li>第一个字节EB是跳转操作码(OPCODE)。跳转的方向由第二个字节的值确定。</li><li>这种跳转是有距离限制的。</li></ul></li></ul><p><strong>有条件跳转指令</strong></p><ul><li><p>CMP A,B 对A 和B 进行比较,根据比较的结果程序将会进行某些操<br>作,如果是另一种结果则进行不同的操作。一般来说,比较会改变标志寄存器EFLAGS的状态,根据这个状态条件跳转指令会决定接下来如何执行。</p><ul><li>如果比较触发标志寄存器 EFLAGS中的Z或者说Zero标记,将执行跳转。在之前的例子中,如果EAX和EBX 相等 将执行跳转。CMP指令类似于SUB指令,只不过不保存计算结果。</li></ul><p><img src="https://img.czruby.eu.org/imgs/%E8%80%8D%E4%B8%80%E8%80%8D%E9%80%86%E5%90%91/image-20240109185441653.png" alt="image-20240109185441653"></p></li><li><p>CALL指令用来调用一个函数。RET指令用来返回调用这个函数的指令的下<br>面一条指令处。</p></li></ul><h1 id="静态分析入门">静态分析入门</h1><p>读 READ®、写 WRITE(W)以及执行 EXECUTION(X)D 和 L 两列分别对应于 DEBUGGER(调试器)和 LOADER(加载器)</p><h1 id="标志寄存器">标志寄存器</h1><h2 id="CARRY-FLAG">CARRY FLAG</h2><ul><li>当运算结果为负或者相加超 过数值上限的时候会被触发</li><li>无符号数运算错误将触发CF标志。</li></ul><h2 id="OVERFLOW-FLAG">OVERFLOW FLAG</h2><ul><li>OF代表有符号数计算出错。</li></ul><h2 id="SIGNED-FLAG">SIGNED FLAG</h2><ul><li>SF标志比较简单。任何操作的结果如果是一个负数就会触发SF。SF只表示<br>结果的符号,并不表示结果是对还是错。</li></ul><h2 id="ZERO-FLAG">ZERO FLAG</h2><ul><li>ZF在以下条件下将会触发:在内部实际是减法的比较指令中,两个操作数<br>相同;增或减导致结果为0;相减结果为0。</li></ul><p><img src="https://img.czruby.eu.org/imgs/%E8%80%8D%E4%B8%80%E8%80%8D%E9%80%86%E5%90%91/image-20240110113809542.png" alt="image-20240110113809542"></p><h1 id="程序脱壳处理">程序脱壳处理</h1><ul><li><p>STUB: 存根入口(STUB ENTRY POINT)</p></li><li><p>OEP:原始程序入口(ORIGINAL ENTRY POINT)</p></li><li><p>IAT 或者说 IMPORT ADDRESS TABLE 是位于可执行文件中用于程序运行的表格。</p></li></ul>]]></content>
<summary type="html"><h1 id="CPU-寄存器以及数据传输指令">CPU 寄存器以及数据传输指令</h1>
<h2 id="寄存器">寄存器</h2>
<p>EAX、ECX、EDX、EBX、ESP、EBP、ESI、EDI 和 EIP</p>
<ul>
<li>EAX (accumulator,累</summary>
<category term="编程学习" scheme="https://www.czruby.eu.org/categories/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/"/>
<category term="逆向" scheme="https://www.czruby.eu.org/categories/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/%E9%80%86%E5%90%91/"/>
<category term="编程学习" scheme="https://www.czruby.eu.org/tags/%E7%BC%96%E7%A8%8B%E5%AD%A6%E4%B9%A0/"/>
<category term="逆向" scheme="https://www.czruby.eu.org/tags/%E9%80%86%E5%90%91/"/>
</entry>
<entry>
<title>计算系统基础期末复习</title>
<link href="https://www.czruby.eu.org/posts/78ef127a.html"/>
<id>https://www.czruby.eu.org/posts/78ef127a.html</id>
<published>2024-01-07T03:55:49.000Z</published>
<updated>2024-01-11T15:33:40.000Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<summary type="html">计算系统基础期末复习资料,通过整合PPT和添加作者思考总结而成</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="计算系统基础" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E8%AE%A1%E7%AE%97%E7%B3%BB%E7%BB%9F%E5%9F%BA%E7%A1%80/"/>
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</entry>
<entry>
<title>柯西-施瓦茨积分不等式的证明及应用</title>
<link href="https://www.czruby.eu.org/posts/cb54442f.html"/>
<id>https://www.czruby.eu.org/posts/cb54442f.html</id>
<published>2024-01-01T09:08:07.000Z</published>
<updated>2024-01-11T13:19:54.000Z</updated>
<content type="html"><![CDATA[<h1 id="柯西-施瓦茨积分不等式的证明及应用">柯西-施瓦茨积分不等式的证明及应用</h1><p><strong>Author:</strong> [扬骁]</p><p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/620322246">https://zhuanlan.zhihu.com/p/620322246</a>]</p><ul><li><strong>定理</strong> 设 $f(x),g(x)$ 在区间[a,b]上连续,则</li></ul><p>$(\int_{a}^{b}f(x)g(x)dx)²≤\int_{a}^{b}f²(x)dx\int_{a}^{b}g²(x)dx$</p><p>等号成立的必要条件是存在常数k使得 $f(x)=kg(x)$</p><h2 id="证明"><strong>证明</strong></h2><h3 id="1-定积分的定义">1.定积分的定义</h3><p>对区间[a,b]等分,等分点为 $x_i=a+\frac{b-a}{n}i,(i=1,2,3,\cdot\cdot\cdot,n)$ 由定积分定义可知</p><p>$\int_{a}^{b}f(x)g(x)dx=\lim_{n \rightarrow \infty}{\sum_{i=1}^{n}{f(x_i)g(x_i)\cdot \frac{b-a}{n}}}\ \int_{a}^{b}f^2(x)dx=\lim_{n \rightarrow \infty}{\sum_{i=1}^{n}{f^2(x_i)\cdot \frac{b-a}{n}}}\ \int_{a}^{b}g^2(x)dx=\lim_{n \rightarrow \infty}{\sum_{i=1}^{n}{g^2(x_i)\cdot \frac{b-a}{n}}}\$</p><p>由柯西不等式 ${\sum_{i=1}^{n}{f(x_i)g(x_i)}}\leq\left( \sum_{i=1}^{n}{f^2(x_i)} \right)\cdot\left( {\sum_{i=1}^{n}{g^2(x_i)}} \right)$</p><p>,以及极限的保号线,定理成立</p><h3 id="2-利用判别式Δ">2.利用判别式Δ</h3><p>对任意t∈R,都有 $[f(x)+tg(x)]²≥0$ ,则 $\int_{a}^{b}[f(x)+tg(x)]²dx≥0$</p><p>挤对任意t∈R有:</p><p>$\int_{a}^{b}[f(x)+tg(x)]²dx=t²\int_{a}^{b}g²(x)dx+2t\int_{a}^{b}f(x)g(x)dx+\int_{a}^{b}f²(x)dx≥0$</p><p>因此上述关于t的一元二次方程判别式Δ≤0,故</p><p>$Δ=[2\int_{a}^{b}f(x)g(x)dx]²-4\int_{a}^{b}g²(x)dx\int_{a}^{b}f²(x)dx≤0$</p><p>即$(\int_{a}^{b}f(x)g(x)dx)²≤\int_{a}^{b}f²(x)dx\int_{a}^{b}g²(x)dx$</p><h3 id="3-构造函数">3.构造函数</h3><p>令 $F(x)=\int_{a}^{x}f²(t)dt\int_{a}^{x}g²(t)dt-[\int_{a}^{x}f(t)g(t)dt]²,F (a)=0$</p><p>$F '(x)=f²(x)\int_{a}^{x}g²(t)dt+g²(x)\int_{a}^{x}f²(t)dt-2f(x)g(x)\int_{a}^{x}f(t)g(t)dt\=\int_{a}^{x}[f²(x)g²(t)dt-2f(x)g(x)f(t)g(t)+g²(x)f²(t)]dt\=\int_{a}^{x}[f(x)g(t)-g(x)f(t)]²dt≥0$</p><p>故F(x)在x≥a上单调递增,F(x)≥F(a)=0,F(b) $\geq$ F(a)=0即证</p><h3 id="4-二重积分">4.二重积分</h3><p>$\int_{a}^{b}f²(x)dx\int_{a}^{b}g²(x)dx-[\int_{a}^{b}f(x)g(x)dx]²\=\int_{a}^{b}f²(x)dx\int_{a}^{b}g²(y)dy-\int_{a}^{b}f(x)g(x)dx\int_{a}^{b}f(y)g(y)dy\=\int_{a}^{b}\int_{a}^{b}[f²(x)g²(y)-f(x)g(x)f(y)g(y)]dxdy\=\frac{1}{2}\int_{a}^{b}\int_{a}^{b}[f²(x)g²(y)-2f(x)g(x)f(y)g(y)+f²(y)g²(x)]dxdy\=\frac{1}{2}\int_{a}^{b}\int_{a}^{b}[f(x)g(y)-f(y)g(x)]²dxdy≥0$</p><h2 id="应用"><strong>应用</strong></h2><h3 id="1-在二元函数上的推广">1.在二元函数上的推广</h3><p>设二元函数f(x,y),g(x,y)在区域D内可积,有</p><p>$(∫∫_Df(x,y)g(x,y)dσ)²≤(∫∫_Df²(x,y)dσ)(∫∫_Dg²(x,y)dσ)$</p><ul><li><strong>证明</strong></li></ul><p>同样利用一元二次方程的判别式证明</p><p>$∫∫_D(f(x,y)+tg(x,y))^2dσ\geq0,t\in R\ ∫∫_Df^2(x,y)dσ+2t∫∫_Df(x,y)g(x,y)dσ=t^2∫∫_Dg^2(x,y)dσ\geq0$</p><p>$\Delta\leq0$ 得证</p><h3 id="2-holder不等式">2.holder不等式</h3><p>设f(x),g(x)在[a,b]上可积,且f(x)>0,g(x)>0,则有</p><p>$\int_{a}^{b}f(x)g(x)dx≤(\int_{a}^{b}f^p(x)dx)^{\frac{1}{p}}(\int_{a}^{b}g^q(x)dx)^{\frac{1}{q}}$</p><p>其中 $p,q>0,{\frac{1}{p}}+{\frac{1}{q}}=1$</p><h3 id="3-习题">3.习题</h3><p>1.设f(x)在[0,1]上连续,且1≤f(x)≤3,证明: $1≤\int_{0}^{1}f(x)dx\int_{0}^{1}\frac{1}{f(x)}dx≤\frac{4}{3}$</p><p>2.证明: $\int_{o}^{\frac{\pi}{2}}\sqrt{cosx}dx\leq \sqrt{\frac{\pi}{2}}$</p><p>3.已知f(x) $\geq$ 0, 在区间[a,b]上连续, $\int_{a}^{b}f(x)dx=1$ , $k$ 为任意实数,试证明:</p><p>$(\int_{a}^{b}f(x)coskxdx)^2+(\int_{a}^{b}f(x)sinkxdx)^2\leq1\$</p><p>4.f(x) >0, 在区间[0,1]上连续,证明: $\frac{\int_{0}^{1}f^3(x)dx}{\int_{0}^{1}f^2(x)dx}\geq \frac{\int_{0}^{1}f^2(x)dx}{\int_{0}^{1}f(x)dx}$</p><p>5.f(x) 在区间[0,1]上二阶连续可导,且 $f(0)=f(1)=f’(1)=1$ ,求证: $\int_{0}^{1}(f’'(x))^2dx\geq4$</p><p>6.设函数f(x)在区间[a,b]上有连续的导函数,且f(a)=0,证明:</p><p>${\int_{a}^{b} f^2(x)dx\leq\frac{(b-a)^2}{2}\int_{a}^{b}f’^2(x)dx.}\$</p><h3 id="4-练习证明答案">4.练习证明答案</h3><p>1.根据柯西一施瓦茨不等式,有</p><p>$\int_{0}^{1}f(x)dx\int_{0}^{1}\frac{1}{f(x)}dx\geq(\int_{0}^{1}\sqrt{f(x)\frac{1}{f(x)}}dx)^2=1\$ 由基本不等式可知</p><p>$\int_{0}^{1}f(x)dx\int_{0}^{1}\frac{3}{f(x)}dx\leq\frac{1}{4}(\int_{0}^{1}f(x)dx+\int_{0}^{1}\frac{3}{f(x)}dx)^2$</p><p>1≤f(x)≤3可得 $(f(x)-1)(f(x)-3)\leq0$ ,则</p><p>$\frac{3}{f(x)}+f(x)\leq4\Rightarrow\int_{0}^{1}(\frac{3}{f(x)}+f(x))dx\leq4$ 得证</p><p>2.