Skip to content

Commit

Permalink
Deployed 8cf7923 with MkDocs version: 1.5.3
Browse files Browse the repository at this point in the history
  • Loading branch information
@TyrannosaurusLjx
TyrannosaurusLjx committed Apr 5, 2024
1 parent 27b6054 commit 83854d2
Show file tree
Hide file tree
Showing 15 changed files with 1,970 additions and 54 deletions.
64 changes: 64 additions & 0 deletions Notebook/操作系统/05-设备管理/index.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -528,6 +528,15 @@
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#_2" class="md-nav__link">
<span class="md-ellipsis">
输入输出程序接口,设备驱动程序接口
</span>
</a>

</li>

</ul>
Expand Down Expand Up @@ -1191,6 +1200,61 @@ <h2 id="io_2">I/O控制方式</h2>
<li><img alt="总结" src="../img/image-19.png" /></li>
</ul>
<h2 id="io_3">I/O软件层次结构</h2>
<ul>
<li>用户层软件</li>
<li>一般是提供与 I/O 相关的库函数</li>
<li>使用设备独立性软件向上提供的系统调用服务<ul>
<li>ex:print 会被翻译成 write 系统调用</li>
</ul>
</li>
<li>设备独立性软件</li>
<li>又被称为设备无关性软件</li>
<li>需要提供系统调用</li>
<li>实现设备保护<ul>
<li>设备被看作特殊的文件,因此有访问权限</li>
<li>差错处理(硬件故障)</li>
<li>设备的分配与回收</li>
<li>数据缓冲区管理</li>
<li>建立逻辑设备名到物理设备名的映射,以及调用相关的驱动程序</li>
</ul>
</li>
<li>设备驱动程序</li>
<li>由设备制造商提供</li>
<li>中断处理软件</li>
<li>I/O 应答中断</li>
<li>硬件</li>
<li>执行 I/O 操作</li>
</ul>
<h2 id="_2">输入输出程序接口,设备驱动程序接口</h2>
<ul>
<li>输入输出程序接口
在执行系统调用的时候,由于底层硬件的不同,需要提供不同的接口,比如硬盘支持序列但是键盘显然是不支持的,因此不可能提供一个单一的系统调用接口给上层用户
<img alt="alt text" src="../img/image-20.png" /></li>
<li>字符设备接口<ul>
<li>get/put系统调用</li>
</ul>
</li>
<li>块设备接口<ul>
<li>read/write 系统调用,seek修改读写指针位置</li>
</ul>
</li>
<li>网络设备接口<ul>
<li>网络套接字接口:socket接口.把套接字绑定到某个本地端口上</li>
<li><img alt="网络设备接口" src="../img/image-22.png" /></li>
</ul>
</li>
<li>
<p>概念:什么是阻塞/非阻塞 I/O</p>
<ul>
<li>阻塞 I/O:当一个进程发出一个 I/O 请求后,进程会一直等待直到 I/O 完成,比如从键盘读取字符</li>
<li>非阻塞 I/O:当一个进程发出一个 I/O 请求后,系统调用会被迅速返回,进程无需阻塞等待,比如往磁盘写数据</li>
</ul>
</li>
<li>
<p>设备驱动程序接口</p>
</li>
<li>!q驱动程序接口](img/image-23.png)</li>
</ul>



Expand Down
Binary file added Notebook/操作系统/img/image-20.png
View file
Loading
Sorry, something went wrong. Reload?
Sorry, we cannot display this file.
Sorry, this file is invalid so it cannot be displayed.
Binary file added Notebook/操作系统/img/image-21.png
View file
Loading
Sorry, something went wrong. Reload?
Sorry, we cannot display this file.
Sorry, this file is invalid so it cannot be displayed.
Binary file added Notebook/操作系统/img/image-22.png
View file
Loading
Sorry, something went wrong. Reload?
Sorry, we cannot display this file.
Sorry, this file is invalid so it cannot be displayed.
Binary file added Notebook/操作系统/img/image-23.png
View file
Loading
Sorry, something went wrong. Reload?
Sorry, we cannot display this file.
Sorry, this file is invalid so it cannot be displayed.
257 changes: 256 additions & 1 deletion Notebook/数学分析/06-不定积分/index.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -606,6 +606,17 @@



