diff --git a/WFFC.tex b/WFFC.tex new file mode 100644 index 0000000..94113ae --- /dev/null +++ b/WFFC.tex @@ -0,0 +1,263 @@ +%模板 +\documentclass[12pt,a4paper]{book} + +\usepackage{ctex} +\usepackage{graphicx}%插入图片的宏包 +\usepackage{makeidx} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb}%数学符号宏包 +\usepackage{textcomp}%树叶图案在这个包里 +\usepackage{bbding}%很多漂亮的图案 +\usepackage[dvipsnames, svgnames, x11names]{xcolor}%导入了所有颜色配置文件的宏包 +\usepackage{enumerate}%使用改宏包优化罗列环境 +\usepackage{CJKfntef} +\usepackage{ulem}%下划线宏包用法和样式如下： +%\uuline{双下划线} +%\uwave{波浪线} +%\sout{中间删除线} +%\xout{斜删除线} +%\dashuline{虚线} +%\dotuline{加点} +\usepackage{geometry}%页边距调整 +\geometry{left=1.5cm,right=1.5cm,bottom=1.8cm,top=1.8cm} +\usepackage{titletoc}%目录页的宏包 +\usepackage{titlesec}%改变章节或标题的样式的宏包 +\usepackage[bookmarks=true,colorlinks,linkcolor=black]{hyperref} + +%各类设置 +%章节或标题的样式 +\titleformat{\section}%设置section的样式 +{\raggedright\Large\bfseries}{\includegraphics[width=0.03\textwidth]{sigma.eps}\thesection\hspace*{0.6cm}}{0pt}{} +\titlespacing*{\subsection}{3em}{3em}{2em}[1em] +\titleformat{\subsection}%设置section的样式 +{\raggedright\large\bfseries}{\hspace*{1cm}\thesubsection\hspace*{0.6cm}}{0pt}{} +\titlespacing*{\subsection}{3em}{2em}{2em}[1em] +%格式如下：\titlespacing*{章节名称}{左间距}{(前)行间距}{(后)行间距}[右间距(一般都没用，填0.1em即可，但不能不填)] +\titlespacing*{\subsubsection}{2em}{1em}{1em}[1em] + +%制作索引 +\makeindex + +%公式编号设置 +\numberwithin{equation}{section} + +%字体设置 +\setCJKmainfont[BoldFont=STZHONGS.ttf]{STZHONGS.ttf}%需要查看电脑字体查找对应字体的文件英文文件名 +\setCJKfamilyfont{song}{STZHONGS.ttf} +\setCJKfamilyfont{kai}{simkai.ttf}%都是用来定义字体的(此处使用电脑自带楷书) + +%定义区域 +\definecolor{zs}{HTML}{2a7ae2}%定义某个颜色，对应颜色代号查表 +\definecolor{dy}{HTML}{FF359A} +\definecolor{dl}{HTML}{4B0091} +\definecolor{ff}{HTML}{007500} +\definecolor{bt}{HTML}{5B00AE} + + %定义环境 +\newtheorem{method}{\hspace*{0.3cm}\color{ff}\textleaf 方法}[section] +\newtheorem{defination}{\hspace*{0.3cm}\color{dy}\FiveFlowerOpen \hspace*{0.2cm}定义}[section] +\newtheorem{feature}{\color{1a9850}$ \square $性质}[section] +\newtheorem{inference}{\color{00ba38}$ \square $推论}[section] +\newtheorem{theorem}{\hspace*{0.3cm}\color{dl} $ \square $定理}[section] +\newtheorem{example}{\color{53a9ab}$ \square $例}[section] +\newtheorem{proof}{证明}[chapter] + +%文章标题 + \title{微分方程总结}\author{易鹏} + +%调整间距（倍数） +\linespread{1.5} + +%标题及目录 +\begin{document} +\maketitle %显示标题 +\newpage +\pagenumbering{Roman} +\setcounter{page}{0}%强行设置起始页码 +\tableofcontents +\thispagestyle{empty} + + +%正文部分 +\newpage +\pagenumbering{arabic} +\setcounter{page}{1} +\chapter{微分方程} +\section{微分方程的基本概念} +\vspace*{0.3cm} + +\begin{defination} + \hspace*{0.3cm}一般地，凡表示未知函数、未知函数的导数与自变量之间的关系的方程，叫做\uuline{微分方程}\index{微分方程!微分方程}，有时也简称\uuline{方程}. +\end{defination} + +\begin{defination} + \hspace*{0.3cm}微分方程中所出现的未知函数的最高阶导数的阶数，叫做\uuline{微分方程的阶}\index{微分方程!微分方程的阶}.例如三阶微分方程$x^3y'''+x^2y''-4xy'=3x^2+\sin{2x}$.一般地，$n$阶微分方程的形式是 + \begin{equation} + F(x,y,y',\cdots ,y^{(n)})=0\label{n阶微分方程} + \end{equation} + 其中$y^{(n)}$是必须有的，而其余项$x,y,y',\cdots ,y^{(n-1)}$可有可无. +\end{defination} + +\begin{defination} + \hspace*{0.3cm}找出这样的函数，把这个函数代入微分方程\eqref{n阶微分方程}能使该方程成为恒等式.这个函数就叫做该\uuline{微分方程的解}\index{微分方程!微分方程的解}.设函数$y=\varphi(x)$在区间$I$上有$n$阶连续导数,如果在区间$I$上, + $$F\left[x,\varphi(x),\varphi'(x),\cdots ,\varphi^{(n)}(x)\right]\equiv 0$$ + 那么函数就叫做微分方程\eqref{n阶微分方程}在区间$I$上的解. +\end{defination} + +\begin{defination} + \hspace*{0.3cm}如果微分方程的解中含有任意常数，且任意常数的个数与微分方程的阶数相同\footnote{这里所说的任意常数是相互独立的， 就是说，它们不能合并而使得任意常数的个数减少(参见本章第六节关于函数组的线性相关性)}，这样的解叫做\uuline{微分方程的通解}\index{微分方程!微分方程的通解}. +\end{defination} + +\hspace*{0.6cm}通解中含有任意常数，所以它还不能完全确定地反映某一客观事物的规律性.