根据柯西一施瓦茨不等式,有</p><p>$(\int_{o}^{\frac{\pi}{2}}\sqrt{cosx}dx)^2\leq\int_{o}^{\frac{\pi}{2}}(\sqrt{cosx})^2dx\cdot\int_{o}^{\frac{\pi}{2}}1^2dx=\frac{\pi}{2}\$</p><p>3.利用柯西一施瓦茨不等式,有</p><p>$(\int_{a}^{b}f(x)coskxdx)^2\leq\int_{a}^{b}f(x)dx\int_{a}^{b}f(x)cos^2kxdx=\int_{a}^{b}f(x)cos^2kxdx\$ 同理可得 $(\int_{a}^{b}f(x)sinkxdx)^2\leq\int_{a}^{b}f(x)dx\int_{a}^{b}f(x)sin^2kxdx=\int_{a}^{b}f(x)sin^2kxdx\$ 两式相加得证</p><p>4.由柯西一施瓦茨不等式,有</p><p>$\int_{0}^{1}f^3(x)dx\int_{0}^{1}f(x)dx=\int_{0}^{1}(f^{\frac{3}{2}}(x))^2dx\int_{0}^{1}(f^{\frac{1}{2}}(x))^2dx\\geq\int_{0}^{1}(f^{\frac{3}{2}}(x)dxf^{\frac{1}{2}}(x)dx)^2=(\int_{0}^{1}f^2(x)dx)^2$</p><p>得证</p><p>5.给定一个函数g(x),由柯西一施瓦茨不等式:</p><p>$\int_{0}^{1}(f’‘(x))^2dx\int_{0}^{1}(g(x))^2dx\geq(\int_{0}^{1}(f’‘(x)g(x))dx)^2=(\int_{0}^{1}g(x)df’(x))^2\ =([g(x)f’(x)]<em>0^1-\int</em>{0}^{1}f’(x)d(g(x))=(g(1)-\int_{0}^{1}f’(x)g’(x)dx)^2\ =(g(1)-\int_{0}^{1}g’(x)d(f(x)))^2=(g(1)-[g’(x)f(x)]<em>0^1+\int</em>{0}^{1}f(x)d(g’(x)))^2\ =(g(1)+\int_{0}^{1}f(x)g’'(x)dx)^2$</p><p>然后令g(x)=x+k, 代入上式有 $\int_{0}^{1}(f”(x))dxz\geq\frac{(1+k)^2}{ \int_{0}^{1}(x+k)^2dx}=\frac{3(k+1)^2}{3k^2+3k+1}$ ,</p><p>此时当 $\frac{3(k+1)^2}{3k^2+3k+1}=4$ 则k=-1/3,即当g(x)=x-1/3等式成立.</p><p>6.证明:由牛顿一莱布尼茨公式,有</p><p>$f(x)-f(a)=\int_{a}^{x} f’(t)dt.\$</p><p>根据柯西一施瓦茨不等式,注意到f(a)=0,有</p><p>$f^2(x)=(\int_{a}^{x} f’(t)dt)^2<\leq\int_{a}^{x} f’^2(t)dt\cdot\int_{a}^{x} 1^2dt\=(x-a)\int_{a}^{x} f’^2(t)dt\leq(x-a)\int_{a}^{b} f’^2(t)dt.\$</p><p>再由定积分性质(保序性)知,</p><p>$\int_{a}^{b} f^2(x)dx\leq\int_{a}^{b}(x-a)dx\cdot\int_{a}^{b} f’^2(t)dt.\$</p><p>而 $\int_{a}^{b}(x-a)dx\cdot\int_{a}^{b} f’^2(t)dt=\frac{(b-a)^2}{2}\int_{a}^{b}f’^2(x)dx.$</p>]]></content>
<summary type="html"><h1 id="柯西-施瓦茨积分不等式的证明及应用">柯西-施瓦茨积分不等式的证明及应用</h1>
<p><strong>Author:</strong> [扬骁]</p>
<p><strong>Link:</strong> [<a href="https://zhuanlan.</summary>
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</entry>
<entry>
<title>微积分往年好题</title>
<link href="https://www.czruby.eu.org/posts/e88ac744.html"/>
<id>https://www.czruby.eu.org/posts/e88ac744.html</id>
<published>2024-01-01T07:17:59.000Z</published>
<updated>2024-01-11T15:33:26.000Z</updated>
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<summary type="html">有东西被加密了, 请输入密码查看.</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/tags/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
<category term="期末" scheme="https://www.czruby.eu.org/tags/%E6%9C%9F%E6%9C%AB/"/>
</entry>
<entry>
<title>马原复习</title>
<link href="https://www.czruby.eu.org/posts/290ad7f9.html"/>
<id>https://www.czruby.eu.org/posts/290ad7f9.html</id>
<published>2023-12-30T14:18:46.000Z</published>
<updated>2024-01-11T13:19:54.000Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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<div class="hbe hbe-content"> <div class="hbe hbe-input hbe-input-xray"> <input class="hbe hbe-input-field hbe-input-field-xray" type="password" id="hbePass"> <label class="hbe hbe-input-label hbe-input-label-xray" for="hbePass"> <span class="hbe hbe-input-label-content hbe-input-label-content-xray">您好, 这里需要密码.</span> </label> <svg class="hbe hbe-graphic hbe-graphic-xray" width="300%" height="100%" viewBox="0 0 1200 60" preserveAspectRatio="none"> <path d="M0,56.5c0,0,298.666,0,399.333,0C448.336,56.5,513.994,46,597,46c77.327,0,135,10.5,200.999,10.5c95.996,0,402.001,0,402.001,0"></path> <path d="M0,2.5c0,0,298.666,0,399.333,0C448.336,2.5,513.994,13,597,13c77.327,0,135-10.5,200.999-10.5c95.996,0,402.001,0,402.001,0"></path> </svg> </div> </div></div><script data-pjax src="/lib/hbe.js"></script><link href="/css/hbe.style.css" rel="stylesheet" type="text/css">]]></content>
<summary type="html">马原复习资料</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="马原" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E9%A9%AC%E5%8E%9F/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="加密" scheme="https://www.czruby.eu.org/tags/%E5%8A%A0%E5%AF%86/"/>
<category term="期末" scheme="https://www.czruby.eu.org/tags/%E6%9C%9F%E6%9C%AB/"/>
<category term="马原" scheme="https://www.czruby.eu.org/tags/%E9%A9%AC%E5%8E%9F/"/>
</entry>
<entry>
<title>实用网站收录</title>
<link href="https://www.czruby.eu.org/posts/502e2c39.html"/>
<id>https://www.czruby.eu.org/posts/502e2c39.html</id>
<published>2023-12-28T15:40:28.000Z</published>
<updated>2024-01-11T13:19:54.000Z</updated>
<content type="html"><![CDATA[<h1 id="呢喃的一些网站">呢喃的一些网站</h1><ol><li><a href="https://ndwy.nju.edu.cn/dztml/admin">南京大学五育项目管理系统</a><br><img src="https://img.czruby.eu.org/imgs/%E5%AE%9E%E7%94%A8%E7%BD%91%E7%AB%99%E6%94%B6%E5%BD%95/image.png" alt="Alt text"></li><li><a href="https://youth.nju.edu.cn/tw/admin">南京大学-学生第二课堂</a><br><img src="https://img.czruby.eu.org/imgs/%E5%AE%9E%E7%94%A8%E7%BD%91%E7%AB%99%E6%94%B6%E5%BD%95/image-1.png" alt="Alt text"></li><li><a href="https://p.nju.edu.cn">网络认证</a></li><li><a href="https://xk.nju.edu.cn/">选课系统</a></li><li><a href="https://itsc.nju.edu.cn/main.htm">信息化建设管理中心</a></li><li><a href="https://ehall.nju.edu.cn/ywtb-portal/official/index.html#/hall">南京大学网上办事服务大厅</a></li><li><a href="https://box.nju.edu.cn/">南大云盘</a></li><li><a href="https://table.nju.edu.cn/">南大表格</a></li><li><a href="https://mirror.nju.edu.cn/">南大镜像</a></li><li><a href="http://test.nju.edu.cn/">南大测速</a></li><li><a href="http://elite.nju.edu.cn/jiaowu/exit.do">教务系统</a></li><li><a href="http://eol.nju.edu.cn/">大外网</a></li></ol>]]></content>
<summary type="html"><h1 id="呢喃的一些网站">呢喃的一些网站</h1>
<ol>
<li><a href="https://ndwy.nju.edu.cn/dztml/admin">南京大学五育项目管理系统</a><br>
<img src="https://img.czruby.eu.or</summary>
<category term="资料" scheme="https://www.czruby.eu.org/categories/%E8%B5%84%E6%96%99/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="网站" scheme="https://www.czruby.eu.org/tags/%E7%BD%91%E7%AB%99/"/>
</entry>
<entry>
<title>英语听说口语</title>
<link href="https://www.czruby.eu.org/posts/2afeca0a.html"/>
<id>https://www.czruby.eu.org/posts/2afeca0a.html</id>
<published>2023-12-26T08:06:01.000Z</published>
<updated>2024-01-11T15:33:25.000Z</updated>