<label class="md-nav__link md-nav__link--active" for="__toc">


<span class="md-ellipsis">
不定积分
</span>


<span class="md-nav__icon md-icon"></span>
</label>

<a href="./" class="md-nav__link md-nav__link--active">


Expand All @@ -616,6 +627,172 @@

</a>



<nav class="md-nav md-nav--secondary" aria-label="Table of contents">






<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
Table of contents
</label>
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>

<li class="md-nav__item">
<a href="#_2" class="md-nav__link">
<span class="md-ellipsis">
不定积分的概念和运算法则
</span>
</a>

<nav class="md-nav" aria-label="不定积分的概念和运算法则">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#_3" class="md-nav__link">
<span class="md-ellipsis">
不定积分,微分的逆运算
</span>
</a>

<nav class="md-nav" aria-label="不定积分,微分的逆运算">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#611" class="md-nav__link">
<span class="md-ellipsis">
定义 6.1.1
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#612" class="md-nav__link">
<span class="md-ellipsis">
定义 6.1.2
</span>
</a>

</li>

</ul>
</nav>

</li>

<li class="md-nav__item">
<a href="#_4" class="md-nav__link">
<span class="md-ellipsis">
不定积分的线性性质
</span>
</a>

<nav class="md-nav" aria-label="不定积分的线性性质">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#611_1" class="md-nav__link">
<span class="md-ellipsis">
定理 6.1.1
</span>
</a>

</li>

</ul>
</nav>

</li>

</ul>
</nav>

</li>

<li class="md-nav__item">
<a href="#_5" class="md-nav__link">
<span class="md-ellipsis">
换元积分法和分部积分法
</span>
</a>

<nav class="md-nav" aria-label="换元积分法和分部积分法">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#_6" class="md-nav__link">
<span class="md-ellipsis">
换元积分法
</span>
</a>

<nav class="md-nav" aria-label="换元积分法">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#_7" class="md-nav__link">
<span class="md-ellipsis">
第一类换元积分法
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#_8" class="md-nav__link">
<span class="md-ellipsis">
第二类换元积分法
</span>
</a>

</li>

</ul>
</nav>

</li>

<li class="md-nav__item">
<a href="#_9" class="md-nav__link">
<span class="md-ellipsis">
分部积分法
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#_10" class="md-nav__link">
<span class="md-ellipsis">
基本积分表
</span>
</a>

</li>

</ul>
</nav>

</li>

<li class="md-nav__item">
<a href="#_11" class="md-nav__link">
<span class="md-ellipsis">
有理函数的不定积分及其应用
</span>
</a>