所以为了完全确定地反映客观事物的规律性，必须确定这些常数的值.为此要根据问题的实际情况，提出确定这些常数的条件.例如设一阶微分方程中的未知函数为$y=\varphi (x)$，通常给出的条件为$x=x_0,y=y_0$，也记为${y|}_{x=x_0}=y_0$. + +\begin{defination} + \hspace*{0.3cm}上述给出的条件就称为\uuline{初值条件}\index{微分方程!初值条件}.确定了通解中的任意常数以后，得到的解就叫做\uuline{微分方程的特解}\index{微分方程!微分方程的特解}.求微分方程$y'=f(x ,y)$满足初值条件${y|}_{x=x_0}=y_0$的特解这样一个问题，叫做一阶微分方程的\uuline{初值问题}\index{微分方程!初值问题}，记作 + \begin{equation} + \left\lbrace + \begin{array}{l} + y' = f(x,y) \\ + y\left | {_{x = {x_0}} = {y_0}} \right. + \end{array} + \right. + \label{初值问题1} + \end{equation} +\end{defination} + +\begin{defination} + 微分方程的解的图形是一条曲线，叫做微分方程的\uuline{积分曲线}\index{微分方程!积分曲线}.初值问题\eqref{初值问题1}的几何意义，就是求微分方程\eqref{初值问题1}的通过点$(x_0,y_0)$的那条积分曲线，二阶微分方程的初值问题 + \begin{equation} + \left\lbrace + \begin{array}{l} + y''= f(x,y,y') \\ + y\left | {_{x = {x_0}} = {y_0}} \right. \\ + y'\left | {_{x = {x_0}} = {y_0'}} \right. + \end{array} + \right. + \label{初值问题2} + \end{equation} + 的几何意义是求微分方程\eqref{初值问题2}的通过点$(x_0,y_0)$的且在该点处的切线斜率为$y_0'$的那条积分曲线. +\end{defination} + +\section{各种微分方程的求解} +\subsection{可分离变量的微分方程} + +\begin{defination} + \hspace*{0.3cm}一般地，如果一个微分方程能写成 + \begin{equation} + g(y)\,\mathrm{d}y=f(x)\,\mathrm{d}x + \label{可分离变量的微分方程} + \end{equation} + 的形式，就是说，能把微分方程写成一端只含$y$的函数和$\mathrm{d}y$，另一端写成只含$x$的函数和$\mathrm{d}x$的形式，那么原方程就称为\uuline{可分离变量的微分方程}\index{微分方程!可分离变量的微分方程}. +\end{defination} + +\begin{method} + \hspace*{0.3cm}方程\eqref{可分离变量的微分方程}的解法，只需要两边同时积分即可，即 + \begin{equation} + \int g(y)\,\mathrm{d}y=\int f(x)\,\mathrm{d}x + \label{可分离变量的微分方程2} + \end{equation} + 设$G(y)$和$F(x)$分别为$g(y)$和$f(x)$的原函数，那么 + \begin{equation} + G(y)=F(x)+C + \label{可分离变量的微分方程3} + \end{equation} +\end{method} + +\begin{defination} + \hspace*{0.3cm}如果方程\eqref{可分离变量的微分方程3}表示的是隐函数$y=\varphi (x)$，那么式\eqref{可分离变量的微分方程3}是微分方程\eqref{可分离变量的微分方程}的\uuline{隐式解}\index{微分方程!隐式解}.又由于关系式\eqref{可分离变量的微分方程3}中含有任意常数，因此式\eqref{可分离变量的微分方程3}微分方程\eqref{可分离变量的微分方程}的通解，也叫做\uuline{隐式通解}\index{微分方程!隐式通解}. +\end{defination} +\vspace{0.3cm} + +\subsection{齐次方程} + +\begin{defination} + \hspace*{0.3cm}如果一阶微分方程可以化成 + + \begin{equation} + \frac{\mathrm{d}y}{\mathrm{d}x}=\varphi (\frac{y}{x}) + \label{齐次方程1} + \end{equation} + + 的形式，那么这个微分方程就叫做\uuline{齐次方程}\index{微分方程!齐次方程}. +\end{defination} + +\begin{method} + \hspace*{0.3cm}求解齐次微分方程\eqref{齐次方程1}的方法如下：\\ + 第一步：换元 + $$u=\frac{y}{x}$$ + 其中$u$是关于$x$的一个新函数，所以 + $$y=ux,\frac{\mathrm{d}y}{\mathrm{d}x}=u+\frac{\mathrm{d}u}{\mathrm{d}x}x$$ + 第二步：代入式\eqref{齐次方程1}，得 + $$u+\frac{\mathrm{d}u}{\mathrm{d}x}x=\varphi (u)$$ + 第三步：分离变量,得到 + $$\frac{1}{\varphi (u)-u}\,\mathrm{d}u=\frac{1}{x}\,\mathrm{d}x$$ + 第四步：两边同时积分,得到 + $$\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u=\ln|x|\,$$ + 即 + $$x=\mathrm{e}^{\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u}$$ +\end{method} + +\begin{theorem} + \hspace{0.3cm}齐次方程\eqref{齐次方程1}的通解为 + \begin{equation} + x=\mathrm{e}^{\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u} + \label{齐次方程的通解} + \end{equation} + +\end{theorem} + + +\subsection{可化为齐次的方程} +\begin{method} + \hspace{0.3cm}方程 + \begin{equation} + \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2} + \label{非齐次方程} + \end{equation} + 当$c_1=c_2=0$时方程\eqref{非齐次方程}是齐次的，否则不是齐次的.在非齐次的情形，可以用变换法和待定系数法使得其变为齐次方程.令 + $$x=X+h,y=Y+k,$$ + 于是， + $$\mathrm{d}x=\mathrm{d}X,\mathrm{d}y=\mathrm{d}Y,$$ + 带入方程\eqref{非齐次方程}得到 + $$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{a_1X+b_1Y+a_1h+b_1k+c_1}{a_2X+b_2Y+a_2h+b_2k+c_2}.$$ + 那么要使方程\eqref{非齐次方程}是齐次方程，那么需要满足方程组 + $$\left\lbrace + \begin{array}{l} + a_1h+b_1k+c_1=0 \\ + a_2h+b_2k+c_2=0 + \end{array} + \right.