<content type="html"><![CDATA[<h1 id="第一单元">第一单元</h1><h2 id="talk-about-charity">talk about charity</h2><p>● Talk about one of the charity events you’ve heard of or got involved in.</p><ul><li>Hope project<ul><li>what: donate books to underprivileged areas</li><li>how: school organized<ol><li>advocate us to donate books (post posters)</li><li>collect books (in every class)</li><li>send to the poor students (on our own)</li></ol></li><li>influence: make such a big difference to these children , feel proud</li></ul></li></ul><p>● In Text 2, we have learnt how students raise money for charity in Cambridge. What do you think of students raising money for charity?</p><ul><li>useful (can help a lot of people)</li><li>creative (by punting)(user their labor to earn money which makes it with two sides value)</li><li>kind (spend their time and labority to help people in need)</li><li>well worth us to learn from them</li><li>useful expression:<br>If you ask me, …<br>I’d say that …<br>As I see it, …<br>In my opinion, …<br>As far as I’m concerned, …<br>I’d like to point out that …<br>The point I’m making is that …</li></ul><div class="row"> <embed src="http://view.officeapps.live.com/op/view.aspx?src=https://www.czruby.eu.org/2023/12/26/https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/opinions%20and%20(dis)agreements.doc" width="100%" height="550" type="application/pdf"></div><h1 id="第二单元">第二单元</h1><h2 id="talking-about-hobbies-leisure-activities">talking about hobbies/ leisure activities</h2><p>eg. My favorite entertainment is basketball.</p><ol><li><p>basketball has become part of my life.<br>when, where & with whom to play it<br>how I feel when I play it:<br>(exciting; relaxing, relieve pressure, get refreshed , be energetic, … )</p></li><li><p>I can benefit a lot from basketball.<br>build a strong body & a strong will<br>make friends who really love basketball<br>develop team spirits(helpful to me)</p></li><li><p>I’m also crazy about watching basketball games.<br>I think all the basketball players are very (amazing, cool, strong)<br>have awesome skills</p></li></ol><h2 id="the-changes-in-leisure-activities-over-the-years-in-China-less-possibility">the changes in leisure activities over the years in China(less possibility)</h2><ul><li>read newspapers<ul><li>the main way to get information</li><li>news stand are get fewer and fewer(how to fix the problem)</li></ul></li><li>attend dance meeting<ul><li>fashion social activities</li><li>make friendship</li></ul></li><li>watch opening opera<ul><li>every big events or festivals have this kind activities</li><li>cannot miss it</li></ul></li></ul><div class="row"> <embed src="http://view.officeapps.live.com/op/view.aspx?src=https://www.czruby.eu.org/2023/12/26/https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/talk%20about%20likes%20dislikes.doc" width="100%" height="550" type="application/pdf"></div><h1 id="第三单元">第三单元</h1><h2 id="Which-kind-of-pets-do-you-like-Talk-about-it">Which kind of pets do you like?Talk about it.</h2><p>eg. I like keeping a cats.</p><ul><li>Why i like cats<ul><li>it’s cute ; easy for me to stroke it which is a very comfortable thing.</li><li>it’s helpful. help us to catch mice</li><li>it’s easy to keep. Compare to dogs, it is more docile.<ul><li>Dogs like to wolf and a bit more dangerous</li><li>Cats are more safe</li></ul></li></ul></li><li>My pets<ul><li>I have a cat whose name is ‘Mimi’</li><li>I get her three years ago</li><li>She is very pretty and cute. (sweet-tempered)</li><li>be scared of a mouse once time</li></ul></li></ul><h2 id="What’s-good-about-keeping-pets-What-are-the-problems">What’s good about keeping pets ? What are the problems?</h2><ul><li>good<ul><li>company ; make us not alone<ul><li>eg: Every time I come back from school. It rushed out to welcome me.</li></ul></li><li>have a lot of happy time with them<ul><li>eg: I often take it out and we always have a good time playing together</li></ul></li></ul></li><li>problem<ul><li>hygiene<ul><li>smell bad if don’t have a bath for a long time</li><li>make water and shit everywherefeedback</li></ul></li><li>food<ul><li>don’t know what and how many to feed them<br>don’t feed properly<br>may be hurt by it</li></ul></li></ul></li></ul><h1 id="第四单元">第四单元</h1><h2 id="What-do-you-think-is-the-best-means-of-communication-And-why-do-you-think-so">What do you think is the best means of communication? And why do you think so?</h2><ul><li>Instant Messaging and Chat Apps:<ul><li>Advantages:<ul><li>Real-time</li><li>allows for multimedia sharing</li><li>and often supports group conversations.</li></ul></li><li>Disadvantages<ul><li>less possibilty to make face-to-face communication</li></ul></li></ul></li></ul><h2 id="Do-you-think-you-can-live-without-your-mobile-phone-What-do-you-usually-use-it-for-What-do-you-think-of-the-roles-that-mobile-phones-play-in-our-life">Do you think you can live without your mobile phone? What do you usually use it for? What do you think of the roles that mobile phones play in our life?</h2><ul><li>NO My mobile phone has become part of my life. I can hardly survive without it.</li><li>I usually use it to serve my life need and get information</li><li>Mobile phons role<ul><li>Communication</li><li>get information</li><li>Entertainment</li></ul></li></ul><h2 id="It’s-very-important-for-us-to-know-how-to-communicate-with-others-According-to-you-how-can-we-achieve-effective-communication">It’s very important for us to know how to communicate with others. According to you, how can we achieve effective communication?</h2><ul><li>We human beings are social creatures. One can hardly survive without socializing with others. So it’s very important for us to know how to achieve effective communication.</li><li>active listening<ul><li>when you concentrate on others’ opinion, get a lot of useful information</li></ul></li><li>clarity and conciseness(简明)<ul><li>pass your opinion to others clearly and fastly</li></ul></li><li>choose the right way to communicate<ul><li>mobile phone; letters; emails;…</li><li>consider their features comprehensively</li></ul></li></ul><h2 id="Do-you-think-there-is-a-generation-gap-between-you-and-your-parents-According-to-you-how-can-we-bridge-the-gap">Do you think there is a generation gap between you and your parents? According to you, how can we bridge the gap?</h2><ul><li>yes<ul><li>No language shared with parents</li><li>they couldn’t understand my meaning many times</li><li>fall behind time(don’t konw the hot events)</li></ul></li><li>share the hot events with them more often<ul><li>teach them how to use bilibili</li></ul></li><li>help them follow the age</li></ul><h1 id="第五单元">第五单元</h1><h2 id="How-do-people-meet-someone-to-date-in-today’s-Chaina-And-what-do-you-think-is-the-best-way-to-find-that-special-someone">How do people meet someone to date in today’s Chaina? And what do you think is the best way to find that special someone?</h2><ul><li>Online Platform<ul><li>provide a good way to meet new friends</li><li>get in touch with others without geographical restrictions</li></ul></li><li>Friend Introductions<ul><li>your friends are outgoing by coincidence</li></ul></li><li>Interest clubs<ul><li>You can always have a talk with them without difficulty because you have the same hobby. And as time passes, you may generate feeling on him or her.