├── .gitignore ├── WFFC.tex ├── calculus.pdf ├── calculus.tex ├── cover.png ├── covertitle.eps ├── covertitle.jpg ├── shevonNotebook.sty └── sigma.eps /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | .*.lb 14 | 15 | ## Intermediate documents: 16 | *.dvi 17 | *.xdv 18 | *-converted-to.* 19 | # these rules might exclude image files for figures etc. 20 | # *.ps 21 | # *.eps 22 | # *.pdf 23 | 24 | ## Generated if empty string is given at "Please type another file name for output:" 25 | # *.pdf 26 | 27 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 28 | *.bbl 29 | *.bcf 30 | *.blg 31 | *-blx.aux 32 | *-blx.bib 33 | *.run.xml 34 | 35 | ## Build tool auxiliary files: 36 | *.fdb_latexmk 37 | *.synctex 38 | *.synctex(busy) 39 | *.synctex.gz 40 | *.synctex.gz(busy) 41 | *.pdfsync 42 | 43 | ## Build tool directories for auxiliary files 44 | # latexrun 45 | latex.out/ 46 | 47 | ## Auxiliary and intermediate files from other packages: 48 | # algorithms 49 | *.alg 50 | *.loa 51 | 52 | # achemso 53 | acs-*.bib 54 | 55 | # amsthm 56 | *.thm 57 | 58 | # beamer 59 | *.nav 60 | *.pre 61 | *.snm 62 | *.vrb 63 | 64 | # changes 65 | *.soc 66 | 67 | # comment 68 | *.cut 69 | 70 | # cprotect 71 | *.cpt 72 | 73 | # elsarticle (documentclass of Elsevier journals) 74 | *.spl 75 | 76 | # endnotes 77 | *.ent 78 | 79 | # fixme 80 | *.lox 81 | 82 | # feynmf/feynmp 83 | *.mf 84 | *.mp 85 | *.t[1-9] 86 | *.t[1-9][0-9] 87 | *.tfm 88 | 89 | #(r)(e)ledmac/(r)(e)ledpar 90 | *.end 91 | *.?end 92 | *.[1-9] 93 | *.[1-9][0-9] 94 | *.[1-9][0-9][0-9] 95 | *.[1-9]R 96 | *.[1-9][0-9]R 97 | *.[1-9][0-9][0-9]R 98 | *.eledsec[1-9] 99 | *.eledsec[1-9]R 100 | *.eledsec[1-9][0-9] 101 | *.eledsec[1-9][0-9]R 102 | *.eledsec[1-9][0-9][0-9] 103 | *.eledsec[1-9][0-9][0-9]R 104 | 105 | # glossaries 106 | *.acn 107 | *.acr 108 | *.glg 109 | *.glo 110 | *.gls 111 | *.glsdefs 112 | *.lzo 113 | *.lzs 114 | 115 | # uncomment this for glossaries-extra (will ignore makeindex's style files!) 116 | # *.ist 117 | 118 | # gnuplottex 119 | *-gnuplottex-* 120 | 121 | # gregoriotex 122 | *.gaux 123 | *.glog 124 | *.gtex 125 | 126 | # htlatex 127 | *.4ct 128 | *.4tc 129 | *.idv 130 | *.lg 131 | *.trc 132 | *.xref 133 | 134 | # hyperref 135 | *.brf 136 | 137 | # knitr 138 | *-concordance.tex 139 | # TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files 140 | # *.tikz 141 | *-tikzDictionary 142 | 143 | # listings 144 | *.lol 145 | 146 | # luatexja-ruby 147 | *.ltjruby 148 | 149 | # makeidx 150 | *.idx 151 | *.ilg 152 | *.ind 153 | 154 | # minitoc 155 | *.maf 156 | *.mlf 157 | *.mlt 158 | *.mtc[0-9]* 159 | *.slf[0-9]* 160 | *.slt[0-9]* 161 | *.stc[0-9]* 162 | 163 | # minted 164 | _minted* 165 | *.pyg 166 | 167 | # morewrites 168 | *.mw 169 | 170 | # newpax 171 | *.newpax 172 | 173 | # nomencl 174 | *.nlg 175 | *.nlo 176 | *.nls 177 | 178 | # pax 179 | *.pax 180 | 181 | # pdfpcnotes 182 | *.pdfpc 183 | 184 | # sagetex 185 | *.sagetex.sage 186 | *.sagetex.py 187 | *.sagetex.scmd 188 | 189 | # scrwfile 190 | *.wrt 191 | 192 | # sympy 193 | *.sout 194 | *.sympy 195 | sympy-plots-for-*.tex/ 196 | 197 | # pdfcomment 198 | *.upa 199 | *.upb 200 | 201 | # pythontex 202 | *.pytxcode 203 | pythontex-files-*/ 204 | 205 | # tcolorbox 206 | *.listing 207 | 208 | # thmtools 209 | *.loe 210 | 211 | # TikZ & PGF 212 | *.dpth 213 | *.md5 214 | *.auxlock 215 | 216 | # todonotes 217 | *.tdo 218 | 219 | # vhistory 220 | *.hst 221 | *.ver 222 | 223 | # easy-todo 224 | *.lod 225 | 226 | # xcolor 227 | *.xcp 228 | 229 | # xmpincl 230 | *.xmpi 231 | 232 | # xindy 233 | *.xdy 234 | 235 | # xypic precompiled matrices and outlines 236 | *.xyc 237 | *.xyd 238 | 239 | # endfloat 240 | *.ttt 241 | *.fff 242 | 243 | # Latexian 244 | TSWLatexianTemp* 245 | 246 | ## Editors: 247 | # WinEdt 248 | *.bak 249 | *.sav 250 | 251 | # Texpad 252 | .texpadtmp 253 | 254 | # LyX 255 | *.lyx~ 256 | 257 | # Kile 258 | *.backup 259 | 260 | # gummi 261 | .*.swp 262 | 263 | # KBibTeX 264 | *~[0-9]* 265 | 266 | # TeXnicCenter 267 | *.tps 268 | 269 | # auto folder when using emacs and auctex 270 | ./auto/* 271 | *.el 272 | 273 | # expex forward references with \gathertags 274 | *-tags.tex 275 | 276 | # standalone packages 277 | *.sta 278 | 279 | # Makeindex log files 280 | *.lpz 281 | 282 | # xwatermark package 283 | *.xwm 284 | 285 | # REVTeX puts footnotes in the bibliography by default, unless the nofootinbib 286 | # option is specified. Footnotes are the stored in a file with suffix Notes.bib. 287 | # Uncomment the next line to have this generated file ignored. 