$$ + 如果上述方程组的系数行列式 + $\left| \begin{array}{cc} + a_1 & b_1 \\ + a_2 & b_2 + \end{array} \right| \ne 0$ + ，即$\displaystyle\frac{a_2}{a_1}\ne \frac{b_1}{b_2} $，那么这个方程组存在唯一的$h$和$k$使得上述方程组成立.那么，方程\eqref{非齐次方程}便化为齐次方程 + $$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{a_1X+b_1Y}{a_2X+b_2Y}.$$ + 求出这个齐次方程的解以后在通解中要记得将元换回来，即代入$X=x-h,Y=y-k$.\vspace*{0.2cm}\\ + + \hspace*{0.5cm}特别地，当$\displaystyle\frac{a_2}{a_1}= \frac{b_1}{b_2}$时，无法求出$h$和$k$.这时令$\displaystyle\frac{a_2}{a_1}= \frac{b_1}{b_2}=\lambda $，那么方程\eqref{非齐次方程}可以写为 + $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a_1x+b_1y+c_1}{\lambda (a_1x+b_1y)+c_2}$$ + 引入新变量$v=a_1x+b_1y$，那么 + $$\frac{\mathrm{d}v}{\mathrm{d}x}=a_1+b_1 \, \frac{\mathrm{d}y}{\mathrm{d}x} \qquad \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{b_1}\left(\frac{\mathrm{d}v}{\mathrm{d}x}-a_1 \right) $$ + 代入方程\eqref{非齐次方程}得到 + $$\frac{1}{b_1}\left(\frac{\mathrm{d}v}{\mathrm{d}x}-a_1 \right)=\frac{v+c_1}{\lambda v+c_2}$$ + 这就变成了一个可以分离变量的方程，很容易就可以求解. +\end{method} + +\subsection{一阶线性微分方程} + + + + +%打印索引————————————— +\newpage +\addcontentsline{toc}{chapter}{附录} +\addcontentsline{toc}{section}{索引} +\appendix +\kaishu +\printindex +%——————————————— + +\end{document} \ No newline at end of file diff --git a/calculus.aux b/calculus.aux new file mode 100644 index 0000000..890d9d0 --- /dev/null +++ b/calculus.aux @@ -0,0 +1,49 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\bbl@cs{beforestart} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline 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b/calculus.synctex.gz differ diff --git a/calculus.tex b/calculus.tex new file mode 100644 index 0000000..64cb6e7 --- /dev/null +++ b/calculus.tex @@ -0,0 +1,267 @@ +\documentclass[10pt,a4paper]{article} +\usepackage{shevonNotebook} +\begin{document} +\numberwithin{equation}{subsection} +\allowdisplaybreaks%强制自动换行 +\maketitle +\thispagestyle{empty} +\newpage +\tableofcontents +\thispagestyle{empty} +\newpage +\setcounter{page}{1} +\section{常用三角函数公式} +\subsection{积化和差公式} +\begin{align} + \sin \alpha \cos \beta & =\frac{1}{2}[\sin (\alpha+\beta)+\sin(\alpha-\beta)] \\[7pt] + \cos \alpha \sin \beta & =\frac{1}{2}[\sin (\alpha+\beta)-\sin(\alpha-\beta)] \\[7pt] + \cos \alpha \cos \beta & =\frac{1}{2}[\cos (\alpha+\beta)+\cos(\alpha-\beta)] \\[7pt] + \sin \alpha \sin \beta & =-\frac{1}{2}[\cos (\alpha+\beta)-\cos(\alpha-\beta)] +\end{align} +\subsection{和差化积公式} +\begin{align} + \sin\alpha+\sin\beta & =2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\[7pt] + \sin\alpha-\sin\beta & =2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\[7pt] + \cos\alpha+\cos\beta & =2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\[7pt] + \cos\alpha-\cos\beta & =-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\[7pt] + \tan\alpha+\tan\beta & =\frac{\sin (\alpha+\beta)}{\cos\alpha\cdot\cos \beta} +\end{align} +\subsection{归一化公式} +\begin{align}\label{gyhgs} + \sin^2 x+\cos^2x & =1 \\[7pt] + \sec^2 x-\tan^2x & =1 \\[7pt] + \cosh^2x-\sinh^2x & =1 +\end{align} +\subsection{倍(半)角公式 \quad 降(升)幂公式} +\begin{align} + \sin^2x & =\frac{1}{2}(1-\cos 2x) \\[7pt] + \cos^2x & =\frac{1}{2}(1+\cos 2x) \\[7pt] + \tan^2x & =\frac{1-\cos 2x}{1+\cos 2x} \\[7pt] + \sin x & =2\sin\frac{x}{2}\cos\frac{x}{2} \\[7pt] + \cos x & =2\cos^2\frac{x}{2}-1=1-2\sin^2\frac{x}{2}=\cos^2\frac{x}{2}-\sin^2\frac{x}{2} \\[7pt] + \tan x & =\frac{2\tan(x/2)}{1-\tan^2(x/2)} +\end{align} +\subsection{万能公式} +令$ u=\tan\dfrac{x}{2} $则 +\begin{align} + \sin x=\frac{2u}{1+u^2} \\[7pt] + \cos x=\frac{1-u^2}{1+u^2} +\end{align} +\section{常用的佩亚诺型余项泰勒公式} +\mybox[53a9ab]{\leftline{\textbf{泰勒公式\small{ $ (\xi\ \mbox{在}x_0\ \mbox{与}x\ \mbox{之间}) $}:}}\vspace{-2em} + \begin{align} + f(x)= & f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o[(x-x_0)^n]\notag \\[7pt] + f(x)= & f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}\notag + \end{align}} + +\leftline\small{由此可得常用的泰勒公式} +\begin{align} + \mathrm{e}^{x} & =1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+\cdots+\frac{1}{n!