1%-*- coding: UTF-8 -*-  2  3%!TEX program = xelatex  4  5\documentclass[UTF8]{ctexart}  6\usepackage{graphicx}  7\usepackage{float}  8\usepackage{amsmath}  9\usepackage{amssymb} 10\usepackage{yhmath} 11\usepackage{amsthm} 12\usepackage{esint} 13\usepackage{geometry} 14\usepackage{bigints} 15\geometry{a4paper,centering,scale=0.8} 16\usepackage[format=hang,font=small,textfont=it]{caption} 17\usepackage[nottoc]{tocbibind} 18\usepackage{setspace}        %使用间距宏包 19\usepackage{ctex} 20\usepackage{fontspec} 21\usepackage{color} 22\usepackage{etoolbox} 23\usepackage{enumitem} 24\usepackage{extarrows} 25\usepackage{tikz} 26\usepackage{pgfplots} 27\newcommand*{\circled}[1]{\lower.7ex\hbox{\tikz\draw (0pt, 0pt)% 28    circle (.5em) node {\makebox[1em][c]{\small #1}};}} 29\robustify{\circled} 30\usepackage{array} 31\usepackage{booktabs} 32\newcounter{rowno} 33 34\usepackage[ colorlinks, 35linkcolor=black, 36anchorcolor=black, 37citecolor=black]{hyperref} 38 39\numberwithin{equation}{section} 40\numberwithin{figure}{section} 41\numberwithin{table}{section} 42\renewcommand{\thefigure}{\arabic{section}-\arabic{figure}} 43\renewcommand{\thetable}{\arabic{section}-\arabic{table}} 44\renewcommand{\theequation}{\arabic{section}-\arabic{equation}} 45 46%带圈数字配置,会改变字体,不用可注释 47%\usepackage{libertine} 48%% \libcirc and \libcircblk display their '0' if the parameter is out of range 49%\newcommand{\libcirc}[1]{\pgfmathparse{ 50%    ifthenelse(#1 > 0 && #1 < 21, Hex(9311+#1), Hex(9450) 51%    }\libertineGlyph{uni\pgfmathresult}} 52%\newcommand{\libcircdbl}[1]{\pgfmathparse{Hex(9460+#1)}\libertineGlyph{uni\pgfmathresult}} 53%\newcommand{\libcircblk}[1]{\pgfmathparse{ 54%    ifthenelse(#1 > 0 && #1 < 11, Hex(10101+#1), 55%        ifthenelse(#1 > 10 && #1 < 21, Hex(9450-10+#1), 56%            Hex(9471) 57%        ) 58%    ) 59%    }\libertineGlyph{uni\pgfmathresult}} 60% 61%\newcommand{\juncirc}[1]{{\fontspec[Ligatures=Discretionary]{Junicode}[#1]}} 62%\newcommand{\juncircdbl}[1]{{\fontspec[Ligatures=Discretionary]{Junicode}[[#1]]}} 63%\newcommand{\juncircblk}[1]{{\fontspec[Ligatures=Discretionary]{Junicode}<#1>}} 64 65\pgfplotsset{compat=newest}% use newest version 66 67%begin画星形线 68\newcommand*{\A}{2} 69\newcommand*{\num}{2.2} 70 71 % X 的参数方程 72\pgfmathdeclarefunction{SolutionX}{1}{% 73    \pgfmathparse{\A*(cos(deg(\t)))^3}% 74} 75 76 % Y 的参数方程 77\pgfmathdeclarefunction{SolutionY}{1}{% 78    \pgfmathparse{\A*(sin(deg(\t)))^3}% 79} 80 81% cosh 的参数方程 82\pgfmathdeclarefunction{Cosh}{1}{% 83    \pgfmathparse{(e^(\x)+e^(-\x))/2}% 84} 85 86% sinh 的参数方程 87\pgfmathdeclarefunction{Sinh}{1}{% 88    \pgfmathparse{(e^(\x)-e^(-\x))/2}% 89} 90 91% tanh 的参数方程 92\pgfmathdeclarefunction{Tanh}{1}{% 93    \pgfmathparse{(e^(\x)-e^(-\x))/(e^(\x)+e^(-\x))}% 94} 95 96% coth 的参数方程 97\pgfmathdeclarefunction{Coth}{1}{% 98    \pgfmathparse{(e^(\x)+e^(-\x))/(e^(\x)-e^(-\x))}% 99}% catenary 的参数方程102\pgfmathdeclarefunction{Catenary}{1}{%103    \pgfmathparse{15*(((e^(\x/15)+e^(-\x/15))/2)-1)}%104}105106% catenary 