288 | #*Notes.bib 289 | 290 | # VSCode coverage 291 | coverage/ 292 | README.html -------------------------------------------------------------------------------- /WFFC.tex: -------------------------------------------------------------------------------- 1 | %模板 2 | \documentclass[12pt,a4paper]{book} 3 | 4 | \usepackage{ctex} 5 | \usepackage{graphicx}%插入图片的宏包 6 | \usepackage{makeidx} 7 | \usepackage{amsmath} 8 | \usepackage{amsfonts} 9 | \usepackage{amssymb}%数学符号宏包 10 | \usepackage{textcomp}%树叶图案在这个包里 11 | \usepackage{bbding}%很多漂亮的图案 12 | \usepackage[dvipsnames, svgnames, x11names]{xcolor}%导入了所有颜色配置文件的宏包 13 | \usepackage{enumerate}%使用改宏包优化罗列环境 14 | \usepackage{CJKfntef} 15 | \usepackage{ulem}%下划线宏包用法和样式如下： 16 | %\uuline{双下划线} 17 | %\uwave{波浪线} 18 | %\sout{中间删除线} 19 | %\xout{斜删除线} 20 | %\dashuline{虚线} 21 | %\dotuline{加点} 22 | \usepackage{geometry}%页边距调整 23 | \geometry{left=1.5cm,right=1.5cm,bottom=1.8cm,top=1.8cm} 24 | \usepackage{titletoc}%目录页的宏包 25 | \usepackage{titlesec}%改变章节或标题的样式的宏包 26 | \usepackage[bookmarks=true,colorlinks,linkcolor=black]{hyperref} 27 | 28 | %各类设置 29 | %章节或标题的样式 30 | \titleformat{\section}%设置section的样式 31 | {\raggedright\Large\bfseries}{\includegraphics[width=0.03\textwidth]{sigma.eps}\thesection\hspace*{0.6cm}}{0pt}{} 32 | \titlespacing*{\subsection}{3em}{3em}{2em}[1em] 33 | \titleformat{\subsection}%设置section的样式 34 | {\raggedright\large\bfseries}{\hspace*{1cm}\thesubsection\hspace*{0.6cm}}{0pt}{} 35 | \titlespacing*{\subsection}{3em}{2em}{2em}[1em] 36 | %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)] 37 | \titlespacing*{\subsubsection}{2em}{1em}{1em}[1em] 38 | 39 | %制作索引 40 | \makeindex 41 | 42 | %公式编号设置 43 | \numberwithin{equation}{section} 44 | 45 | %字体设置 46 | \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 47 | \setCJKfamilyfont{song}{STZHONGS.ttf} 48 | \setCJKfamilyfont{kai}{simkai.ttf}%都是用来定义字体的(此处使用电脑自带楷书) 49 | 50 | %定义区域 51 | \definecolor{zs}{HTML}{2a7ae2}%定义某个颜色，对应颜色代号查表 52 | \definecolor{dy}{HTML}{FF359A} 53 | \definecolor{dl}{HTML}{4B0091} 54 | \definecolor{ff}{HTML}{007500} 55 | \definecolor{bt}{HTML}{5B00AE} 56 | 57 | %定义环境 58 | \newtheorem{method}{\hspace*{0.3cm}\color{ff}\textleaf 方法}[section] 59 | \newtheorem{defination}{\hspace*{0.3cm}\color{dy}\FiveFlowerOpen \hspace*{0.2cm}定义}[section] 60 | \newtheorem{feature}{\color{1a9850}$ \square $性质}[section] 61 | \newtheorem{inference}{\color{00ba38}$ \square $推论}[section] 62 | \newtheorem{theorem}{\hspace*{0.3cm}\color{dl} $ \square $定理}[section] 63 | \newtheorem{example}{\color{53a9ab}$ \square $例}[section] 64 | \newtheorem{proof}{证明}[chapter] 65 | 66 | %文章标题 67 | \title{微分方程总结}\author{易鹏} 68 | 69 | %调整间距（倍数） 70 | \linespread{1.5} 71 | 72 | %标题及目录 73 | \begin{document} 74 | \maketitle %显示标题 75 | \newpage 76 | \pagenumbering{Roman} 77 | \setcounter{page}{0}%强行设置起始页码 78 | \tableofcontents 79 | \thispagestyle{empty} 80 | 81 | 82 | %正文部分 83 | \newpage 84 | \pagenumbering{arabic} 85 | \setcounter{page}{1} 86 | \chapter{微分方程} 87 | \section{微分方程的基本概念} 88 | \vspace*{0.3cm} 89 | 90 | \begin{defination} 91 | \hspace*{0.3cm}一般地，凡表示未知函数、未知函数的导数与自变量之间的关系的方程，叫做\uuline{微分方程}\index{微分方程!微分方程}，有时也简称\uuline{方程}. 92 | \end{defination} 93 | 94 | \begin{defination} 95 | \hspace*{0.3cm}微分方程中所出现的未知函数的最高阶导数的阶数，叫做\uuline{微分方程的阶}\index{微分方程!微分方程的阶}.例如三阶微分方程$x^3y'''+x^2y''-4xy'=3x^2+\sin{2x}$.一般地，$n$阶微分方程的形式是 96 | \begin{equation} 97 | F(x,y,y',\cdots ,y^{(n)})=0\label{n阶微分方程} 98 | \end{equation} 99 | 其中$y^{(n)}$是必须有的，而其余项$x,y,y',\cdots ,y^{(n-1)}$可有可无. 100 | \end{defination} 101 | 102 | \begin{defination} 103 | \hspace*{0.3cm}找出这样的函数，把这个函数代入微分方程\eqref{n阶微分方程}能使该方程成为恒等式.这个函数就叫做该\uuline{微分方程的解}\index{微分方程!微分方程的解}.设函数$y=\varphi(x)$在区间$I$上有$n$阶连续导数,如果在区间$I$上, 104 | $$F\left[x,\varphi(x),\varphi'(x),\cdots ,\varphi^{(n)}(x)\right]\equiv 0$$ 105 | 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. 106 | \end{defination} 107 | 108 | \begin{defination} 109 | \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}. 110 | \end{defination} 111 | 112 | \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${y|}_{x=x_0}=y_0$. 