</li>

</ul>

</nav>

</li>


Expand Down Expand Up @@ -955,9 +1132,87 @@


<h1 id="_1">不定积分</h1>
<h2 id="_2">不定积分的概念和运算法则</h2>
<h3 id="_3">不定积分,微分的逆运算</h3>
<h4 id="611">定义 6.1.1</h4>
<ul>
<li>在某个区间上,函数<span class="arithmatex">\(F(x)\)</span>和<span class="arithmatex">\(f(x)\)</span>成立关系:<span class="arithmatex">\(F'(x)=f(x)\)</span>,或者等价的<span class="arithmatex">\(dF(x)=f(x)dx\)</span>,则称<span class="arithmatex">\(F(x)\)</span>是<span class="arithmatex">\(f(x)\)</span>的一个原函数</li>
</ul>
<h4 id="612">定义 6.1.2</h4>
<ul>
<li>一个函数<span class="arithmatex">\(f(x)\)</span>的原函数全体称为这个函数的不定积分,用<span class="arithmatex">\(\int f(x)dx\)</span>表示</li>
</ul>
<h3 id="_4">不定积分的线性性质</h3>
<h4 id="611_1">定理 6.1.1</h4>
<ul>
<li>若函数<span class="arithmatex">\(f(x)\)</span>和<span class="arithmatex">\(g(x)\)</span>都在区间<span class="arithmatex">\(I\)</span>上有原函数,则<span class="arithmatex">\(\forall k,l\in R\)</span>,函数<span class="arithmatex">\(kf+lg\)</span>的原函数也存在,并且:<span class="arithmatex">\(\int(kf(x)+lg(x))dx=k\int f(x)dx+l\int g(x)dx\)</span></li>
</ul>
<h2 id="_5">换元积分法和分部积分法</h2>
<h3 id="_6">换元积分法</h3>
<h4 id="_7">第一类换元积分法</h4>
<ul>
<li><span class="arithmatex">\(F(x)\)</span>是<span class="arithmatex">\(f(x)\)</span>的一个原函数,则<span class="arithmatex">\(\int f(\varphi(x))\varphi'(x)dx=F(\varphi(x))+C\)</span></li>
</ul>
<h4 id="_8">第二类换元积分法</h4>
<ul>
<li><span class="arithmatex">\(F(t)\)</span>是<span class="arithmatex">\(f(\varphi(t))\varphi'(t)\)</span>的一个原函数,且<span class="arithmatex">\(x = \varphi(t)\)</span>,则<span class="arithmatex">\(\int f(x)dx = \int f(\varphi(t))\varphi'(t)dt = F(\varphi^{-1}(x))+C\)</span></li>
</ul>
<h3 id="_9">分部积分法</h3>
<ul>
<li><span class="arithmatex">\(\int u(x)v'(x)dx = u(x)v(x) - \int u'(x)v(x)dx\)</span></li>
</ul>
<h3 id="_10">基本积分表</h3>
<p><span class="arithmatex">\(\begin{aligned}
&amp; \int x^\alpha \mathrm{d} x =
\begin{cases}
\frac{1}{\alpha+1} x^{\alpha+1}+C, &amp; \alpha \neq -1, \\
\ln |x|+C, &amp; \alpha=-1 ;
\end{cases}
&amp; \int \ln x \mathrm{~d} x = x(\ln x-1)+C ; \\
&amp; \int a^x \mathrm{~d} x = \frac{a^x}{\ln a}+C, \text{ 特别地 } \int \mathrm{e}^x \mathrm{~d} x = \mathrm{e}^x+C ;
&amp; \int \sin x \mathrm{~d} x = -\cos x+C ; \\
&amp; \int \cos x \mathrm{~d} x = \sin x+C \text{;}
&amp; \int \tan x \mathrm{~d} x = -\ln |\cos x|+C ;\\
&amp; \int \cot x \mathrm{~d} x = \ln |\sin x|+C \text{;}
&amp; \int \sec x \mathrm{~d} x = \ln |\sec x+\tan x|+C ; \\
&amp; \int \csc x \mathrm{~d} x = \ln |\csc x-\cot x|+C ;
&amp; \int \operatorname{sh} x \mathrm{~d} x = \operatorname{ch} x+C ; \\
&amp; \int \operatorname{ch} x \mathrm{~d} x = \operatorname{sh} x+C ;
&amp; \int \frac{\mathrm{d} x}{\sqrt{a^2-x^2}} = \arcsin \frac{x}{a}+C ; \\
&amp; \int \frac{\mathrm{d} x}{\sqrt{x^2 \pm a^2}} = \ln \left|x+\sqrt{x^2 \pm a^2}\right|+C ;
&amp; \int \frac{\mathrm{d} x}{x^2-a^2} = \frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C ; \\
&amp; \int \frac{\mathrm{d} x}{x^2+a^2} = \frac{1}{a} \arctan \frac{x}{a}+C ;