</li></ul></li><li>Social Events and Activities<ul><li>Meet people who you may never meet before</li><li>Ask contact information</li><li>At least friendship or find the special someone</li></ul></li></ul><h2 id="What-do-you-think-of-internet-dating-What-are-the-advantages-and-disadvantages">What do you think of internet dating? What are the advantages and disadvantages?</h2><ul><li>also known as online dating</li><li>has become a popular way to meet someone special</li><li>advantage<ul><li>Convenience<ul><li>You can learn about others easily by looking at their profile</li></ul></li><li>Reduced Pressure<ul><li>Many people are anxiety when they talk to the opposite sex.</li><li>By online dating you can be relaxed</li></ul></li><li>Increased posiibilty to have the same interest<ul><li>Compare to the way of being introduced by relatives</li><li>you add others usually because you have the same interest</li></ul></li></ul></li><li>disadvantage<ul><li>Safety Concerns<ul><li>The information online is hard to be distinguished</li><li>仙人跳:being cheated by others using online dating</li></ul></li><li>cannot get others’ true feelings<ul><li>couldn’t see others in reality</li></ul></li></ul></li></ul><h2 id="What-do-you-think-of-cyber-love-Do-you-believe-that-true-love-can-be-found-online">What do you think of cyber love? Do you believe that true love can be found online?</h2><ul><li>has become a popular way to meet someone special</li><li>Connection Beyond Geography</li><li>Increased posiibilty to have the same interest<ul><li>Compare to the way of being introduced by relatives</li><li>you add others usually because you have the same interest</li></ul></li><li>Safety Concerns<ul><li>The information online is hard to be distinguished</li><li>仙人跳:being cheated by others using online dating</li></ul></li><li>Balancing Online and Offline Interaction:<ul><li>It’s important to transition from online to offline interactions to deepen the connection</li></ul></li></ul><h1 id="口语考试小tip">口语考试小tip</h1><p><img src="https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/image.png" alt="Alt text"><br><img src="https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/image-1.png" alt="Alt text"><br><img src="https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/image-2.png" alt="Alt text"><br><img src="https://img.czruby.eu.org/imgs/%E8%8B%B1%E8%AF%AD%E5%90%AC%E8%AF%B4%E5%8F%A3%E8%AF%AD/image-3.png" alt="Alt text"></p><p>These are my views on this issue.That’s all.Thank you!</p>]]></content>
<summary type="html"><h1 id="第一单元">第一单元</h1>
<h2 id="talk-about-charity">talk about charity</h2>
<p>● Talk about one of the charity events you’ve heard of or got</summary>
<category term="学习" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/"/>
<category term="英语" scheme="https://www.czruby.eu.org/categories/%E5%AD%A6%E4%B9%A0/%E8%8B%B1%E8%AF%AD/"/>
<category term="学习" scheme="https://www.czruby.eu.org/tags/%E5%AD%A6%E4%B9%A0/"/>
<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
<category term="期末" scheme="https://www.czruby.eu.org/tags/%E6%9C%9F%E6%9C%AB/"/>
<category term="英语" scheme="https://www.czruby.eu.org/tags/%E8%8B%B1%E8%AF%AD/"/>
</entry>
<entry>
<title>微电子期末复习纲要</title>
<link href="https://www.czruby.eu.org/posts/dd7d6d10.html"/>
<id>https://www.czruby.eu.org/posts/dd7d6d10.html</id>
<published>2023-12-25T14:18:04.000Z</published>
<updated>2024-01-11T15:33:27.000Z</updated>
<content type="html"><![CDATA[<div class="hbe hbe-container" id="hexo-blog-encrypt" data-wpm="抱歉, 这个密码看着不太对, 请再试试." data-whm="抱歉, 这个文章不能被校验, 不过您还是能看看解密后的内容."> <script id="hbeData" type="hbeData" 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</entry>
<entry>
<title>Euler积分—B函数与Γ函数</title>
<link href="https://www.czruby.eu.org/posts/87d06c28.html"/>
<id>https://www.czruby.eu.org/posts/87d06c28.html</id>
<published>2023-12-25T08:19:26.000Z</published>
<updated>2024-01-11T13:19:56.000Z</updated>
<content type="html"><![CDATA[<h1 id="Euler积分—B函数与Γ函数">Euler积分—B函数与Γ函数</h1><p><strong>Author:</strong> [BriChen]</p><p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/433589729">https://zhuanlan.zhihu.com/p/433589729</a>]</p><h2 id="前言">前言</h2><p>某些非初等函数可以由含参变量的积分表示,B(beta)函数和Γ(gamma)函数是很重要的两个例子。它们统称为Euler积分,其定义为:</p><p>$\begin{aligned} 定义:\ &B(p,q)=\int_{0}^{1},{x^{p-1}(1-x)^{q-1}},dx \&\Gamma§=\int_{0}^{+\infty},{x^{p-1}e^{-x}},dx \end{aligned}$</p><p>其中B函数定义域为:p>0,q>0;Γ函数定义域为:p>0。</p><hr><p><strong>下面介绍它们的一些性质:</strong></p><ol><li>两个函数的其他表示形式;</li><li>递推性质;</li><li>余元公式;</li><li>B函数与Γ函数的联系公式;</li><li>两类函数在其定义域上连续,并且任意次可微。</li></ol><p><strong>然后,会做一下本节内容的小结</strong>。</p><hr><h3 id="1-两个函数的其他表示形式">1.两个函数的其他表示形式</h3><p>$\begin{aligned} (1)&B(p,q)=2\int_{0}^{\frac{\pi}{2}},{\sin^{2p-1}\theta \cos^{2q-1}\theta},d\theta \(2)&B(p,q)=\int_{0}^{+\infty},{\frac{x^{p-1}}{(1+x)^{p+q}}},dx \(3)&\Gamma§=2\int_{0}^{+\infty},{x^{2p-1}e^{-x^2}},dx \end{aligned}$</p><p>$\begin{aligned} &证明是容易的。 \&(1)做替换x=\sin^2\theta \&(2)做替换x=\frac{x}{1+x} \&(3)做替换x=x^2 \end{aligned}$</p><h3 id="2-递推公式">2.递推公式</h3><p>$\begin{aligned} &(1)B(p+1,q)=\frac{p}{p+q}B(p,q)\ &(2)B(p,q+1)=\frac{q}{p+q}B(p,q)\ &(3)B(p+1,q+1)=\frac{pq}{(p+q+1)(p+q)},B(p,q)\ &(4)\Gamma (p+1)=p,\Gamma§\ \end{aligned}$</p><p>$\begin{aligned} Proof:&\ &(1)B(p+1,q)=\int_{0}^{1}, x^p(1-x)^{q-1},dx\overset{\text{分部积分}}{\implies} \frac{p}{q}\int_{0}^{1}, x^{p-1}(1-x)^{q},dx=\frac{p}{q}\int_{0}^{1}, x^{p-1}(1-x)^{q-1}(1-x),dx\ &\implies \frac{p}{q}\int_{0}^{1}, x^{p-1}(1-x)^{q-1},dx\ -\ \frac{p}{q}\int_{0}^{1}, x^{p}(1-x)^{q-1},dx =\frac{p}{q}B(p,q)-\frac{p}{q}B(p+1,q)\ &因此,有,\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad B(p+1,q)=\frac{p}{p+q}B(p,q)\ \ \&(2)由对称性知,B(q,p)=B(p,q).(或者说对B(p,q)做替换x=1-x)\&\quad B(p,q+1)=B(q+1,p),然后用(1)的结论,有, \ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad B(p,q+1)=\frac{q}{p+q}B(p,q)\ \ \&(3)依次用结论(1),(2)可得\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad B(p+1,q+1)=\frac{pq}{(p+q+1)(p+q)}B(p,q)\ \ \&(4)\Gamma (p+1)=\int_{0}^{+\infty}x^pe^{-x},dx \overset{\text{分部积分}}{\implies}-(x^pe^{-x}|^{+\infty}<em>0-p\int</em>{0}^{+\infty}x^{p-1}e^{-x},dx)=p\int_{0}^{+\infty}x^{p-1}e^{-x},dx=p\Gamma§\ \end{aligned}$</p><h3 id="3-余元公式-0-p-1">3.