113 | 114 | \begin{defination} 115 | \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 116 | \begin{equation} 117 | \left\lbrace 118 | \begin{array}{l} 119 | y' = f(x,y) \\ 120 | y\left | {_{x = {x_0}} = {y_0}} \right. 121 | \end{array} 122 | \right. 123 | \label{初值问题1} 124 | \end{equation} 125 | \end{defination} 126 | 127 | \begin{defination} 128 | 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 129 | \begin{equation} 130 | \left\lbrace 131 | \begin{array}{l} 132 | y''= f(x,y,y') \\ 133 | y\left | {_{x = {x_0}} = {y_0}} \right. \\ 134 | y'\left | {_{x = {x_0}} = {y_0'}} \right. 135 | \end{array} 136 | \right. 137 | \label{初值问题2} 138 | \end{equation} 139 | 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. 140 | \end{defination} 141 | 142 | \section{各种微分方程的求解} 143 | \subsection{可分离变量的微分方程} 144 | 145 | \begin{defination} 146 | \hspace*{0.3cm}一般地，如果一个微分方程能写成 147 | \begin{equation} 148 | g(y)\,\mathrm{d}y=f(x)\,\mathrm{d}x 149 | \label{可分离变量的微分方程} 150 | \end{equation} 151 | 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}. 152 | \end{defination} 153 | 154 | \begin{method} 155 | \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 156 | \begin{equation} 157 | \int g(y)\,\mathrm{d}y=\int f(x)\,\mathrm{d}x 158 | \label{可分离变量的微分方程2} 159 | \end{equation} 160 | 设$G(y)$和$F(x)$分别为$g(y)$和$f(x)$的原函数，那么 161 | \begin{equation} 162 | G(y)=F(x)+C 163 | \label{可分离变量的微分方程3} 164 | \end{equation} 165 | \end{method} 166 | 167 | \begin{defination} 168 | \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}. 169 | \end{defination} 170 | \vspace{0.3cm} 171 | 172 | \subsection{齐次方程} 173 | 174 | \begin{defination} 175 | \hspace*{0.3cm}如果一阶微分方程可以化成 176 | 177 | \begin{equation} 178 | \frac{\mathrm{d}y}{\mathrm{d}x}=\varphi (\frac{y}{x}) 179 | \label{齐次方程1} 180 | \end{equation} 181 | 182 | 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}. 183 | \end{defination} 184 | 185 | \begin{method} 186 | \hspace*{0.3cm}求解齐次微分方程\eqref{齐次方程1}的方法如下：\\ 187 | 第一步：换元 188 | $$u=\frac{y}{x}$$ 189 | 其中$u$是关于$x$的一个新函数，所以 190 | $$y=ux,\frac{\mathrm{d}y}{\mathrm{d}x}=u+\frac{\mathrm{d}u}{\mathrm{d}x}x$$ 191 | 第二步：代入式\eqref{齐次方程1}，得 192 | $$u+\frac{\mathrm{d}u}{\mathrm{d}x}x=\varphi (u)$$ 193 | 第三步：分离变量,得到 194 | $$\frac{1}{\varphi (u)-u}\,\mathrm{d}u=\frac{1}{x}\,\mathrm{d}x$$ 195 | 第四步：两边同时积分,得到 196 | $$\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u=\ln|x|\,$$ 197 | 即 198 | $$x=\mathrm{e}^{\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u}$$ 199 | \end{method} 200 | 201 | \begin{theorem} 202 | \hspace{0.3cm}齐次方程\eqref{齐次方程1}的通解为 203 | \begin{equation} 204 | x=\mathrm{e}^{\int \frac{1}{\varphi (u)-u}\,\mathrm{d}u} 205 | \label{齐次方程的通解} 206 | \end{equation} 207 | 208 | \end{theorem} 209 | 210 | 211 | \subsection{可化为齐次的方程} 212 | \begin{method} 213 | \hspace{0.3cm}方程 214 | \begin{equation} 215 | \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2} 216 | \label{非齐次方程} 217 | \end{equation} 218 | 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 219 | $$x=X+h,y=Y+k,$$ 220 | 于是， 221 | $$\mathrm{d}x=\mathrm{d}X,\mathrm{d}y=\mathrm{d}Y,$$ 222 | 带入方程\eqref{非齐次方程}得到 223 | $$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{a_1X+b_1Y+a_1h+b_1k+c_1}{a_2X+b_2Y+a_2h+b_2k+c_2}.$$ 224 | 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 225 | $$\left\lbrace 226 | \begin{array}{l} 227 | a_1h+b_1k+c_1=0 \\ 228 | a_2h+b_2k+c_2=0 229 | \end{array} 230 | \right.$$ 231 | 如果上述方程组的系数行列式 232 | $\left| \begin{array}{cc} 233 | a_1 & b_1 \\ 234 | a_2 & b_2 235 | \end{array} \right| \ne 0$ 236 | ，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 237 | $$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{a_1X+b_1Y}{a_2X+b_2Y}.$$ 238 | 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}\\ 239 | 240 | \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 241 | $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a_1x+b_1y+c_1}{\lambda (a_1x+b_1y)+c_2}$$ 242 | 引入新变量$v=a_1x+b_1y$，那么 243 | $$\frac{\mathrm{d}v}{\mathrm{d}x}=a_1+b_1 \, \frac{\mathrm{d}y}{\mathrm{d}x} \qquad \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{b_1}\left(\frac{\mathrm{d}v}{\mathrm{d}x}-a_1 \right) $$ 244 | 代入方程\eqref{非齐次方程}得到 245 | $$\frac{1}{b_1}\left(\frac{\mathrm{d}v}{\mathrm{d}x}-a_1 \right)=\frac{v+c_1}{\lambda v+c_2}$$ 246 | 这就变成了一个可以分离变量的方程，很容易就可以求解. 