}x^{n}+o(x^{n}) \\[7pt] + \ln(x+1) & =x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots+(-1)^{n-1}\frac{1}{n}x^{n}+o(x^{n}) +\end{align} +令 $n=2m$ 有, +\begin{align} + \sin x & =x-\frac{1}{6}x^{3}+\frac{1}{120}x^{5}+\cdots+(-1)^{m-1}\frac{1}{(2m-1)!}x^{2m-1}+o(x^{2m}) \\[7pt] + \cos x & =1-\frac{1}{2}x^2+\frac{1}{24}x^4-\cdots+(-1)^m \frac{1}{(2m)!}x^{2m}+o(x^{2m+1}) \\[7pt] + \tan x & =x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\cdots+o(x^{2m-1}) +\end{align} +\small\mbox{此处考虑到tan的泰勒公式其通项公式会出现伯努利数故此处略去其通项 } +\begin{align} + \arcsin x & =x+\frac{1}{6}x^3+\frac{3}{40}x^{5}+\cdots+o(x^{2m}) +\end{align} +常用于近似计算的泰勒公式 +\begin{align} + \frac{1}{1-x} & =1+x+x^2+x^3+\cdots+x^n+o(x^n) \\[7pt] + (1+x)^{\alpha} & =\sum_{i=0}^{n}\frac{\prod_{j=0}^{i-1}{(\alpha-j})}{i!}x^n+o(x^n)\notag \\[7pt] + & =1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2+\cdots+o(x^n) \\[7pt] + \alpha^x & =\sum_{i=0}^{n}\frac{\ln^n \alpha}{n!}x^n+o(x^n)\notag \\[7pt] + & =1+x\ln \alpha+\frac{\ln^2 \alpha}{2}x^2+\cdots+\frac{\ln^n \alpha}{n!}x^n+o(x^n) +\end{align} +\section{基本求导公式} +\begin{multicols}{2} + \begin{equation} + \left( C\right)'=0 + \end{equation} + \begin{equation} + \left( x^{\mu}\right)'=\mu x^{\mu-1} + \end{equation} + \begin{equation} + \left( \sin x\right)'=\cos x + \end{equation} + \begin{equation} + \left( \cos x\right)'=-\sin x + \end{equation} + \begin{equation} + \left( \tan x\right)'=\sec^2 x + \end{equation} + \begin{equation} + \left( \cot x\right)'=-\csc^2 x + \end{equation} + \begin{equation} + \left( \sec x\right)'=\sec x\cdot\tan x + \end{equation} + \begin{equation} + \left( \csc x\right)'=-\csc x\cdot\tan x + \end{equation} + \begin{equation} + \left( a^x\right)'=a^x\ln a\ (a>0,a\neq1) + \end{equation} + \begin{equation} + \left( \log_{a}x\right)'=\frac{1}{x\cdot\ln a}\ (a>0,a\neq1) + \end{equation} + \begin{equation} + \left( \arcsin x\right)'=\frac{1}{\sqrt{1-x^2}} + \end{equation} + \begin{equation} + \left( \arccos x\right)'=-\frac{1}{\sqrt{1-x^2}} + \end{equation} + \begin{equation} + \left( \arctan x\right)'=\frac{1}{1+x^2} + \end{equation} + \begin{equation} + \left( \mathrm{arccot}\, x\right)'=-\frac{1}{1+x^2} + \end{equation} +\end{multicols} +\section{函数图形描述中涉及到的重要公式} +\subsection{常用曲率计算公式} +\mybox[53a9ab]{ + 曲率的定义式$ K=\displaystyle\left|\frac{\mathrm{d}\alpha}{\mathrm{d}s}\right| $} +由定义式我们可以推得 +\begin{enumerate} + \item \textbf{直角坐标}系中的曲线\ $y=y(x)$\ 有曲率表达式 + \begin{equation} + K=\frac{\left|y''\right|}{\left( 1+y^{'2} \right)^{3/2}}\mbox{;} + \end{equation} + \item \textbf{参数方程}表示的曲线\ $x=\varphi(t),y=\psi(t) $\ 有曲率表达式 + \begin{equation} + K=\frac{\left|\varphi'(t)\psi''(t)-\varphi''(t)\psi'(t)\right|}{\left[ \varphi^{'2}(t) +\psi^{'2}(t) \right]^{3/2}}\mbox{;} + \end{equation} + \item \textbf{极坐标}表示的的曲线\ $y=y(x)$\ 有曲率表达式 + \begin{equation} + K=\frac{\left|r^2+2r^{'2}-r\cdot r''\right|}{\left(r^2+r^{'2}\right)^{3/2}}\mbox{;} + \end{equation} + \item 曲线在对应点\ $M(x,y)$\ 的曲率中心\ $D(\alpha,\beta)$\ 的坐标为 + \begin{equation} + \begin{cases} + \alpha=x-\displaystyle\frac{y'(1+y^{'2})^3}{y^{''2}} \\[7pt] + \beta=y+\displaystyle\frac{1+y^{'2}}{y''} + \end{cases} + \end{equation} + . +\end{enumerate} +\subsection{曲线的渐近线} +\begin{enumerate} + \item 