余元公式(0<p<1)</h3><p>$B(p,1-p)=\frac{\pi}{\sin p\pi}\ \ \ p\in(0,1)\ \Gamma§\Gamma(1-p)=\frac{\pi}{\sin p\pi} p\in(0,1)$</p><p>$\begin{aligned} Proof:&\ &这里我们使用到了B函数的另一种表示形式,\ &B (p,1-p)=\int_{0}^{+\infty}\frac{x^{p-1}}{(1+x)^{p+1-p}},dx =\int_{0}^{+\infty}\frac{\ \ x^{p-1}}{1+x},dx\ &\qquad \quad \quad \ \ ,=\int_{0}^{1}\frac{\ \ x^{p-1}}{1+x},dx+\int_{1}^{+\infty}\frac{\ \ x^{p-1}}{1+x},dx\ &\xrightarrow{第二个式子做替换x=\frac{1}{x}}\int_{0}^{1}\frac{\ \ x^{p-1}}{1+x},dx+\int_{0}^{1}\frac{\ \ x^{-p}}{1+x},dx\ &=\int_{0}^{1}\frac{\ \ x^{p-1}+x^{-p}}{1+x},dx\xrightarrow{展开级数\frac{1}{1+x}}\int_{0}^{1}\sum_{n=0}^{\infty}(-1)^n x^n( x^{p-1}+x^{-p}),dx\ &=\lim_{r\rightarrow1^-}\int_{0}^{r}(\sum_{n=0}^{\infty}(-1)^n x^n( x^{p-1}+x^{-p})),dx(这里的积分号可与和号换序)\ &=\lim_{r\rightarrow1^-}(\sum_{n=0}^{\infty} \frac{(-1)^n}{n+p} r^{p+n}+\sum_{n=0}^{\infty}\frac{(-1)^n}{n-p+1} r^{n+1-p})(极限可以放入和式,不再赘述)\ &=\frac{1}{p}+\sum_{n=1}^{\infty} \frac{(-1)^n}{n+p} +\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n-p} \ &=\frac{1}{p}+\sum_{n=1}^{\infty} \frac{(-1)^n}{p+n} +\sum_{n=1}^{\infty}\frac{(-1)^{n}}{p-n} \ &=\frac{1}{p}+\sum_{n=1}^{\infty} (-1)^n\frac{2p}{p^2-n^2} \ &=\frac{\pi}{\sin p\pi} \& 最后一个等号,联系\frac{1}{\sin x}的Fourier展开式可以推出。 \end{aligned}$</p><h3 id="4-联系公式(p-0-q-0)">4.联系公式(p>0,q>0)</h3><p>$B(p,q)=\frac{\Gamma§\Gamma(q)}{\Gamma(p+q)}$</p><p>$\begin{aligned} Proof:&\ &现设p>1,q>1,先看\Gamma (p+q)=\int_{0}^{+\infty}x^{p+q-1}e^{-x}, dx,\ \ &做替换x=(1+t)u\ (t\ge 0,且先认作为参数),得,\ & \qquad \qquad\quad \frac{\Gamma(p+q)}{(1+t)^{p+q}}=\int_{0}^{+\infty}u^{p+q-1}e^{-(1+t)u},du\ &上式两边乘以t^{p-1},然后对t在(0,+\infty)上积分,得,\ &\int_{0}^{+\infty}\frac{t^{p-1}}{(1+t)^{p+q}}\Gamma(p+q), dt=\int_{0}^{+\infty}dt\int_{0}^{+\infty}t^{p-1}u^{p+q-1}e^{-(1+t)u},du\ &即,\ &B(p,q)\Gamma(p+q)=\int_{0}^{+\infty}dt\int_{0}^{+\infty}t^{p-1}u^{p+q-1}e^{-(1+t)u},du\ &=\int_{0}^{+\infty}\left[\int_{0}^{+\infty}t^{p-1}u^{p+q-1}e^{-(1+t)u} ,dt\right]du\ \ \ (这里的积分换序是合理的稍后说明)\ &=\int_{0}^{+\infty}\left[u^{p+q-1}e^{-u}\int_{0}^{+\infty}t^{p-1}e^{-tu} ,dt\right]du\xrightarrow{里层积分做替换t=\frac{t}{u}}\ &\int_{0}^{+\infty}\left[u^{q-1}e^{-u}\int_{0}^{+\infty}t^{p-1}e^{-t} ,dt\right]du\ =\int_{0}^{+\infty}\left[u^{q-1}e^{-u}\Gamma§\right]du=\Gamma§\int_{0}^{+\infty}u^{q-1}e^{-u}du\ &=\Gamma§\Gamma(q),故有, \&\qquad \qquad\qquad\qquad\qquad\qquad\frac{\Gamma§\Gamma(q)}{\Gamma(p+q)}=B(p,q). \&现在说明上面两个无穷积分换序是合理的,因为被积函数在t>0,u>0连续,非负;\&积分\int_{0}^{+\infty}t^{p-1}u^{p+q-1}e^{-(1+t)u} ,du=\frac{\Gamma(p+q)}{(1+t)^{p+q}}t^{p-1}在t>0上连续;\ &积分\int_{0}^{+\infty}t^{p-1}u^{p+q-1}e^{-(1+t)u} ,dt=u^{q-1}e^{-u}\Gamma§在u>0上连续;\ &又有积分估计\ &\int_{0}^{+\infty}\left[\int_{0}^{+\infty}u^{p+q-1}e^{-u}t^{p-1}e^{-tu} ,dt\right]du=\Gamma§\Gamma(q)<+\infty 故上面的广义积分号可以换序。\ \ \ &现在考虑p>0,q>0.前面证明了B函数与Γ函数的联系公式,和递推公式,\ &B(p+1,q+1)=\frac{\Gamma(p+1)\Gamma(q+1)}{\Gamma(p+q+2)}\xrightarrow{两边都应用递推公式}B(p,q)=\frac{\Gamma§\Gamma(q)}{\Gamma(p+q)}\ \end{aligned}$</p><h3 id="5-两类函数的连续性和任意次可微性">5.两类函数的连续性和任意次可微性</h3><p>因为B函数可以由Γ函数表示(联系公式),所以我们只要讨论Γ函数的性质即可。</p><p>要说明Γ函数在定义域上连续且任意次可微。只需要证明其导数存在就够了(因为它是一元的函数)。</p><p>$\Gamma’§=\int_{0}^{+\infty}x^{p-1}\ln{x},e^{-x},dx=\int_{0}^{1}x^{p-1}\ln{x},e^{-x},dx+\int_{1}^{+\infty}x^{p-1}\ln{x},e^{-x},dx$</p><p>根据含参数变量广义积分的可微性定理,我们只需要验证上面的式子是一致收敛的(其他条件就自己验证吧XD)。</p><p>上面积分的第一个式子可以由Abel判别法得知其在(0,1)内闭一致收敛,第二个也一样,做法稍有不同。</p><p>同样的我们可以证明更加高阶的情形,所以这个函数是任意次可微的。</p><hr><h2 id="小结:">小结:</h2><ul><li>通过联系公式可以知道,B函数的值可以归结到Γ函数的计算上来(在定义域内);</li><li>余元公式帮助我们了解到了两类函数在(0,1)上的取值;</li><li>递推公式帮助我们把(0,1)上的值推向了整个正实数轴上;</li><li>而最开始的两类函数的不同表示方式可以帮助我们理解后面理论的推导;</li><li>有意思的是这两类函数的递推性质,可以类比一下Γ(n+1)与n!;Γ(n+1)一直往后递推会得到类似阶乘的结果!或者说Γ函数是正实数域上的“阶乘”。</li></ul>]]></content>
<summary type="html"><h1 id="Euler积分—B函数与Γ函数">Euler积分—B函数与Γ函数</h1>
<p><strong>Author:</strong> [BriChen]</p>
<p><strong>Link:</strong> [<a href="https://zhuanlan</summary>
<category term="转载" scheme="https://www.czruby.eu.org/categories/%E8%BD%AC%E8%BD%BD/"/>
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<category term="复习" scheme="https://www.czruby.eu.org/tags/%E5%A4%8D%E4%B9%A0/"/>
<category term="微积分" scheme="https://www.czruby.eu.org/tags/%E5%BE%AE%E7%A7%AF%E5%88%86/"/>
<category term="资料" scheme="https://www.czruby.eu.org/tags/%E8%B5%84%E6%96%99/"/>
</entry>
<entry>
<title>C语言结构体内存对齐规则</title>
<link href="https://www.czruby.eu.org/posts/c021c1e0.html"/>
<id>https://www.czruby.eu.org/posts/c021c1e0.html</id>
<published>2023-12-22T07:00:39.000Z</published>
<updated>2024-01-11T15:33:24.000Z</updated>
<content type="html"><![CDATA[<p>author:zhangtao<br>link:[<a href="https://blog.csdn.net/weixin_45157820/article/details/112755832">https://blog.csdn.net/weixin_45157820/article/details/112755832</a>]</p><h2 id="C语言结构体对齐步骤">C语言结构体对齐步骤:</h2><ol><li>结构体各成员对齐.</li><li>结构体总体对齐</li></ol><h2 id="C语言结构体对齐规则">C语言结构体对齐规则:</h2><ol><li>结构体(struct)的数据成员,第一个数据成员存放的地址为结构体变量偏移量为0的地址处.</li><li>其他结构体成员自身对齐时,存放的地址为min{有效对齐值为自身对齐值, 指定对齐值} 的最小整数倍的地址处. 注:自身对齐值:结构体变量里每个成员的自身大小 注:指定对齐值:有宏 #pragma pack(N) 指定的值,这里面的 N一定是2的幂次方.如1,2,4,8,16等.如果没有通过宏那么在32位Linux主机上默认指定对齐值为4,64位的默认对齐值为8,AMR CPU默认指定对齐值为8; 注:有效对齐值:结构体成员自身对齐时有效对齐值为自身对齐值与指定对齐值中 较小的一个.</li><li>总体对齐时,字节大小是min{所有成员中自身对齐值最大的, 指定对齐值} 的整数倍. #pragma pack(N) 每个特定平台上的编译器都有自己的默认“对齐系数”(也叫对齐模数)。程序员可以通过预编译命令#pragma pack(n),n=1,2,4,8,16来改变这一系数,其中的n就是你要指定的“对齐系数”。</li></ol><h2 id="实例解析">实例解析:</h2><h3 id="例一">例一</h3><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//此代码在64位Linux下编写</span></span><br><span class="line">typedef struct _st_struct1</span><br><span class="line"> {</span><br><span class="line"> </span><br><span class="line"><span class="type">char</span> a;</span><br><span class="line"><span class="type">short</span> b;</span><br><span class="line"><span class="type">int</span>c;</span><br><span class="line"> }st_struct1;</span><br><span class="line"> </span><br><span class="line"> printf(<span class="string">"%ld\n"</span>,sizeof(st_struct1));</span><br></pre></td></tr></table></figure><p>打印结果为:8. 让我们分析一下为什么结果为8. a是char类型,占1个字节.第一个数据成员,放在结构体变量偏移量为0 的地址处. b是short类型,占2个字节,接下来我们要看结构体对齐规则,b的有效对齐值为min{2, 8}=2. 依次查看2的整数倍地址是否可以存放俩个字节.2×0=0 //此地址处已经存放a. 2×1=2 //此地址为空,我们将占有俩个字节的b存放在地址偏移量为2和3处. c是int类型,占4个字节,我们根据结构体对齐规则可知,c的有效对齐值为4.对齐到4的整数倍地址,即地址偏移量为4处.在内存中存放的位置为4,5,6,7. 结构体总对齐字节大小为min{4, 8}=4 的整数倍.此时内存中共占8个字节,正好是4的整数倍,所以sizeof(st_struct1)=8. 占用内存空间如下: <img src="https://img.czruby.eu.org/imgs/C%E8%AF%AD%E8%A8%80%E7%BB%93%E6%9E%84%E4%BD%93%E5%86%85%E5%AD%98%E5%AF%B9%E9%BD%90%E8%A7%84%E5%88%99/1703228316253.png" alt="img"></p><h3 id="例二">例二</h3><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//此代码在64位Linux下编写</span></span><br><span class="line">typedef struct _st_struct2</span><br><span class="line"> {</span><br><span class="line"> </span><br><span class="line"> <span class="type">char</span> a;</span><br><span class="line"><span class="type">int</span>c;</span><br><span class="line"><span class="type">short</span> b;</span><br><span class="line"> }st_struct2;</span><br><span class="line"> </span><br><span class="line"> printf(<span class="string">"%ld\n"</span>,sizeof(st_struct2));</span><br></pre></td></tr></table></figure><p>打印结果为:12. 让我们分析一下为什么结果为12. a是char类型,占1个字节.第一个数据成员,放在结构体变量偏移量为0 的地址处. c是int类型,占4个字节,我们根据结构体对齐规则可知,c的有效对齐值为4.对齐到4的整数倍地址,即地址偏移量为4处.