247 | \end{method} 248 | 249 | \subsection{一阶线性微分方程} 250 | 251 | 252 | 253 | 254 | %打印索引————————————— 255 | \newpage 256 | \addcontentsline{toc}{chapter}{附录} 257 | \addcontentsline{toc}{section}{索引} 258 | \appendix 259 | \kaishu 260 | \printindex 261 | %——————————————— 262 | 263 | \end{document} -------------------------------------------------------------------------------- /calculus.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ShevonKuan/Calculus-Summarize/772b4017055564b2ca16035c6d5f7df95f1d31f6/calculus.pdf -------------------------------------------------------------------------------- /calculus.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,a4paper]{article} 2 | \usepackage{shevonNotebook} 3 | \begin{document} 4 | \numberwithin{equation}{subsection} 5 | \allowdisplaybreaks%强制自动换行 6 | \maketitle 7 | \thispagestyle{empty} 8 | \newpage 9 | \tableofcontents 10 | \thispagestyle{empty} 11 | \newpage 12 | \setcounter{page}{1} 13 | \section{常用三角函数公式} 14 | \subsection{积化和差公式} 15 | \begin{align} 16 | \sin \alpha \cos \beta & =\frac{1}{2}[\sin (\alpha+\beta)+\sin(\alpha-\beta)] \\[7pt] 17 | \cos \alpha \sin \beta & =\frac{1}{2}[\sin (\alpha+\beta)-\sin(\alpha-\beta)] \\[7pt] 18 | \cos \alpha \cos \beta & =\frac{1}{2}[\cos (\alpha+\beta)+\cos(\alpha-\beta)] \\[7pt] 19 | \sin \alpha \sin \beta & =-\frac{1}{2}[\cos (\alpha+\beta)-\cos(\alpha-\beta)] 20 | \end{align} 21 | \subsection{和差化积公式} 22 | \begin{align} 23 | \sin\alpha+\sin\beta & =2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\[7pt] 24 | \sin\alpha-\sin\beta & =2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\[7pt] 25 | \cos\alpha+\cos\beta & =2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\[7pt] 26 | \cos\alpha-\cos\beta & =-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\[7pt] 27 | \tan\alpha+\tan\beta & =\frac{\sin (\alpha+\beta)}{\cos\alpha\cdot\cos \beta} 28 | \end{align} 29 | \subsection{归一化公式} 30 | \begin{align}\label{gyhgs} 31 | \sin^2 x+\cos^2x & =1 \\[7pt] 32 | \sec^2 x-\tan^2x & =1 \\[7pt] 33 | \cosh^2x-\sinh^2x & =1 34 | \end{align} 35 | \subsection{倍(半)角公式 \quad 降(升)幂公式} 36 | \begin{align} 37 | \sin^2x & =\frac{1}{2}(1-\cos 2x) \\[7pt] 38 | \cos^2x & =\frac{1}{2}(1+\cos 2x) \\[7pt] 39 | \tan^2x & =\frac{1-\cos 2x}{1+\cos 2x} \\[7pt] 40 | \sin x & =2\sin\frac{x}{2}\cos\frac{x}{2} \\[7pt] 41 | \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] 42 | \tan x & =\frac{2\tan(x/2)}{1-\tan^2(x/2)} 43 | \end{align} 44 | \subsection{万能公式} 45 | 令$ u=\tan\dfrac{x}{2} $则 46 | \begin{align} 47 | \sin x=\frac{2u}{1+u^2} \\[7pt] 48 | \cos x=\frac{1-u^2}{1+u^2} 49 | \end{align} 50 | \section{常用的佩亚诺型余项泰勒公式} 51 | \mybox[53a9ab]{\leftline{\textbf{泰勒公式\small{ $ (\xi\ \mbox{在}x_0\ \mbox{与}x\ \mbox{之间}) $}:}}\vspace{-2em} 52 | \begin{align} 53 | 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] 54 | 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 55 | \end{align}} 56 | 57 | \leftline\small{由此可得常用的泰勒公式} 58 | \begin{align} 59 | \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] 60 | \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}) 61 | \end{align} 62 | 令 $n=2m$ 有, 63 | \begin{align} 64 | \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] 65 | \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] 66 | \tan x & =x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\cdots+o(x^{2m-1}) 67 | \end{align} 68 | \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 } 69 | \begin{align} 70 | \arcsin x & =x+\frac{1}{6}x^3+\frac{3}{40}x^{5}+\cdots+o(x^{2m}) 71 | \end{align} 72 | 常用于近似计算的泰勒公式 73 | \begin{align} 74 | \frac{1}{1-x} & =1+x+x^2+x^3+\cdots+x^n+o(x^n) \\[7pt] 75 | (1+x)^{\alpha} & =\sum_{i=0}^{n}\frac{\prod_{j=0}^{i-1}{(\alpha-j})}{i!}x^n+o(x^n)\notag \\[7pt] 76 | & =1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2+\cdots+o(x^n) \\[7pt] 77 | \alpha^x & =\sum_{i=0}^{n}\frac{\ln^n \alpha}{n!}x^n+o(x^n)\notag \\[7pt] 78 | & =1+x\ln \alpha+\frac{\ln^2 \alpha}{2}x^2+\cdots+\frac{\ln^n \alpha}{n!