若\ $\lim\limits_{ x\rightarrow \infty }f(x)=b$\ ,则称\ $y=b$\ 为曲线\ $f(x)$\ 的\textbf{水平渐近线} + \item 若\ $\lim\limits_{ x\rightarrow x_0 }f(x)=\infty$\ ,则称\ $x=x_0$\ 为曲线\ $f(x)$\ 的\textbf{垂直渐近线} + \item 若\ $\lim\limits_{ x\rightarrow \infty }[f(x)-(ax+b)]=0$\ ,其中 + $ + \begin{cases} + a=\displaystyle \lim\limits_{x\to \infty}\frac{f(x)}{x} \\[7pt] + b=\displaystyle \lim\limits_{x\to \infty}[f(x)-ax] + \end{cases} + $ + 则称\ $y=ax+b$\ 为曲线\ $f(x)$\ 的\textbf{斜渐近线} +\end{enumerate} +\section{基本积分公式} +\begin{align} + & \int k \,\mathrm{d}x=kx+C \ \mbox{(其中}k\mbox{为常数)} \\[7pt] + & \int x^\mu\,\mathrm{d}x=\frac{x^{\mu+1}}{\mu+1}+C\ (\mu\neq-1) \\[7pt] + & \int \frac{1}{x}\,\mathrm{d}x=\ln|x|+C \\[7pt] + & \int\frac{\mathrm{d}x}{1+x^2}=\arctan x+C \\[7pt] + & \int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin x+C_1=-\arccos x+C_2 \\[7pt] + & \int \sin x\,\mathrm{d}x=-\cos x+C \\[7pt] + & \int\cos x \,\mathrm{d}x=\sin x +C \\[7pt] + & \int\tan x\,\mathrm{d}x=-\ln |\cos x|+C \\[7pt] + & \int\cot x\,\mathrm{d}x=\ln |\sin x|+C \\[7pt] + & \int\csc x\,\mathrm{d}x=\int\frac{1}{\sin{x}}\,\mathrm{d}x=\frac{1}{2} + \ln{\left|\frac{1-\cos{x}}{1+\cos{x}}\right|}+C=\ln{\left|\tan{\frac{x}{2}}\right|}+C=\ln{\left|\csc{x}-\cot{x}\right|}+C \\[7pt] + & \int\sec x\,\mathrm{d}x=\int\frac{1}{\cos{x}}\,\mathrm{d}x=\frac{1}{2} + \ln{\left|\frac{1+\sin{x}}{1-\sin{x}}\right|}+C=\ln{\left|\sec{x}+\tan{x}\right|}+C \\[7pt] + & \int\sec^2 x\,\mathrm{d}x=\tan x +C \\[7pt] + & \int \csc^2 x\,\mathrm{d}x=-\cot x +C \\[7pt] + & \int \sec x\cdot\tan x \,\mathrm{d}x=\sec x+C \\[7pt] + & \int\csc x \cdot\cot x \,\mathrm{d}x=-\csc x+C \\[7pt] + & \int \mathrm{e}^x \,\mathrm{d}x=\mathrm{e}^x+C \\[7pt] + & \int a^x\,\mathrm{d}x=\frac{a^x}{\ln a}+C \\[7pt] + & \int \sinh x\,\mathrm{d}x=\cosh x+C \\[7pt] + & \int \cosh x\,\mathrm{d}x=\sinh x+C \\[7pt] + & \int \frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan\frac{x}{a}+C \\[7pt] + & \int \frac{1}{a^2-x^2}\,\mathrm{d}x=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C \\[7pt] + & \int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin\frac{x}{a}+C \\[7pt] + & \int \frac{1}{\sqrt{x^2\pm a^2}}\,\mathrm{d}x=\ln \left|x+\sqrt{x^2\pm a^2}\right|+C +\end{align} +\section{基本积分方法} +\subsection{第一类换元法} +\subsubsection{三角函数之积的积分} +\begin{enumerate} + \item 一般地,对于$ \sin^{2k+1}x\cos^n x $ 或$ \sin^n x \cos^{2k+1}x $ (其中$ k\in\mathbb{N} $)型函数的积分,总可依次作变换 $ u=\cos x $或$ u=\sin x $ ,从而求得结果; + \item 一般地,对于$ \sin^{2k}x\cos^{2l}x $ 或 (其中$ k,l\in \mathbb{N} $)型函数的积分,总是利用降幂公式$ \sin^2=\dfrac{1}{2}(1-\cos 2x), + \cos^2=\dfrac{1}{2}(1+\cos 2x) $化成$ \cos 2x $的多项式 ,从而求得结果; + \item 一般地,对于$ \tan^{n}x\sec^{2k} x $ 或$ \tan^{2k-1} x \sec^{n}x $ (其中$ n,k\in\mathbb{N}_{+} $)型函数的积分,总可依次作变换 $ u=\tan x $或$ u=\sec x $ ,从而求得结果; +\end{enumerate} +\subsubsection{常见的凑微分类型} +\begin{align} + & \int {f( ax + b){\rm{d}}x = }\frac{1}{a}\int {f(ax+b){\mathrm{d}}(ax + b)\;(a \neq 0)} \\[7pt] + & \int {f(a{x^{m + 1}} + b){x^m}{\rm{d}}x} = \frac{1}{{a(m + 1)}}\int {f(a{x^{m + 1}} + b){\rm{d}}(a{x^{m + 1}} + b)} \\[7pt] + & \int {f\left( \frac{1}{x}\right) \frac{{{\rm{d}}x}}{{{x^2}}}\;} = - \int {f\left( \frac{1}{x}\right) {\rm{d}}\left( \frac{{\rm{1}}}{x}\right) \;} \\[7pt] + & \int {f(\ln x)\frac{1}{x}} {\rm{d}}x = \int {f(\ln x){\rm{d(}}\ln x)} \\[7pt] + & \int {f({\mathrm{e}^x})} {\mathrm{e}^x}{\rm{d}}x = \int {f({\mathrm{e}^x}} ){\rm{d(}}{\mathrm{e}^x}) \\[7pt] + & \int {f(\sqrt x } )\frac{{{\rm{d}}x}}{{\sqrt x }} = 2\int {f(\sqrt x } ){\rm{d}}(\sqrt x ) \\[7pt] + & \int {f(\sin x)\cos x{\rm{d}}x = } \int {f(\sin