在内存中存放的位置为4,5,6,7. b是short类型,占2个字节,我们根据结构体对齐规则可知,b的有效对齐值为2.对齐到2的整数倍地址,即地址偏移量为8处.在内存中存放的位置为8,9. 结构体总对齐字节大小为min{4, 8}=4 的整数倍.此时内存中共占10个字节,又要求是4的整数倍,所以sizeof(st_struct1)=12. <img src="https://img.czruby.eu.org/imgs/C%E8%AF%AD%E8%A8%80%E7%BB%93%E6%9E%84%E4%BD%93%E5%86%85%E5%AD%98%E5%AF%B9%E9%BD%90%E8%A7%84%E5%88%99/1703228316411.png" alt="img"></p><p><strong>我们如果想要得到的是类型所占字节之和.例如:char a; int b; short 3;结构体总对齐长度为"1+4+2=7"应该如何定义呢?在例三中可得到答案</strong></p><h3 id="例三">例三</h3><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//此代码在64位Linux下编写</span></span><br><span class="line">#pragma <span class="title function_">pack</span><span class="params">(<span class="number">1</span>)</span></span><br><span class="line">typedef struct _st_struct3</span><br><span class="line"> {</span><br><span class="line"> </span><br><span class="line"> <span class="type">char</span> a;</span><br><span class="line"><span class="type">int</span>c;</span><br><span class="line"><span class="type">short</span> b;</span><br><span class="line"> }st_struct3;</span><br><span class="line"></span><br><span class="line"> printf(<span class="string">"%ld\n"</span>,sizeof(st_struct3));</span><br></pre></td></tr></table></figure><p><strong>#pragma pack(1):设置结构体的边界对齐为1个字节,也就是所有数据在内存中是连续存储的。</strong> 打印结果为:7. 让我们分析一下为什么结果为7. a是char类型,占1个字节.第一个数据成员,放在结构体变量偏移量为0 的地址处. c是int类型,占4个字节,我们根据结构体对齐规则可知,c的有效对齐值为min{4,1}.对齐到1的整数倍地址,即地址偏移量为1处.在内存中存放的位置为1,2,3,4. b是short类型,占2个字节,我们根据结构体对齐规则可知,b的有效对齐值为1,对齐到1的整数倍地址,即地址偏移量为5处.在内存中存放的位置为5,6. 结构体总对齐字节大小为min{4, 1}=1 的整数倍.此时内存中共占7个字节,所以sizeof(st_struct3)=7. <strong>特别注意一下:在使用pragma pack (1)之后,我们最好要unpack. 否则在他之后定义的所有结构体都会按1字节对齐!</strong> <img src="https://img.czruby.eu.org/imgs/C%E8%AF%AD%E8%A8%80%E7%BB%93%E6%9E%84%E4%BD%93%E5%86%85%E5%AD%98%E5%AF%B9%E9%BD%90%E8%A7%84%E5%88%99/1703228316464.png" alt="img"></p><h3 id="例四">例四</h3><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//此代码在64位Linux下编写</span></span><br><span class="line">#pragma <span class="title function_">pack</span><span class="params">(<span class="number">2</span>)</span></span><br><span class="line">typedef struct _st_struct4</span><br><span class="line"> {</span><br><span class="line"> </span><br><span class="line"> <span class="type">char</span> a;</span><br><span class="line"><span class="type">int</span>c;</span><br><span class="line"><span class="type">short</span> b;</span><br><span class="line"> }st_struct4;</span><br><span class="line"></span><br><span class="line"> printf(<span class="string">"%ld\n"</span>,sizeof(st_struct4));</span><br></pre></td></tr></table></figure><p>打印结果为:8. 让我们分析一下为什么结果为8. <strong>#pragma pack(2):设置结构体的边界对齐为2个字节。</strong> a是char类型,占1个字节.第一个数据成员,放在结构体变量偏移量为0 的地址处. c是int类型,占4个字节,我们根据结构体对齐规则可知,c的有效对齐值为min{4,2}.对齐到2的整数倍地址,即地址偏移量为2处.在内存中存放的位置为2,3,4,5 b是short类型,占2个字节,我们根据结构体对齐规则可知,b的有效对齐值为2,对齐到2的整数倍地址,即地址偏移量为6处.在内存中存放的位置为6,7. 结构体总对齐字节大小为min{4, 2}=2 的整数倍.此时内存中共占8个字节,正好是2的整数倍,所以sizeof(st_struct4)=8.</p><p><img src="https://img.czruby.eu.org/imgs/C%E8%AF%AD%E8%A8%80%E7%BB%93%E6%9E%84%E4%BD%93%E5%86%85%E5%AD%98%E5%AF%B9%E9%BD%90%E8%A7%84%E5%88%99/1703228316535.png" alt="img"></p>]]></content>
<summary type="html"><p>author:zhangtao<br>
link:[<a href="https://blog.csdn.net/weixin_45157820/article/details/112755832">https://blog.csdn.net/weixin_45157820</summary>
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<category term="CSDN" scheme="https://www.czruby.eu.org/categories/%E8%BD%AC%E8%BD%BD/CSDN/"/>
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<category term="C语言" scheme="https://www.czruby.eu.org/tags/C%E8%AF%AD%E8%A8%80/"/>
</entry>
<entry>
<title>关于e的负x平方的积分</title>
<link href="https://www.czruby.eu.org/posts/5af9a9a9.html"/>
<id>https://www.czruby.eu.org/posts/5af9a9a9.html</id>
<published>2023-12-21T10:51:58.000Z</published>
<updated>2024-01-11T15:33:39.000Z</updated>
<content type="html"><![CDATA[<h1 id="关于e的负x平方的积分">关于e的负x平方的积分</h1><p><strong>Author:</strong> [GGGGWHW]</p><p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/651305078?utm_id=0">https://zhuanlan.zhihu.com/p/651305078?utm_id=0</a>]</p><h2 id="简介"><img src="https://img.czruby.eu.org/imgs/%E5%85%B3%E4%BA%8Ee%E7%9A%84%E8%B4%9Fx%E5%B9%B3%E6%96%B9%E7%9A%84%E7%A7%AF%E5%88%86/v2-c1069bddfb3259c678a7a11d04d19857_r.jpg" alt=""><br>简介</h2><p>在统计学和物理学中经常要对形如 $f(x)=e^{-x^2}$ 这样的函数进行积分, 相当于求上图中函数曲线与 $x$ 轴之间所夹的面积. 本文将展示该函数和类似函数的积分过程.</p><h2 id="考虑幂级数展开">考虑幂级数展开</h2><p>考虑到 $F(x)=\int e^{-x^2}\mathrm{d}x$ 并不能得到一个简单的函数表达式, 所以不妨将被积函数先根据泰勒公式展开.</p><p>$$\begin{aligned} & \because e^x=\sum_{n=0}^{+\infty}\frac{x^n}{n!} \& \therefore e^{-x^2}=\sum_{n=0}^{+\infty}\frac{(-x^2)^n}{n!}= \sum_{n=0}^{+\infty}\frac{(-1)^nx^{2n}}{n!} \&\therefore F(x)=\int e^{-x^2}\mathrm{d}x = \sum_{n=0}^{+\infty}\int \frac{(-1)^nx^{2n}}{n!}\mathrm{d}x = \sum_{n=0}^{+\infty} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}+C \end{aligned}$$</p><p>令 $C=0$ , 即 $F(0)=0$ , 则 $F(x)= \sum_{n=0}^{+\infty} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$ . 这个幂级数的收敛半径是 $+\infty$ , 但是收敛速度随着 $x$ 的变大迅速变慢. 比如:</p><p>当 $x=1$ 时, $\sum_{n=0}^{8} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$和$F(x)$的误差为 $1.33\times10^{-7}$ .</p><p>当 $x=2$ 时, $\sum_{n=0}^{8} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$和$F(x)$的误差为 $0.0555$ .</p><p>当 $x=2$ 时, $\sum_{n=0}^{17} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$和$F(x)$的误差为 $4.83\times10^{-7}$ .</p><p>当 $x=3$ 时, $\sum_{n=0}^{17} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$和$F(x)$的误差为 $1.305$ .</p><p>当 $x=3$ 时, $\sum_{n=0}^{26} \frac{(-1)^nx^{2n+1}}{n!(2n+1)}$和$F(x)$的误差为 $2.219\times10^{-4}$ .</p><p>所以这个幂级数对于求积分上限为较大数字的定积分不是很实用, 还不如直接通过分割被积函数, 然后进行数值积分方便.</p><h2 id="考虑函数对称性">考虑函数对称性</h2><p>虽然不知道 $F(x)$ 的简单表达式, 但是不影响我们求一个特殊的定积分: $\int_{-\infty} ^{+\infty}e^{-x^2}\mathrm{d}x$ .</p><p>考虑到被积函数是一个偶函数,所以 $\int_{-\infty} ^{+\infty}e^{-x^2}\mathrm{d}x =2\int_{0} ^{+\infty}e^{-x^2}\mathrm{d}x$ .<br>变形后, 由于不能直接求被积函数的原函数 $F(x)$ , 所以上面的操作对于我们解决问题依然没有帮助.</p><h2 id="考虑在高维空间解决">考虑在高维空间解决</h2><p>再换一个思路, 有理数域解决不了的问题, 放到实数域说不定就解决了; 实数域解决不了的问题, 放到复数域说不定就解决了. 同样地, 一元函数解决不了的问题, 换成二元函数说不定就解决了.</p><p>将文章开头的函数图像绕着纵轴旋转 $180°$ , 旋转过程中函数曲线在空间画出一个曲面, 如下图所示:</p><p><img src="https://img.czruby.eu.org/imgs/%E5%85%B3%E4%BA%8Ee%E7%9A%84%E8%B4%9Fx%E5%B9%B3%E6%96%B9%E7%9A%84%E7%A7%AF%E5%88%86/v2-c6c13a89fabc06ac6751e6e8281a9c09_r.jpg" alt=""><br>曲面的函数表达式为 $f(x,y)=e^{-x^2-y^2}$ 现在我们试着求上图中函数曲面与 $xOy$ 平面之间所夹的体积.</p><p>$$\begin{aligned}&\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y\=& \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^2}\cdot e^{-y^2}\mathrm{d}x\mathrm{d}y\=& \int_{-\infty}^{+\infty}e^{-x^2}\mathrm{d}x\cdot\int_{-\infty}^{+\infty}e^{-y^2}\mathrm{d}y\=& (\int_{-\infty}^{+\infty}e^{-x^2}\mathrm{d}x)^2 \end{aligned}$$</p><p>惊喜地发现, 这个曲面积分结果恰好是我们要求的曲线积分的平方, 可是我们依然不知道这两个积分的结果.