}x^n+o(x^n) 79 | \end{align} 80 | \section{基本求导公式} 81 | \begin{multicols}{2} 82 | \begin{equation} 83 | \left( C\right)'=0 84 | \end{equation} 85 | \begin{equation} 86 | \left( x^{\mu}\right)'=\mu x^{\mu-1} 87 | \end{equation} 88 | \begin{equation} 89 | \left( \sin x\right)'=\cos x 90 | \end{equation} 91 | \begin{equation} 92 | \left( \cos x\right)'=-\sin x 93 | \end{equation} 94 | \begin{equation} 95 | \left( \tan x\right)'=\sec^2 x 96 | \end{equation} 97 | \begin{equation} 98 | \left( \cot x\right)'=-\csc^2 x 99 | \end{equation} 100 | \begin{equation} 101 | \left( \sec x\right)'=\sec x\cdot\tan x 102 | \end{equation} 103 | \begin{equation} 104 | \left( \csc x\right)'=-\csc x\cdot\tan x 105 | \end{equation} 106 | \begin{equation} 107 | \left( a^x\right)'=a^x\ln a\ (a>0,a\neq1) 108 | \end{equation} 109 | \begin{equation} 110 | \left( \log_{a}x\right)'=\frac{1}{x\cdot\ln a}\ (a>0,a\neq1) 111 | \end{equation} 112 | \begin{equation} 113 | \left( \arcsin x\right)'=\frac{1}{\sqrt{1-x^2}} 114 | \end{equation} 115 | \begin{equation} 116 | \left( \arccos x\right)'=-\frac{1}{\sqrt{1-x^2}} 117 | \end{equation} 118 | \begin{equation} 119 | \left( \arctan x\right)'=\frac{1}{1+x^2} 120 | \end{equation} 121 | \begin{equation} 122 | \left( \mathrm{arccot}\, x\right)'=-\frac{1}{1+x^2} 123 | \end{equation} 124 | \end{multicols} 125 | \section{函数图形描述中涉及到的重要公式} 126 | \subsection{常用曲率计算公式} 127 | \mybox[53a9ab]{ 128 | 曲率的定义式$ K=\displaystyle\left|\frac{\mathrm{d}\alpha}{\mathrm{d}s}\right| $} 129 | 由定义式我们可以推得 130 | \begin{enumerate} 131 | \item \textbf{直角坐标}系中的曲线\ $y=y(x)$\ 有曲率表达式 132 | \begin{equation} 133 | K=\frac{\left|y''\right|}{\left( 1+y^{'2} \right)^{3/2}}\mbox{;} 134 | \end{equation} 135 | \item \textbf{参数方程}表示的曲线\ $x=\varphi(t),y=\psi(t) $\ 有曲率表达式 136 | \begin{equation} 137 | K=\frac{\left|\varphi'(t)\psi''(t)-\varphi''(t)\psi'(t)\right|}{\left[ \varphi^{'2}(t) +\psi^{'2}(t) \right]^{3/2}}\mbox{;} 138 | \end{equation} 139 | \item \textbf{极坐标}表示的的曲线\ $y=y(x)$\ 有曲率表达式 140 | \begin{equation} 141 | K=\frac{\left|r^2+2r^{'2}-r\cdot r''\right|}{\left(r^2+r^{'2}\right)^{3/2}}\mbox{;} 142 | \end{equation} 143 | \item 曲线在对应点\ $M(x,y)$\ 的曲率中心\ $D(\alpha,\beta)$\ 的坐标为 144 | \begin{equation} 145 | \begin{cases} 146 | \alpha=x-\displaystyle\frac{y'(1+y^{'2})^3}{y^{''2}} \\[7pt] 147 | \beta=y+\displaystyle\frac{1+y^{'2}}{y''} 148 | \end{cases} 149 | \end{equation} 150 | . 151 | \end{enumerate} 152 | \subsection{曲线的渐近线} 153 | \begin{enumerate} 154 | \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} 155 | \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} 156 | \item 若\ $\lim\limits_{ x\rightarrow \infty }[f(x)-(ax+b)]=0$\ ,其中 157 | $ 158 | \begin{cases} 159 | a=\displaystyle \lim\limits_{x\to \infty}\frac{f(x)}{x} \\[7pt] 160 | b=\displaystyle \lim\limits_{x\to \infty}[f(x)-ax] 161 | \end{cases} 162 | $ 163 | 则称\ $y=ax+b$\ 为曲线\ $f(x)$\ 的\textbf{斜渐近线} 164 | \end{enumerate} 165 | \section{基本积分公式} 166 | \begin{align} 167 | & \int k \,\mathrm{d}x=kx+C \ \mbox{(其中}k\mbox{为常数)} \\[7pt] 168 | & \int x^\mu\,\mathrm{d}x=\frac{x^{\mu+1}}{\mu+1}+C\ (\mu\neq-1) \\[7pt] 169 | & \int \frac{1}{x}\,\mathrm{d}x=\ln|x|+C \\[7pt] 170 | & \int\frac{\mathrm{d}x}{1+x^2}=\arctan x+C \\[7pt] 171 | & \int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin x+C_1=-\arccos x+C_2 \\[7pt] 172 | & \int \sin x\,\mathrm{d}x=-\cos x+C \\[7pt] 173 | & \int\cos x \,\mathrm{d}x=\sin x +C \\[7pt] 174 | & \int\tan x\,\mathrm{d}x=-\ln |\cos x|+C \\[7pt] 175 | & \int\cot x\,\mathrm{d}x=\ln |\sin x|+C \\[7pt] 176 | & \int\csc x\,\mathrm{d}x=\int\frac{1}{\sin{x}}\,\mathrm{d}x=\frac{1}{2} 177 | \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] 178 | & \int\sec x\,\mathrm{d}x=\int\frac{1}{\cos{x}}\,\mathrm{d}x=\frac{1}{2} 179 | \ln{\left|\frac{1+\sin{x}}{1-\sin{x}}\right|}+C=\ln{\left|\sec{x}+\tan{x}\right|}+C \\[7pt] 180 | & \int\sec^2 x\,\mathrm{d}x=\tan x +C \\[7pt] 181 | & \int \csc^2 x\,\mathrm{d}x=-\cot x +C \\[7pt] 182 | & \int \sec x\cdot\tan x \,\mathrm{d}x=\sec x+C \\[7pt] 183 | & \int\csc x \cdot\cot x \,\mathrm{d}x=-\csc x+C \\[7pt] 184 | & \int \mathrm{e}^x \,\mathrm{d}x=\mathrm{e}^x+C \\[7pt] 185 | & \int a^x\,\mathrm{d}x=\frac{a^x}{\ln a}+C \\[7pt] 186 | & \int \sinh x\,\mathrm{d}x=\cosh x+C \\[7pt] 187 | & \int \cosh x\,\mathrm{d}x=\sinh x+C \\[7pt] 188 | & \int \frac{1}{a^2+x^2}\,\mathrm{d}x=\frac{1}{a}\arctan\frac{x}{a}+C \\[7pt] 189 | & \int \frac{1}{a^2-x^2}\,\mathrm{d}x=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C \\[7pt] 190 | & \int \frac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin\frac{x}{a}+C \\[7pt] 191 | & \int \frac{1}{\sqrt{x^2\pm a^2}}\,\mathrm{d}x=\ln \left|x+\sqrt{x^2\pm a^2}\right|+C 192 | \end{align} 