x){\rm{d}}\sin x} \\[7pt] + & \int {f(\cos x)\sin x{\rm{d}}x = } - \int {f(\cos x){\rm{d}}\cos x} \\[7pt] + & \int {f(\tan x){{\sec }^2}} x{\rm{d}}x = \int {f(\tan x){\rm{d}}\tan x} \\[7pt] + & \int {f(\cot x){{\csc }^2}} x{\rm{d}}x = - \int {f(\cot x){\rm{d}}\cot x} \\[7pt] + & \int {f(\arcsin x)\frac{1}{{\sqrt {1 - {x^2}} }}} {\rm{d}}x = \int {f(\arcsin x){\rm{d}}\arcsin x} \\[7pt] + & \int {f(\arctan x)\frac{1}{{1 + {x^2}}}} {\rm{d}}x = \int {f(\arctan x){\rm{d}}\arctan x} \\[7pt] + & \int {\frac{{f'(x)}}{{f(x)}}} {\rm{d}}x = \int {\frac{{{\rm{d}}f(x)}}{{f(x)}}} = \ln \left| f(x)\right| + C +\end{align} +\subsection{有理函数的积分} +\subsubsection{部分分式} +\begin{align} + \frac{{P(x)}}{{Q(x)}} = & \frac{{{A_1}}}{{{{(x - a)}^\alpha }}} + \frac{{{A_2}}}{{{{(x - a)}^{\alpha - 1}}}} + \cdots + \frac{{{A_\alpha }}}{{x - a}} + \notag \\ + & \frac{{{B_1}}}{{{{(x - b)}^\beta }}} + \frac{{{B_2}}}{{{{(x - b)}^{\beta - 1}}}} + \cdots + \frac{{{B_\beta }}}{{x - b}} + \notag \\ + & \frac{{{M_1}x + {N_1}}}{{{{({x^2} + px + q)}^\lambda }}} + \frac{{{M_2}x + {N_2}}}{{{{({x^2} + px + q)}^{\lambda - 1}}}} + \cdots + \frac{{{M_\lambda }x + {N_\lambda }}}{{{x^2} + px + q}} + \notag \\ + & \cdots +\end{align} +\subsubsection{三角函数的特殊定积分} +\begin{align} + I_n & =\int_0^{\frac{\pi}{2}}\sin^nx\,\mathrm{d}x=\int_0^{\frac{\pi}{2}}\cos^nx\,\mathrm{d}x\notag \\ + I_n & =\frac{n-1}{n}I_{n-2}\notag \\ + & =\begin{cases} + \ \dfrac{{n - 1}}{n} \cdot \dfrac{{n - 3}}{{n - 2}} \cdots \dfrac{4}{5} \cdot \dfrac{2}{3}\quad (n\mbox{为大于}1\mbox{的正奇数}),I_1=1 \\[13pt] + \ \dfrac{{n - 1}}{n} \cdot \dfrac{{n - 3}}{{n - 2}} \cdots \dfrac{3}{4} \cdot \dfrac{1}{2} \cdot \dfrac{\pi }{2}\quad (n\mbox{为正偶数}),I_0=\dfrac{\pi}{2} + \end{cases} +\end{align} +\section{多元函数微分} +\subsection{偏导数} +\subsubsection{偏导数记法} +设函数$ z=f(x,y) $在区域$ D $内有偏导数:\[\frac{{\partial z}}{{\partial x}} = {f_x}(x,y),\quad \frac{{\partial z}}{{\partial y}} = {f_y}(x,y)\]他们的偏导数若存在,那么称其偏导数为$ z=f(x,y) $的\md{二阶偏导数}.按照对变量求导次序不同,有如下四个二阶偏导数:\begin{align*} + & \frac{\partial }{{\partial x}}\left( {\frac{{\partial z}}{{\partial x}}} \right) = \frac{{{\partial ^2}z}}{{\partial {x^2}}} = {f_{xx}}(x,y) & \frac{\partial }{{\partial y}}\left( {\frac{{\partial z}}{{\partial x}}} \right) = \frac{{{\partial ^2}z}}{{\partial x\partial y}} = {f_{xy}}(x,y) \\[7pt] + & \frac{\partial }{{\partial x}}\left( {\frac{{\partial z}}{{\partial y}}} \right) = \frac{{{\partial ^2}z}}{{\partial y\partial x}} = {f_{yx}}(x,y) & \frac{\partial }{{\partial y}}\left( {\frac{{\partial z}}{{\partial y}}} \right) = \frac{{{\partial ^2}z}}{{\partial {y^2}}} = {f_{yy}}(x,y). +\end{align*} +\subsection{全微分} + + +\section{微分方程(该部分将会采用详细的讲义样式)} +\subsection{微分方程的基本概念} +\begin{definition}[微分方程的定义] + 一般地,凡表示\md{未知函数},\md{未知函数的导数}与\md{自变量之间的关系的方程},称为\textbf{微分方程}. + 其中未知函数的最高阶导数的阶数,称为\textbf{微分方程的阶}.一般地,$n$阶微分方程的形式是:\[ + F( + x,y,y',\cdots,y^{(n)} + )=0 + \] +\end{definition} +\begin{definition}[微分方程的解] + 设函数$y=\varphi(x)$在区间$I$上有$n$阶连续导数,如果在区间$I$上有:\[ + F[x,\varphi(x),\varphi'(x),\cdots,\varphi^{(n)}(x)]\equiv 0, + \]那么函数$y=\varphi(x)$称为\textbf{微分方程$ F(x,y,y',\cdots,y^{(n)})=0$在区间$I$上的解} + 特别地,如果微分方程的解\md{含有任意常数}\footnote{此处的任意常数必须是相互独立的,或者说他们线性无关.},且任意常数的个数与微分方程的阶数相同,这样的解称为\textbf{微分方程的通解}. + + 通解中时常含有任意常数，所以它还不能完全确定地反映某一客观事物的规律性.所以为了完全确定地反映客观事物的规律性,必须确定这些常数的值.为此要根据问题的实际情况，提出确定这些常数的条件.例如设一阶微分方程中的未知函数为$y=\varphi (x)$，通常给出的条件为$x=x_0,y=y_0$，也记为$y|_{x=x_0}=y_0$. + + 因此我们定义,在实际问题中所给定的能够确定这些常数的条件称为\textbf{初值条件}.