</p><h2 id="考虑坐标变换解决">考虑坐标变换解决</h2><p>经验告诉我们, 对于轴对称图形, 柱坐标系比直角坐标系用起来会更方便一些.</p><p><img src="https://img.czruby.eu.org/imgs/%E5%85%B3%E4%BA%8Ee%E7%9A%84%E8%B4%9Fx%E5%B9%B3%E6%96%B9%E7%9A%84%E7%A7%AF%E5%88%86/v2-a7b57e440700205c29cfb53de581dd7f_r.jpg" alt=""><br>坐标转换:</p><p>$\begin{cases} x=r\cdot\cos\theta\ y=r\cdot\sin\theta\ r=\sqrt{x^2+y^2}\ \mathrm{d}x\mathrm{d}y=r\mathrm{d}r\mathrm{d}\theta\ \end{cases}$</p><p>于是曲面积分</p><p>$$\begin{aligned} & \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y\=& \int_{\theta=0}^{2\pi}\int_{r=0}^{+\infty}e^{-r^2}r\mathrm{d}r\mathrm{d}\theta\=& 2\pi\int_{r=0}^{+\infty}e^{-r^2}r\mathrm{d}r\=& \pi\int_{r^2=0}^{+\infty}e^{-r^2}\mathrm{d}(r^2)\=& -\pi\int_{-r^2=0}^{-\infty}e^{-r^2}\mathrm{d}(-r^2)\=& -\pi\int_{-r^2=0}^{-\infty}\mathrm{d}(e^{-r^2})\=& -\pi e^{-r^2}\mid_{-r^2=0}^{-\infty}\=& -\pi (e^{-\infty}-e^{0})\=& \pi \end{aligned}$$</p><p>之前已经证明曲面积分结果是曲线积分结果的平方, 所以</p><p>$$\begin{aligned}& \int_{-\infty} ^{+\infty}e^{-x^2}\mathrm{d}x=\sqrt\pi\& \int_{0} ^{+\infty}e^{-x^2}\mathrm{d}x=\frac12\int_{-\infty} ^{+\infty}e^{-x^2}\mathrm{d}x=\frac{\sqrt\pi}2\&\end{aligned}$$</p><h2 id="结论">结论</h2><p>$$\begin{aligned} & \int_{-\infty} ^{+\infty}e^{-x^2}\mathrm{d}x=\sqrt\pi\& \int_{0} ^{+\infty}e^{-x^2}\mathrm{d}x=\frac{\sqrt\pi}2\&\end{aligned}$$</p><hr><h2 id="【习题】"><strong>【习题】</strong></h2><h3 id="习题1">习题1.</h3><ol><li>求 $\int_{-\infty} ^{+\infty}xe^{-x^2}\mathrm{d}x$ .</li></ol><p>$$\begin{aligned} 解: & \int_{0} ^{+\infty}xe^{-x^2}\mathrm{d}x\=& \frac{1}{2}\int_{x^2=0} ^{+\infty}e^{-x^2}\mathrm{d}(x^2)\=& -\frac{1}{2}\int_{-x^2=0} ^{-\infty}e^{-x^2}\mathrm{d}(-x^2)\=& -\frac{1}{2}\int_{-x^2=0} ^{-\infty}\mathrm{d}(e^{-x^2})\=& -\frac{1}{2}e^{-x^2}\mid_{-x^2=0} ^{-\infty}\=& \frac{1}{2}\end{aligned}$$</p><h3 id="习题2">习题2.</h3><ol start="2"><li>求 $\int_{-\infty} ^{+\infty}x^2e^{-x^2}\mathrm{d}x$.</li></ol><p>$$\begin{aligned}解: & \int_{0} ^{+\infty}x^2e^{-x^2}\mathrm{d}x\=& \frac{1}{2}\int_{x=0} ^{+\infty}xe^{-x^2}\mathrm{d}(x^2)\=& -\frac{1}{2}\int_{x=0} ^{+\infty}xe^{-x^2}\mathrm{d}(-x^2)\=& -\frac{1}{2}\int_{x=0} ^{+\infty}x\mathrm{d}(e^{-x^2})\=& -\frac{1}{2}(xe^{-x^2}\mid_{x=0} ^{+\infty}-\int_{x=0} ^{+\infty}e^{-x^2}\mathrm{d}x)\=& \frac{1}{2}\int_{x=0} ^{+\infty}e^{-x^2}\mathrm{d}x\=& \frac{1}{2}\times\frac{\sqrt{\pi}}{2}\=& \frac{\sqrt{\pi}}{4}\end{aligned}$$</p><h1 id="补充证明为什么-dxdy-rdrdθ?">补充证明为什么 dxdy=rdrdθ?</h1><h1 id="直角坐标与极坐标的互化中,为什么-dxdy-rdrdθ?">直角坐标与极坐标的互化中,为什么 dxdy=rdrdθ?</h1><p><strong>Author:</strong> [予一人]</p><p><strong>Link:</strong> [<a href="https://www.zhihu.com/question/368888687/answer/992671463">https://www.zhihu.com/question/368888687/answer/992671463</a>]</p><p>这是一个好问题,而要真正理解这个问题,你首先需要在头脑里消除一种误会:</p><blockquote><p>${\rm d}x{\rm d}y,{\rm d}r{\rm d}\theta$ 这样的记号并不代表通常意义上的乘法,因此不能按照通常的多项式乘法来处理。事实上,它们是所谓的<strong>楔形积</strong>(wedge product),相当于**外积。**严格来讲,这记号应该写作 ${\rm d}x\wedge{\rm d}y,{\rm d}r\wedge{\rm d}\theta.$</p></blockquote><p>这种运算规定:</p><p>$a \wedge b=-(b \wedge a),~~~ a \wedge a=0\$ 于是</p><p>$\mathrm{d}r\wedge\mathrm{d}r=0,\quad \mathrm{d}\theta\wedge\mathrm{d}r=-(\mathrm{d}r\wedge\mathrm{d}\theta),\quad \mathrm{d}\theta\wedge\mathrm{d}\theta=0\$</p><p>于是<br>$$\begin{aligned}{\rm d}x\wedge{\rm d}y&={\rm d}(r\cos\theta)\wedge{\rm d}(r\sin\theta)\&=(-r\sin\theta{\rm d}\theta+\cos\theta{\rm d}r)\wedge(r\cos\theta{\rm d}\theta+\sin\theta{\rm d}r)\ &=-r\sin^2\theta ({\rm d}\theta \wedge{\rm d} r)+r\cos^2\theta ({\rm d}r \wedge{\rm d} \theta)\ &=r\sin^2\theta ({\rm d}r \wedge{\rm d} \theta)+r\cos^2\theta ({\rm d}r \wedge{\rm d} \theta)\ &=r(\sin^2 \theta+\cos^2 \theta) ({\rm d}r \wedge{\rm d} \theta)\ &=r({\rm d} r\wedge{\rm d} \theta)\\end{aligned}$$</p>]]></content>
<summary type="html"><h1 id="关于e的负x平方的积分">关于e的负x平方的积分</h1>
<p><strong>Author:</strong> [GGGGWHW]</p>
<p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.c</summary>
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</entry>
<entry>
<title>变限积分函数的求导</title>
<link href="https://www.czruby.eu.org/posts/418f6e6f.html"/>
<id>https://www.czruby.eu.org/posts/418f6e6f.html</id>
<published>2023-12-21T07:38:41.000Z</published>
<updated>2024-01-11T13:41:25.000Z</updated>
<content type="html"><![CDATA[<h1 id="变限积分函数的求导">变限积分函数的求导</h1><p><strong>Author:</strong> [羽墨志]</p><p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/376629429">https://zhuanlan.zhihu.com/p/376629429</a>]</p><h3 id="一、定义">一、定义</h3><p>设函数 $f(x)$ 在区间 $[a,b]$ 上连续,设 $x$ 为区间 $[a,b]$ 上的一点,考察定积分</p><p>$\int _a^xf(x)dx=\int _a^xf(t)dt$</p><p>如果上限x在区间 $[a,b]$ 上任意变动,则对于每一个取定的 $x$ 值,定积分 $\int _a^xf(t)dt$ 都有一个对应值,所以它在区间 $[a,b]$ 上定义了一个函数,记为</p><p>$\Phi(x)=\int _a^xf(t)dt$</p><p>该函数就是积分上限函数。</p><h3 id="二、变限积分函数求导公式">二、变限积分函数求导公式</h3><p>如果函数 $f(x)$ 连续, $\phi(x)$ 和 $\varphi(x)$ 可导,那么变限积分函数的求导公式可表示为</p><p>$\Phi’(x)=\frac{d}{dx}\int_{\phi(x)}^{\varphi(x)}f(t)dt=f[\varphi(x)]\varphi’(x)-f[\phi(x)]\phi’(x)$</p><p>[推导过程]</p><p>记函数f(x)的原函数为F(x),则有 $F’(x)=f(x)$ 或 $\int f(x)dx=F(x)+C$ 。则对 $\Phi(x)=\int_{\phi(x)}^{\varphi(x)}f(t)dt$ 应用牛顿-莱布尼茨公式$\int_a^bf(x)=F(x)|_a^b=F(b)-F(a)$</p><p>可得 $\Phi(x)=\int_{\phi(x)}^{\varphi(x)}f(t)dt=F(x)|_{\phi(x)}^{\varphi(x)}=F[\varphi(x)]-F[\phi(x)]$ 。</p><p>由函数和的求导法则 $[u(x)\pm v(x)]‘=u’(x)\pm v’(x)$ 可得</p><p>$\Phi^{‘}(x)=\frac{d}{dx}\int_{\phi(x)}^{\varphi(x)}f(t)dt={F[\varphi(x)]-F[\phi(x)]}’={F[\varphi(x)]}‘-{F[\phi(x)]}’$</p><p>由复合函数的求导法则 ${f[g(x)]}‘=f’[g(x)]g’(x)$ 可得</p><p>$\Phi^{‘}(x)={F[\varphi(x)]}’-{F[\phi(x)]}‘=F’[\varphi(x)]\varphi’(x)-F’[\phi(x)]\phi’(x)$</p><p>由 $F’(x)=f(x)$ 可知 $F’[\varphi(x)]=f[\varphi(x)]$ , $F’[\phi(x)]=f[\phi(x)]$ ,则上式可改写为</p><p>$\Phi^{‘}(x)=F’[\varphi(x)]\varphi’(x)-F’[\phi(x)]\phi’(x)=f[\varphi(x)]\varphi’(x)-f[\phi(x)]\phi’(x)$</p><h3 id="三、定理">三、定理</h3><p>定理1 如果函数 $f(x)$ 在区间 $[a,b]$ 上连续,则积分上限函数 $\Phi(x)=\int _a^xf(t)dt$ 在 $[a,b]$ 上具有导数,且导数为 $\Phi^{'}(x)=\frac{d}{dx}\int _a^xf(t)dt=f(x)$ 。</p><h3 id="四、应用">四、应用</h3><p>【例】求极限 $\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}$ 。</p><p>【解】令函数 $f(x)=\int_x^{2x}e^{t^2}dt$ ,则函数 $f(x)$ 在 $x=0$ 处连续,运用洛必达法则(L’Hôpital’s rule)则有</p><p>$\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}=\lim_{x \to 0} \frac{f’(x)}{x’}=\lim_{x \to 0} f’(x)$</p><p>这是一个典型的变限积分函数的求导,根据变限积分函数求导公式可得</p><p>$f’(x)=\frac{d}{dx}\int_x^{2x}e^{t^2}dt=e^{(2x)^2}(2x)‘-e^{x^2}(x)’=2e^{4x^2}-e^{x^2}$</p><p>则有 $\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}=\lim_{x \to 0}2e^{4x^2}-e^{x^2}=1$ 。</p><h3 id="五、扩展:变限积分复合函数导数">五、扩展:变限积分复合函数导数</h3><p>【情形一】 $F(x)=\int_{a}^{x}g(x)f(t)dt$</p><p>对于【情形一】,由于积分变量为 $t$ ,可将函数 $g(x)$ 提出到积分前面,则有 $F(x)=g(x)\cdot\int_{a}^{x}f(t)dt$ ,即原有的式子可变为 $g(x)$ 与 $\int_{a}^{x}f(t)dt$ 两项之积。</p><p>根据求导基本法则中的乘法法则可知,函数 $F(x)=f(x)\cdot g(x)$ 的导数为 $F(x)=f’(x) g(x)+f(x) g’(x)$ ,则有</p><p>$F’(x)=g’(x)\cdot\int_{a}^{x}f(t)dt + g(x)\cdot(\int_{a}^{x}f(t)dt)‘= g’(x)\int_{a}^{x}f(t)dt+g(x)f(x)$ 。</p><p>【例】已知函数 $F(x)=\int_{a}^{x}xtdt$ ,求函数 $F(x)$ 的导数。</p><p>【解】 $F’(x)=(x)‘\int_{a}^{x}tdt + x(\int_{a}^{x}tdt)’=\int_{a}^{x}tdt+x\cdot x={[\frac{1}{2}t^2]}_{a}^{x}+x^2=\frac{1}{2}(3x^2-a^2)$</p><p>【情形二】 $F(x)=\int_{a}^{x}f(t,x)dt$</p><p>对于【情形二】, $t$ 和 $x$ 无法分离,需要采用换元法将 $x$ 分离出来。</p><p>【例】求函数 $F(x)=\int_{a}^{x}tf(x^2-t^2)dt$ 的导数。</p><p>【解】令 $u=x^2-t^2$ ,则 $du=-2tdt\Rightarrow dt=-\frac{du}{2t}$,带入原式可得</p><p>$F(x)=\int_{x^2-a^2}^{x^2-x^2}tf(u)\frac{du}{-2t}=-\frac{1}{2}\int_{x^2-a^2}^{0}f(u)du$ ,即 $F(x)=\frac{1}{2}\int_{0}^{x^2-a^2}f(u)du$</p><p>则 $F’(x)=\frac{1}{2}f(x^2-a^2)(x^2-a^2)'=xf(x^2-a^2)$ 。</p>]]></content>