193 | \section{基本积分方法} 194 | \subsection{第一类换元法} 195 | \subsubsection{三角函数之积的积分} 196 | \begin{enumerate} 197 | \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; 198 | \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), 199 | \cos^2=\dfrac{1}{2}(1+\cos 2x) $化成$ \cos 2x $的多项式 ,从而求得结果; 200 | \item 一般地,对于$ \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; 201 | \end{enumerate} 202 | \subsubsection{常见的凑微分类型} 203 | \begin{align} 204 | & \int {f( ax + b){\rm{d}}x = }\frac{1}{a}\int {f(ax+b){\mathrm{d}}(ax + b)\;(a \neq 0)} \\[7pt] 205 | & \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] 206 | & \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] 207 | & \int {f(\ln x)\frac{1}{x}} {\rm{d}}x = \int {f(\ln x){\rm{d(}}\ln x)} \\[7pt] 208 | & \int {f({\mathrm{e}^x})} {\mathrm{e}^x}{\rm{d}}x = \int {f({\mathrm{e}^x}} ){\rm{d(}}{\mathrm{e}^x}) \\[7pt] 209 | & \int {f(\sqrt x } )\frac{{{\rm{d}}x}}{{\sqrt x }} = 2\int {f(\sqrt x } ){\rm{d}}(\sqrt x ) \\[7pt] 210 | & \int {f(\sin x)\cos x{\rm{d}}x = } \int {f(\sin x){\rm{d}}\sin x} \\[7pt] 211 | & \int {f(\cos x)\sin x{\rm{d}}x = } - \int {f(\cos x){\rm{d}}\cos x} \\[7pt] 212 | & \int {f(\tan x){{\sec }^2}} x{\rm{d}}x = \int {f(\tan x){\rm{d}}\tan x} \\[7pt] 213 | & \int {f(\cot x){{\csc }^2}} x{\rm{d}}x = - \int {f(\cot x){\rm{d}}\cot x} \\[7pt] 214 | & \int {f(\arcsin x)\frac{1}{{\sqrt {1 - {x^2}} }}} {\rm{d}}x = \int {f(\arcsin x){\rm{d}}\arcsin x} \\[7pt] 215 | & \int {f(\arctan x)\frac{1}{{1 + {x^2}}}} {\rm{d}}x = \int {f(\arctan x){\rm{d}}\arctan x} \\[7pt] 216 | & \int {\frac{{f'(x)}}{{f(x)}}} {\rm{d}}x = \int {\frac{{{\rm{d}}f(x)}}{{f(x)}}} = \ln \left| f(x)\right| + C 217 | \end{align} 218 | \subsection{有理函数的积分} 219 | \subsubsection{部分分式} 220 | \begin{align} 221 | \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 \\ 222 | & \frac{{{B_1}}}{{{{(x - b)}^\beta }}} + \frac{{{B_2}}}{{{{(x - b)}^{\beta - 1}}}} + \cdots + \frac{{{B_\beta }}}{{x - b}} + \notag \\ 223 | & \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 \\ 224 | & \cdots 225 | \end{align} 226 | \subsubsection{三角函数的特殊定积分} 227 | \begin{align} 228 | I_n & =\int_0^{\frac{\pi}{2}}\sin^nx\,\mathrm{d}x=\int_0^{\frac{\pi}{2}}\cos^nx\,\mathrm{d}x\notag \\ 229 | I_n & =\frac{n-1}{n}I_{n-2}\notag \\ 230 | & =\begin{cases} 231 | \ \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] 232 | \ \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} 233 | \end{cases} 234 | \end{align} 235 | \section{多元函数微分} 236 | \subsection{偏导数} 237 | \subsubsection{偏导数记法} 238 | 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) $的\md{二阶偏导数}.按照对变量求导次序不同,有如下四个二阶偏导数:\begin{align*} 239 | & \frac{\partial }{{\partial x}}\left( {\frac{{\partial z}}{{\partial x}}} \right) = \frac{{{\partial ^2}z}}{{\partial {x^2}}} = {f_{xx}}(x,y) & \frac{\partial }{{\partial y}}\left( {\frac{{\partial z}}{{\partial x}}} \right) = \frac{{{\partial ^2}z}}{{\partial x\partial y}} = {f_{xy}}(x,y) \\[7pt] 240 | & \frac{\partial }{{\partial x}}\left( {\frac{{\partial z}}{{\partial y}}} \right) = \frac{{{\partial ^2}z}}{{\partial y\partial x}} = {f_{yx}}(x,y) & \frac{\partial }{{\partial y}}\left( {\frac{{\partial z}}{{\partial y}}} \right) = \frac{{{\partial ^2}z}}{{\partial {y^2}}} = {f_{yy}}(x,y). 241 | \end{align*} 242 | \subsection{全微分} 243 | 244 | 245 | \section{微分方程(该部分将会采用详细的讲义样式)} 246 | \subsection{微分方程的基本概念} 247 | \begin{definition}[微分方程的定义] 248 | 一般地,凡表示\md{未知函数},\md{未知函数的导数}与\md{自变量之间的关系的方程},称为\textbf{微分方程}. 249 | 其中未知函数的最高阶导数的阶数,称为\textbf{微分方程的阶}.一般地,$n$阶微分方程的形式是:\[ 250 | F( 251 | x,y,y',\cdots,y^{(n)} 252 | )=0 253 | \] 254 | \end{definition} 255 | \begin{definition}[微分方程的解] 256 | 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:\[ 257 | F[x,\varphi(x),\varphi'(x),\cdots,\varphi^{(n)}(x)]\equiv 0, 258 | \]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$上的解} 259 | 特别地,如果微分方程的解\md{含有任意常数}\footnote{此处的任意常数必须是相互独立的,或者说他们线性无关.},且任意常数的个数与微分方程的阶数相同,这样的解称为\textbf{微分方程的通解}. 260 | 261 | 通解中时常含有任意常数，所以它还不能完全确定地反映某一客观事物的规律性.所以为了完全确定地反映客观事物的规律性,必须确定这些常数的值.为此要根据问题的实际情况，提出确定这些常数的条件.例如设一阶微分方程中的未知函数为$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$. 262 | 263 | 因此我们定义,在实际问题中所给定的能够确定这些常数的条件称为\textbf{初值条件}.由初值条件确定了常数的值进而可以得到微分方程的\textbf{特解}. 