由初值条件确定了常数的值进而可以得到微分方程的\textbf{特解}. +\end{definition} +\section{可分离变量的微分方程} +本节我们将讨论一阶微分方程$y'=f(x,y)$ +\end{document} diff --git a/calculus.thm b/calculus.thm new file mode 100644 index 0000000..e69de29 diff --git a/calculus.toc b/calculus.toc new file mode 100644 index 0000000..c7f709a --- /dev/null +++ b/calculus.toc @@ -0,0 +1,28 @@ +\babel@toc {nil}{} +\contentsline {section}{\numberline {1}常用三角函数公式}{1}{section.1}% +\contentsline {subsection}{\numberline {1.1}积化和差公式}{1}{subsection.1.1}% +\contentsline {subsection}{\numberline {1.2}和差化积公式}{1}{subsection.1.2}% +\contentsline {subsection}{\numberline {1.3}归一化公式}{1}{subsection.1.3}% +\contentsline {subsection}{\numberline {1.4}倍(半)角公式 \hskip 1em\relax 降(升)幂公式}{1}{subsection.1.4}% +\contentsline {subsection}{\numberline {1.5}万能公式}{2}{subsection.1.5}% +\contentsline {section}{\numberline {2}常用的佩亚诺型余项泰勒公式}{2}{section.2}% +\contentsline {section}{\numberline {3}基本求导公式}{2}{section.3}% +\contentsline {section}{\numberline {4}函数图形描述中涉及到的重要公式}{3}{section.4}% +\contentsline {subsection}{\numberline {4.1}常用曲率计算公式}{3}{subsection.4.1}% +\contentsline {subsection}{\numberline {4.2}曲线的渐近线}{3}{subsection.4.2}% +\contentsline {section}{\numberline {5}基本积分公式}{3}{section.5}% +\contentsline {section}{\numberline {6}基本积分方法}{4}{section.6}% +\contentsline {subsection}{\numberline {6.1}第一类换元法}{5}{subsection.6.1}% +\contentsline {subsubsection}{\numberline {6.1.1}三角函数之积的积分}{5}{subsubsection.6.1.1}% +\contentsline {subsubsection}{\numberline {6.1.2}常见的凑微分类型}{5}{subsubsection.6.1.2}% +\contentsline {subsection}{\numberline {6.2}有理函数的积分}{5}{subsection.6.2}% +\contentsline {subsubsection}{\numberline {6.2.1}部分分式}{5}{subsubsection.6.2.1}% +\contentsline {subsubsection}{\numberline {6.2.2}三角函数的特殊定积分}{6}{subsubsection.6.2.2}% +\contentsline {section}{\numberline {7}多元函数微分}{6}{section.7}% +\contentsline {subsection}{\numberline {7.1}偏导数}{6}{subsection.7.1}% +\contentsline {subsubsection}{\numberline {7.1.1}偏导数记法}{6}{subsubsection.7.1.1}% +\contentsline {subsection}{\numberline {7.2}全微分}{6}{subsection.7.2}% +\contentsline {section}{\numberline {8}微分方程(该部分将会采用详细的讲义样式)}{6}{section.8}% +\contentsline {subsection}{\numberline {8.1}微分方程的基本概念}{6}{subsection.8.1}% +\contentsline {section}{\numberline {9}可分离变量的微分方程}{7}{section.9}% +\contentsfinish diff --git a/cover.png b/cover.png new file mode 100644 index 0000000..e5abcf5 Binary files /dev/null and b/cover.png differ diff --git a/covertitle.eps b/covertitle.eps new file mode 100644 index 0000000..7f30c4a Binary files /dev/null and b/covertitle.eps differ diff --git a/covertitle.jpg b/covertitle.jpg new file mode 100644 index 0000000..deb7caf Binary files /dev/null and b/covertitle.jpg differ diff --git a/shevonNotebook.sty b/shevonNotebook.sty new file mode 100644 index 0000000..67571e6 --- /dev/null +++ b/shevonNotebook.sty @@ -0,0 +1,180 @@ + % |-----------------------------------------------------------------------------------------------|% + % | Shevon Kuan 课程笔记基本模板 |% + % | 版本号:3.15b |% + % | 页面配置:A4 paper侧栏设计 |% + % |-----------------------------------------------------------------------------------------------|% +\ProvidesPackage{shevonNotebook} +\usepackage{babel} +\usepackage{ctex} +\usepackage[dvipsnames, svgnames, x11names]{xcolor} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb}%数学符号宏包 +\usepackage{geometry} +\usepackage{fancyhdr} +\usepackage{framed} +\usepackage{fontspec} +\usepackage{ntheorem} +\usepackage{nomencl} +\usepackage{nicematrix} +\usepackage{multicol} +\usepackage{indentfirst}%首行缩进 + \setlength{\parindent}{2em} +\usepackage[bookmarks=true,colorlinks,linkcolor=black]{hyperref} +\usepackage{CJKfntef} +\usepackage{amsmath,bm}%重要宏包,粗斜体\bm +\usepackage{makeidx}%重要宏包，用于添加索引 +\usepackage{pgf,tikz,pgfplots}%一般绘图宏包 +\pgfplotsset{compat=1.15} +\usepackage{mathrsfs} +\usetikzlibrary{arrows} +\usepackage{tikz-3dplot}%3d绘图宏包 +\usepackage{tcolorbox}%box宏包 +\usepackage{subfig} +\usepackage{autobreak} +\usepackage{enumerate} +\usepackage{wasysym} 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+\newcommand{\theorem}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[2a7ae2]{\color{2a7ae2}{\CJKfamily{heiti}定理\ \con{A}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } +\newcommand{\definition}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[53a9ab]{\color{53a9ab}{\CJKfamily{heiti}定义\ \con{B}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } +\newcommand{\feature}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[ffa725]{\color{ffa725}{\CJKfamily{heiti}性质\ \con{C}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } +\newcommand{\inference}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[1a9850]{\color{1a9850}{\CJKfamily{heiti}推论\ \con{D}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } +\newcommand{\method}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[6a3d9a]{\color{6a3d9a}{\CJKfamily{heiti}方法\ \con{E}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } +\newcommand{\example}[1][]{\vspace{0.5em}\par\noindent\hspace{-8pt}\myboxp[f8766d]{\color{f8766d}{\CJKfamily{heiti}例题\ \con{F}\hspace{1em}{\CJKfamily{songbold}#1} } }\vspace{0.1em}\par } + +%符号设置------------------------------- +{\color{titlepurple}} +%标题配置————————————— +\title{\vspace{5cm} + \begin{center} + \includegraphics[width=16cm]{covertitle.eps} + %\includegraphics[width=8cm]{cover2.png} + %\includegraphics[width=8cm]{cover3.png} + %\includegraphics[width=12cm]{cover4.png} + \end{center} +\vspace{5cm}} +\author{\large\color{titlepurple}关舒文\\{\color{titlepurple}\kaishu{华南理工大学}}} +\date{\color{titlepurple}\small{Latest Update\ :\ \today}} + +%章节或标题的样式------------------- +%\titleformat{\chapter}{\CJKfamily{title}\huge\color{titlepurple}}{\includegraphics[height=2cm]{sigmaformal.eps} 第\ \thechapter\ 章\ }{0pt}{} +\titleformat{\section}{\CJKfamily{hei}\large\color{titlepurple}}{\bfseries{\thesection}\quad }{0pt}{} +\titleformat{\subsection}{\CJKfamily{hei}\large\color{titlepurpleb}}{\bfseries{\thesubsection}\quad }{0pt}{} +\titleformat{\subsubsection}{\CJKfamily{hei}\large\color{titlepurplec}}{\bfseries{\thesubsubsection}\quad }{0pt}{} +\renewcommand{\nomname}{符号说明} +%自定义优化命令----------------------- +\renewcommand{\a}{\ensuremath A}% +\renewcommand{\b}{\ensuremath B}% +\renewcommand{\c}{\ensuremath C}% +\renewcommand{\o}{\ensuremath \varnothing}% +\renewcommand{\textbf}[1]{{\CJKfamily{heiti}#1}}% +\newcommand{\md}[1]{{\,\color{purple}#1\,}}% +\newcommand{\cf}[1]{\textit{cf.Fig~\ref{#1}}}% +\newcommand{\margin}[1]{{\marginpar{\footnotesize\kaishu\hspace*{2em}#1}}} +%\renewcommand{\chapter}[1]{{\chapter{#1}\thispagestyle{empty}} }% +%使用了自定义页眉页脚--------------- +\pagestyle{fancy} +%\renewcommand{\chaptermark}[1]{\markboth{\;第\ \thechapter\ 章\quad#1\;}{}} +\renewcommand{\sectionmark}[1]{\markright{\;\thesection\ #1\;}} +\fancyhf{} +%\fancyfoot[C]{\bfseries\thepage} +\fancyhead[LO]{\small\CJKfamily{heilight}\rightmark} +\fancyhead[RE]{\small\CJKfamily{heilight}\leftmark} +\fancyhead[RO,LE]{\;\thepage\;} +\fancyfoot[RO,LE]{\small\CJKfamily{heilight}{微积分 重要公式}} +\fancyfoot[RE,LO]{\small\CJKfamily{heilight}The Important Equations of Calculus} +\renewcommand{\headrulewidth}{0.4pt} % 注意不用\setlength +%\renewcommand{\footrulewidth}{0pt} diff --git a/sigma.eps b/sigma.eps new file mode 100644 index 0000000..4452f2c Binary files /dev/null and b/sigma.eps differ