<summary type="html"><h1 id="变限积分函数的求导">变限积分函数的求导</h1>
<p><strong>Author:</strong> [羽墨志]</p>
<p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/376</summary>
<category term="转载" scheme="https://www.czruby.eu.org/categories/%E8%BD%AC%E8%BD%BD/"/>
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</entry>
<entry>
<title>分部积分法之列表积分法</title>
<link href="https://www.czruby.eu.org/posts/e955e5ba.html"/>
<id>https://www.czruby.eu.org/posts/e955e5ba.html</id>
<published>2023-12-20T06:50:23.000Z</published>
<updated>2024-01-11T15:21:35.000Z</updated>
<content type="html"><![CDATA[<h1 id="分部积分法→列表积分法">分部积分法→列表积分法</h1><p><strong>Author:</strong> [秋分丿]</p><p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p/81024770">https://zhuanlan.zhihu.com/p/81024770</a>]</p><p>翻了一下教材,似乎国内的教材很少有说列表积分法的。全都是到了分部积分就戛然而止了。虽然这样也不是不行,但是在后面的习题中都会有连续几次的分部积分法。这即使不让人感到头大也让人感到繁杂无比。</p><p>除此之外,还有那些分部积分到一半就开始用代数方法解出积分的情况。更是让人摸不着头脑:我怎么知道要这么做?我怎么知道到这一步要开始用代数方法?</p><p>实际上这一切,前者用列表积分便可以迎刃而解,后者用“半个”列表积分法(自己取的名字…)也能很简单。不过对于后者,还会有一些巧妙的变化。</p><p>内容出自《托马斯微积分》7.2分部积分一节。</p><hr><h2 id="正文"><strong>正文</strong></h2><p>首先来看一下普通的分部积分法:</p><p>对于$\int f(x)g(x)\text{d}x$ :</p><p>因为分部积分法是 $\int u\text{d}v=uv-\int v\text{d}u$</p><p>分出待会要处理的 $f(x)$ 和 $g(x)\text{d}x$ 。为了方便起见,就写出代表的 $u$ 和 $v$ :</p><p>$u=f(x)\quad \text{d}v=g(x)\text{d}x\$</p><p>则:</p><p>$\text{d}u=f^\prime (x)\text{d}x\quad v=G(x)\$</p><p>按照分部积分法的公式带入,则有:</p><p>$\int f(x)g(x)\text{d}x=f(x)G(x)-\int G(x)f^\prime(x)\text{d}x\$</p><p>如果要再来一次分部积分法呢?</p><p>显然,如果取 $u=G(x)\quad \text{d}v=f^\prime(x)\text{d}x$ ,那么待会取 $\text{d}u$ 和 $v$ 的时候就回到之前的积分去了。于是只能之前取了微分的继续微分,取了原函数的继续原函数。那接下去再来一次。</p><p>为了使结论明显直观,把 $g(x)$ 的原函数**不规范(不规范!不规范!不规范!!!)**的用 $\int g(x)$ 来表示求了一次原函数, $\iint g(x)$ 表示求了两次原函数。则第一次分部积分有:</p><p>$\int f(x)g(x)\text{d}x=f(x)[\int g(x)]-\int [\int g(x)]f^\prime(x)\text{d}x\$</p><p>接下来确定 $u$ 和 $v$ :</p><p>$u=f^\prime(x)\quad \text{d}v=[\int g(x)]\text{d}x\$</p><p>则:</p><p>$\text{d}u=f^{\prime\prime} (x)\text{d}x\quad v=\iint g(x)\$</p><p>按照分部积分法的公式带入,则有:</p><p>$\int f(x)g(x)\text{d}x=f(x)[\int g(x)]-f^\prime(x)[\iint g(x)]+ \int [\iint g(x)]f^{\prime\prime}(x)\text{d}x\$</p><p>可以注意到几点:</p><p>1)、选择出谁求导谁积分之后,求导的会一直求导,积分的会一直积分。</p><p>2)、函数直接相乘的项,求导的次数始终比积分的次数少一,或者说积分的次数比求导的次数多一。</p><p>3)、由于有一个负号,那么函数直接相乘的项的负号就应该是: $±±±…$</p><p>4)、剩下一个未完成的积分项,这个积分项中求导的函数的次数和积分的函数的次数相等。</p><p>第4)点得注意了,如果一个求导到零了,那么它就不存在了。就前面函数相乘的结果就是积分的结果。如果最后未完成的积分恰好是要求的积分式!那么就可以用代数的方法解出来。</p><hr><p>在上一段话中先考虑前者,即两个函数中有一个能求导到零。那么对另一个就是能积分“用于求导的函数求导到零的次数”那么多次。比如某一个函数第三次求导到零,那么另外一个函数应当能“坚持”积分三次。满足这些要求就可以用列表积分法了。然后将函数按照以下方式列出来:</p><p>$\frac{\text{d}}{\text{d}x}\downarrow$ $f(x)\ (1)\[2ex] f^\prime(x)\ (2)\[2ex] f^{\prime\prime}(x)\ (3)\[2ex] 0\ (4)\$ (空格空格)$(a)\ g(x)\[2ex] (b)\ \int g(x)\[2ex] ©\ \iint g(x)\[2ex] (d)\ \iiint g(x)$ $\int \downarrow$</p><p>那么,原积分就等于:</p><p>$原式=+(1)(b)-(2)©+(3)(d)\$</p><p>将上面四行每行对其排列着,然后划箭头: $(1)\rightarrow(b)$ 这样连起来,然后在上面标上 $+$ ,这样会变得很明显:</p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-c01efd6656c3c6da596ae9774647cde2_r.jpg" alt=""><br>这只是一个三次求导到零的情况,四次五次都可以。</p><p>这么这么看来,选择进行列表积分条件就很清晰了:</p><p><strong>1)、某一个函数能求导到零</strong></p><p><strong>2)、另一个函数能比较简单的积分和求导的函数那么多次。</strong></p><p>可见:</p><p>$\int x^n\sin {ax}\text{d}x$ 、 $\int x^n\text{e}^{ax}\text{d}x$</p><p>都是这样的经典类型(函数类型无论正弦还是余弦都可以,不再分别举例,下同)。除此之外当然还有其他类型的,比如 $\int x\tan^2x\text{d}x$ 、 $\int \arctan^2x\text{d}x$ .具体的可参见我的文章:</p><p><a href="https://zhuanlan.zhihu.com/p/78976041">https://zhuanlan.zhihu.com/p/78976041</a></p><p><strong>例</strong>:</p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-eb4e591da810d909137363c63585cf0b_r.jpg" alt=""></p><hr><p>对于后者,即最后未完成的积分式恰好是需要求的积分式,则使用半个列表积分法。</p><p>什么情况下最后未完成的积分式恰好是需要求的积分式?最经典的情况就是:</p><p>$\int \text{e}^x\sin x \text{d}x$</p><p>稍微思考一下就知道, $\text{e}^x$ 无论求导还是积分都是不会变的,而 $\sin x$ 两次积分或者求导之后便会变回之前的样子,这样的话就有了未完成的积分式恰好是需要求的。半个分部积分的操作模板与上面的很像,只是最后多了一个横着的“ $\rightarrow$ ”表示最后那一个积分。</p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-d9b87f4b43e24b4a9b1e955e42ccbf8e_r.jpg" alt=""><br>如此看来,使用半个列表积分的条件也很明显了:</p><p><strong>要么一个函数求导之后不变,另一个函数积分之后函数形式循环(积分和求导的函数可交换);要么两个函数积分和求导都循环。</strong></p><p>经典的题目类型就是: $\int \text{e}^{ax}\sin{bx}\text{d}x$</p><p>除此之外还有 $\int \sin(ax)\cos(bx)\text{d}x$ ,不过这类的题目也可以不用列表积分法求解,而是用积化和差求解。具体的可见我的文章:</p><p><a href="https://zhuanlan.zhihu.com/p/77893101">https://zhuanlan.zhihu.com/p/77893101</a></p><p><strong>例:</strong></p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-70c7a50e3e34568363156748f8cd8472_r.jpg" alt=""></p><hr><p>似乎在题头的时候说了“不过对于后者,还会有一些巧妙的变化。”</p><p>实际上,半个列表积分法不只是可以处理最后未完成的积分式与要求的积分式一样的情况。更深层的,前两种方法实际上都是在对最后一项未完成的积分项做文章:要么最后一项积分项为零,要么最后一项积分项和所求相同。</p><p>但是忽视了最简单的一种情况,(似乎)也是分部积分最初的目标:最后一项积分项能积、易积即可。最后一项积分项为零当然是易积的,但是易积绝不止为此。如接下来这题,就很巧妙:</p><p>$\int x\ln x\text{d}x\$</p><p>若以 $x$ 求导 $\ln x$ 积分,则有:</p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-e665eb1a284086ae2ff9b990633b6685_r.jpg" alt=""><br>积分的项再积分的话发现正是要求的积分,不能再往下积了。但是左边的恰好又是 $1$ ,这样一来,就又出现了要求的积分。半个列表积分法就安排的妥妥的。</p><p>若以$x$ 积分 $\ln x$ 求导,则有:</p><p><img src="https://img.czruby.eu.org/imgs/%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95%E4%B9%8B%E5%88%97%E8%A1%A8%E7%A7%AF%E5%88%86%E6%B3%95/v2-fce7d8a964a8435dedbe9404470f67b8_r.jpg" alt=""><br>求导的发现求导无止尽,越求导离“0”和原来的函数形式越远。但是骤然发现,求导和积分剩下的都是幂函数!这明显是很容易积分的。那么,果断使用半个积分法。</p><p>除此之外, $\int \ln x\text{d}x$ 也可以同样的用这样的方法求解,对 $\ln x$ 求导,对 “$1$ ”积分,用上的半个列表积分法也能同样解决问题。但是完全没有必要了…</p><p>读者可以尝试 $\int x^2\ln x\text{d}x$ ,分别尝试选用不同的函数求导和积分,与 $\int x\ln x\text{d}x$ 有异曲同工之妙。</p><hr><p>一眼看出到底是用求导到零的列表积分法,还是循环的半个列表积分法固然是好,这也是一个求积分的方向。特别是当看见某个乘积中的函数可以求导到零时,看见出现三角函数指数函数时,应当敏感的找到这个方向。</p><p>但是“巧妙的半个列表积分法”可以让人意识到,列表积分法应该是随心所欲的。熟练的话,就完全可以抛掉那些条条框框,只需要做到:**尝试一下列表积分法吧。**然后选择两个准备要求导和积分的函数,在草稿本或者脑子里求一列导,求一列积分。<strong>观察一下两列的函数变化以及关系</strong>,就可以从容的选择出接下来要做的步骤。</p>]]></content>
<summary type="html"><h1 id="分部积分法→列表积分法">分部积分法→列表积分法</h1>
<p><strong>Author:</strong> [秋分丿]</p>
<p><strong>Link:</strong> [<a href="https://zhuanlan.zhihu.com/p</summary>
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