264 | \end{definition} 265 | \section{可分离变量的微分方程} 266 | 本节我们将讨论一阶微分方程$y'=f(x,y)$ 267 | \end{document} 268 | 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-------------------------------------------------------------------------------- 1 | % |-----------------------------------------------------------------------------------------------|% 2 | % | Shevon Kuan 课程笔记基本模板 |% 3 | % | 版本号:3.15b |% 4 | % | 页面配置:A4 paper侧栏设计 |% 5 | % |-----------------------------------------------------------------------------------------------|% 6 | \ProvidesPackage{shevonNotebook} 7 | \usepackage{babel} 8 | \usepackage{ctex} 9 | \usepackage[dvipsnames, svgnames, x11names]{xcolor} 10 | \usepackage{amsmath} 11 | \usepackage{amsfonts} 12 | \usepackage{amssymb}%数学符号宏包 13 | \usepackage{geometry} 14 | \usepackage{fancyhdr} 15 | \usepackage{framed} 16 | \usepackage{fontspec} 17 | \usepackage{ntheorem} 18 | \usepackage{nomencl} 19 | \usepackage{nicematrix} 20 | \usepackage{multicol} 21 | \usepackage{indentfirst}%首行缩进 22 | \setlength{\parindent}{2em} 23 | \usepackage[bookmarks=true,colorlinks,linkcolor=black]{hyperref} 24 | \usepackage{CJKfntef} 25 | \usepackage{amsmath,bm}%重要宏包,粗斜体\bm 26 | \usepackage{makeidx}%重要宏包，用于添加索引 27 | \usepackage{pgf,tikz,pgfplots}%一般绘图宏包 28 | \pgfplotsset{compat=1.15} 29 | \usepackage{mathrsfs} 30 | \usetikzlibrary{arrows} 31 | \usepackage{tikz-3dplot}%3d绘图宏包 32 | \usepackage{tcolorbox}%box宏包 33 | \usepackage{subfig} 34 | \usepackage{autobreak} 35 | \usepackage{enumerate} 36 | \usepackage{wasysym} 37 | \usepackage{textcomp} 38 | \usepackage{marginnote} 39 | \usepackage{ulem}%下划线宏包用法和样式如下： 40 | %\uuline{双下划线} 41 | %\uwave{波浪线} 42 | %\sout{中间删除线} 43 | %\xout{斜删除线} 44 | %\dashuline{虚线} 45 | %\dotuline{加点} 46 | \usepackage{titletoc}%目录页的宏包 47 | \usepackage[center]{titlesec}%改变章节或标题的样式的宏包 48 | \makenomenclature 49 | %\titleformat{command}[shape]{format}{label}{sep}{before}[after] 50 | 51 | %1.command 是要重新定义的各种标题命令,比如 \part，\chapter，\section，\subsection，\subsubsection，\paragraph，\subparagraph等； 52 | %2.shape 是用来设定段落形状的,可选的参数有 hang 、 block 、 display 等,详见 titlesec 文档,位于：TEXLIVE/VERSION/texmf-dist/doc/latex/titlesec 53 | %3.format 用于定义标题外观,比如使标题居中、字体加粗等； 54 | %4.label 用于定义定义标题的标签,就是标题内容前面的标号； 55 | %5.sep 定义标题的标签与标题内容之间的间隔距离。 56 | %6.before 用于在标题内容前再加些内容； 57 | %7.after 用于在标题内容后再加些内容。 58 | 59 | \usetikzlibrary{shapes,arrows} 60 | \makeindex%添加索引 61 | %缺省页面 62 | \geometry{inner=2cm,outer=2cm,bottom=1.5cm,top=2cm,twoside} 63 | %\geometry{showframe,showcrop}%此句用于显示排版文本框 64 | %字体设置------------------------------- 65 | \setCJKmainfont[BoldFont={SourceHanSansHC-Medium}]{SourceHanSerifSC-Regular} 66 | \setCJKfamilyfont{song}{SourceHanSerifSC-Regular} 67 | \setCJKfamilyfont{hei}{SourceHanSansHC-Regular} 68 | \setCJKfamilyfont{heiti}{SourceHanSansHC-Medium} 69 | \setCJKfamilyfont{heilight}{SourceHanSansHC-Light} 70 | \setCJKfamilyfont{title}{SourceHanSansHC-Regular} 71 | \setCJKfamilyfont{songbold}{TsukuBRdGothicStd-Regular} 72 | \theorembodyfont{\CJKfamily{song}} 73 | %\setmainfont{Times New Roman} 74 | %颜色设置-—————————— 75 | \definecolor{f8766d}{HTML}{EA7500} 76 | \definecolor{1a9850}{HTML}{0097e6} 77 | \definecolor{ffa725}{HTML}{1289A7} 78 | \definecolor{2a7ae2}{HTML}{7158e2} 79 | \definecolor{6a3d9a}{HTML}{ED4C67} 80 | \definecolor{53a9ab}{HTML}{007500} 81 | \definecolor{titlepurple}{HTML}{5758BB} 82 | \definecolor{titlepurpleb}{HTML}{833471} 83 | \definecolor{titlepurplec}{HTML}{006266} 84 | \definecolor{md}{HTML}{EA2027} 85 | \definecolor{background}{HTML}{f5f5ed} 86 | %geogebra颜色 87 | \definecolor{zzttqq}{rgb}{0.6,0.2,0} 88 | \definecolor{uuuuuu}{rgb}{0.26666666666666666,0.26666666666666666,0.26666666666666666} 89 | \definecolor{ududff}{rgb}{0.30196078431372547,0.30196078431372547,1} 90 | \definecolor{xdxdff}{rgb}{0.49019607843137253,0.49019607843137253,1} 91 | %定理定义环境设置-------------------- 92 | 93 | %\newtcbox{\mybox}[1][]{on line, 94 | % arc=0pt,outer arc=0pt,colback=#1!10!white,colframe=#1, 95 | % boxsep=0pt,left=3pt,right=3pt,top=2pt,bottom=2pt, 96 | % boxrule=0pt,leftrule=1.5pt} 97 | 98 | \newcommand{\mybox}[2][]{ 99 | \begin{tcolorbox}[on line, 100 | arc=0pt,outer arc=0pt,colback=#1!7!white,colframe=#1, 101 | boxsep=0pt,left=3pt,right=3pt,top=5pt,bottom=5pt, 102 | boxrule=0pt,leftrule=1.5pt,halign=flush center]#2 103 | \end{tcolorbox}} 104 | 105 | \newcommand{\myboxp}[2][]{ 106 | \begin{tcolorbox}[on line, 107 | arc=0pt,outer arc=0pt,colback=white,colframe=#1, 108 | boxsep=0pt,left=3pt,right=3pt,top=5pt,bottom=5pt, 109 | boxrule=0pt,leftrule=1.5pt]#2 110 | \end{tcolorbox}} 111 | \newcounter{A}[section] 112 | \newcounter{B}[section] 113 | \newcounter{C}[section] 114 | \newcounter{D}[section] 115 | \newcounter{E}[section] 116 | \newcounter{F}[section] 117 | 118 | \newcommand{\con}[1]{{\bfseries\refstepcounter{#1}\thesection.