&amp; \int \sqrt{a^2-x^2} \mathrm{~d} x = \frac{1}{2} x \sqrt{a^2-x^2}+\frac{a^2}{2} \arcsin \frac{x}{a}+C ; \\
&amp; \int \sqrt{x^2 \pm a^2} \mathrm{~d} x = \frac{1}{2}\left(x \sqrt{x^2 \pm a^2} \pm a^2 \ln \left|x+\sqrt{x^2 \pm a^2}\right|\right)+C .
\end{aligned}\)</span></p>
<ul>
<li>函数 </li>
<li>一些证明:</li>
<li><span class="arithmatex">\(\int \sec(x)dx = \int \frac{1}{\cos(x)}dx = \int \frac{1}{\cos^2(x)}d\sin(x) \overset{t=\sin(x)}{=} \int \frac{1}{2}\ln(|\frac{t+1}{t-1}|) = \ln(|\frac{1+\sin(x)}{\cos(x)}|) = \ln(|\sec(x)+\tan(x)|) +C\)</span></li>
<li><span class="arithmatex">\(\int \frac{dx}{\sqrt{a^2-x^2}} = \int \frac{dx}{\sqrt{a^2(1-\frac{x^2}{a^2})}} = \int \frac{dx}{a\sqrt{1-(\frac{x}{a})^2}} \overset{t=\frac{x}{a}}{=} \int \frac{dt}{\sqrt{1-t^2}} = \arcsin(t)+C = \arcsin(\frac{x}{a}) + C\)</span></li>
<li><span class="arithmatex">\(\int\frac{dx}{\sqrt{x^2\pm a^2}} = \int\frac{dx}{\sqrt{a^2(1\pm\frac{x^2}{a^2})}} = \int\frac{dx}{a\sqrt{1\pm(\frac{x}{a})^2}} \overset{t=\frac{x}{a}}{=} \int\frac{dt}{\sqrt{1\pm t^2}} = \ln(|t+\sqrt{1+t^2}|) = \ln(|\frac{x}{a}+\sqrt{1+(\frac{x}{a})^2}|) + C = \ln(|x+\sqrt{x^2\pm a^2}|) + C\)</span></li>
<li>
<p><span class="arithmatex">\(\int\sqrt{a^2-x^2}dx\overset{x = a\cos(\theta)}{=} -\int |a^2\sin^2(\theta)d\theta = -a^2\int \frac{1-2\cos(2\theta)}{2}d\theta = -\frac{1}{2}a^2\theta+\frac{1}{2}\sin(\theta)\cos(\theta)+C = -\frac{1}{2}a^2\cos(\frac{x}{a})+\frac{1}{2}\frac{x}{a}\sin(\frac{x}{a}) + C= \frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin(\frac{x}{a})+C\)</span></p>
</li>
<li>
<p><span class="arithmatex">\(\begin{aligned}
&amp; \sqrt{x^2-a^2}=a \tan t, \Rightarrow d x=a \sec t \tan t d t , \\
&amp; \int \sqrt{x^2-a^2} d x=a^2 \int \sec t \tan ^2 t d t=a^2 \int \sec t\left(\sec ^2 t-1\right) d t \\
&amp; =a^2 \int \sec ^3 t d t-a^2 \int \sec t d t \\
&amp; =a^2 \int \sec ^3 t d t-a^2 \ln |\sec t+\tan t|+C_1 \\
&amp; \int \sec ^3 t=\int \frac{d(\sin t)}{\left(1-\sin ^2 t\right)^2}=\frac{1}{4} \int\left(\frac{1}{1+\sin t}+\frac{1}{1-\sin t}\right)^2 d(\sin t) \\
&amp; =\frac{1}{4} \int \frac{d(1+\sin t)}{(1+\sin t)^2}-\frac{1}{4} \int \frac{d(1-\sin t)}{(1-\sin t)^2}+\frac{1}{2} \int \frac{d(\sin t)}{1-\sin ^2 t} \\
&amp; =\frac{1}{4}\left(\frac{1}{1-\sin t}-\frac{1}{1+\sin t}\right)+\frac{1}{4} \ln \left(\frac{1+\sin t}{1-\sin t}\right)+C_2 \\
&amp; =\frac{1}{2} \tan t \sec t+\frac{1}{2} \ln (\sec t+\tan t)+C_2 \\
&amp; \int \sqrt{x^2-a^2} d x=\frac{a^2}{2} \tan t \sec t+\frac{a^2}{2} \ln |\sec t+\tan t|-a^2 \ln |\sec t+\tan t|+C \\
&amp; =\frac{a^2}{2} \tan t \sec t-\frac{a^2}{2} \ln |\sec t+\tan t|+C \\
&amp; =\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \ln \left|\frac{x}{a}+\frac{\sqrt{x^2-a^2}}{a}\right|+C
\end{aligned}\)</span></p>
</li>
<li>
<p><span class="arithmatex">\(\int\sqrt{x^2+a^2}dx\)</span>同理</p>
</li>
</ul>
<h2 id="_11">有理函数的不定积分及其应用</h2>



Expand Down
Loading

0 comments on commit 83854d2

Please sign in to comment.