\arabic{#1}}} 119 | 120 | %\theoremindent0.2cm 121 | %\theoremheaderfont{\CJKfamily{songbold}} 122 | %\theoremstyle{break} 123 | 124 | %\newtheorem*{theorem}{\hspace{-0.16cm}\color{2a7ae2}\mybox[2a7ae2]{\color{2a7ae2}定理\addtocounter{A}{1} \thesection.\arabic{A}}} 125 | %\newtheorem*{defination}{\hspace{-0.16cm}\color{f8766d}\mybox[f8766d]{\color{f8766d}定义\addtocounter{B}{1} \thesection.\arabic{B}}} 126 | %\newtheorem*{feature}{\hspace{-0.16cm}\color{ffa725}\mybox[ffa725]{\color{ffa725}性质\addtocounter{C}{1} \thesection.\arabic{C}}} 127 | %\newtheorem*{inference}{\hspace{-0.16cm}\color{1a9850}\mybox[1a9850]{\color{1a9850}推论\addtocounter{D}{1} \thesection.\arabic{D}}} 128 | %\newtheorem*{method}{\hspace{-0.16cm}\color{6a3d9a}\mybox[6a3d9a]{\color{6a3d9a}方法\addtocounter{E}{1} \thesection.\arabic{E}}} 129 | %\newtheorem*{example}{\hspace{-0.16cm}\color{53a9ab}\mybox[53a9ab]{\color{53a9ab}例题\addtocounter{F}{1} \thesection.\arabic{F}}} 130 | 131 | \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 } 132 | \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 } 133 | \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 } 134 | \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 } 135 | \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 } 136 | \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 } 137 | 138 | %符号设置------------------------------- 139 | {\color{titlepurple}} 140 | %标题配置————————————— 141 | \title{\vspace{5cm} 142 | \begin{center} 143 | \includegraphics[width=16cm]{covertitle.eps} 144 | %\includegraphics[width=8cm]{cover2.png} 145 | %\includegraphics[width=8cm]{cover3.png} 146 | %\includegraphics[width=12cm]{cover4.png} 147 | \end{center} 148 | \vspace{5cm}} 149 | \author{\large\color{titlepurple}关舒文\\{\color{titlepurple}\kaishu{华南理工大学}}} 150 | \date{\color{titlepurple}\small{Latest Update\ :\ \today}} 151 | 152 | %章节或标题的样式------------------- 153 | %\titleformat{\chapter}{\CJKfamily{title}\huge\color{titlepurple}}{\includegraphics[height=2cm]{sigmaformal.eps} 第\ \thechapter\ 章\ }{0pt}{} 154 | \titleformat{\section}{\CJKfamily{hei}\large\color{titlepurple}}{\bfseries{\thesection}\quad }{0pt}{} 155 | \titleformat{\subsection}{\CJKfamily{hei}\large\color{titlepurpleb}}{\bfseries{\thesubsection}\quad }{0pt}{} 156 | \titleformat{\subsubsection}{\CJKfamily{hei}\large\color{titlepurplec}}{\bfseries{\thesubsubsection}\quad }{0pt}{} 157 | \renewcommand{\nomname}{符号说明} 158 | %自定义优化命令----------------------- 159 | \renewcommand{\a}{\ensuremath A}% 160 | \renewcommand{\b}{\ensuremath B}% 161 | \renewcommand{\c}{\ensuremath C}% 162 | \renewcommand{\o}{\ensuremath \varnothing}% 163 | \renewcommand{\textbf}[1]{{\CJKfamily{heiti}#1}}% 164 | \newcommand{\md}[1]{{\,\color{purple}#1\,}}% 165 | \newcommand{\cf}[1]{\textit{cf.Fig~\ref{#1}}}% 166 | \newcommand{\margin}[1]{{\marginpar{\footnotesize\kaishu\hspace*{2em}#1}}} 167 | %\renewcommand{\chapter}[1]{{\chapter{#1}\thispagestyle{empty}} }% 168 | %使用了自定义页眉页脚--------------- 169 | \pagestyle{fancy} 170 | %\renewcommand{\chaptermark}[1]{\markboth{\;第\ \thechapter\ 章\quad#1\;}{}} 171 | \renewcommand{\sectionmark}[1]{\markright{\;\thesection\ #1\;}} 172 | \fancyhf{} 173 | %\fancyfoot[C]{\bfseries\thepage} 174 | \fancyhead[LO]{\small\CJKfamily{heilight}\rightmark} 175 | \fancyhead[RE]{\small\CJKfamily{heilight}\leftmark} 176 | \fancyhead[RO,LE]{\;\thepage\;} 177 | \fancyfoot[RO,LE]{\small\CJKfamily{heilight}{微积分 重要公式}} 178 | \fancyfoot[RE,LO]{\small\CJKfamily{heilight}The Important Equations of Calculus} 179 | \renewcommand{\headrulewidth}{0.4pt} % 注意不用\setlength 180 | %\renewcommand{\footrulewidth}{0pt} 181 | -------------------------------------------------------------------------------- /sigma.eps: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ShevonKuan/Calculus-Summarize/772b4017055564b2ca16035c6d5f7df95f1d31f6/sigma.eps --------------------------------------------------------------------------------