的参数方程107\pgfmathdeclarefunction{Tractrix}{1}{%108    \pgfmathparse{15*ln((15+sqrt(15^2-(\x)^2))/\x)-sqrt(15^2-(\x)^2)}%109}110111% ferry 的参数方程112\pgfmathdeclarefunction{Ferry}{1}{%113    \pgfmathparse{0.5*\x*(((\x/20)^(-0.5))-((\x/20)^(0.5)))}%114}115116% rho_1 的参数方程117\pgfmathdeclarefunction{Rho1}{1}{%118    \pgfmathparse{sqrt(2)*10*e^(pi/4-\x*pi/180)}%119}120121% rho_2 的参数方程122\pgfmathdeclarefunction{Rho2}{1}{%123    \pgfmathparse{sqrt(2)*10*e^(3*pi/4-\x*pi/180)}%124}125126% rho_3 的参数方程127\pgfmathdeclarefunction{Rho3}{1}{%128    \pgfmathparse{sqrt(2)*10*e^(5*pi/4-\x*pi/180)}%129}130131% rho_4 的参数方程132\pgfmathdeclarefunction{Rho4}{1}{%133    \pgfmathparse{sqrt(2)*10*e^(7*pi/4-\x*pi/180)}%134}135136% roadx 的参数方程137\pgfmathdeclarefunction{RoadX}{1}{%138    \pgfmathparse{8*e^(-8*\x)*cos(\x)}%139}140141% roady 的参数方程142\pgfmathdeclarefunction{RoadY}{1}{%143    \pgfmathparse{8*e^(-8*\x)*sin(\x)}%144}145146% roadz 的参数方程147\pgfmathdeclarefunction{RoadZ}{1}{%148    \pgfmathparse{15*(1-e^(-15*\x))}%149}150151 % define elegant style152\tikzset{elegant/.style={smooth, red, thick, samples=101}}153154%end画星形线155156%%使section后面也有引导点,暂时无此需要157\usepackage{titletoc} 158%\titlecontents{section}[0pt]{\addvspace{2pt}\filright}159%{\contentspush{\thecontentslabel\ }}160%{}{\titlerule*[8pt]{.}\contentspage}161162%新命令163\newcommand\dif{\mathrm{d}}164\newcommand\no{\noindent}165\newcommand\dis{\displaystyle}166\newcommand\ls{\leqslant}167\newcommand\gs{\geqslant}168169\newcommand\limit{\dis\lim\limits}170\newcommand\limn{\dis\lim\limits_{n\to\infty}}171\newcommand\limxz{\dis\lim\limits_{x\to0}}172\newcommand\limxi{\dis\lim\limits_{x\to\infty}}173\newcommand\limxpi{\dis\lim\limits_{x\to+\infty}}174\newcommand\limxni{\dis\lim\limits_{x\to-\infty}}175176\newcommand\sumn{\dis\sum\limits_{n=1}^{\infty}}177\newcommand\sumnz{\dis\sum\limits_{n=0}^{\infty}}178\newcommand\sumk{\dis\sum\limits_{k=1}^{\infty}}179\newcommand\sumkz{\dis\sum\limits_{k=0}^{\infty}}180\newcommand\sumin{\dis\sum\limits_{i=1}^{n}}181\newcommand\sumizn{\dis\sum\limits_{i=0}^{n}}182\newcommand\sumkn{\dis\sum\limits_{k=0}^n}183\newcommand\sumkfn{\dis\sum\limits_{k=1}^n}184185\newcommand\pzx{\dis\frac{\partial z}{\partial x}}186\newcommand\pzy{\dis\frac{\partial z}{\partial y}}187188\newcommand\pfx{\dis\frac{\partial f}{\partial x}}189\newcommand\pfy{\dis\frac{\partial f}{\partial x}}190191\newcommand\pzxx{\dis\frac{\partial^2 z}{\partial x^2}}192\newcommand\pzxy{\dis\frac{\partial^2 z}{\partial x\partial y}}193\newcommand\pzyx{\dis\frac{\partial^2 z}{\partial y\partial x}}194\newcommand\pzyy{\dis\frac{\partial^2 z}{\partial y^2}}195196\newcommand\pfxx{\dis\frac{\partial^2 f}{\partial x^2}}197\newcommand\pfxy{\dis\frac{\partial^2 f}{\partial x\partial y}}198\newcommand\pfyx{\dis\frac{\partial^2 f}{\partial y\partial x}}199\newcommand\pfyy{\dis\frac{\partial^2 f}{\partial y^2}}200201\newcommand\intzi{\dis\int_{0}^{+\infty}}202\newcommand\intd{\dis\int}203\newcommand\intab{\dis\int_a^b}204205%调整分数数字与分数线的间距206%\renewcommand{\dfrac}[2]{{207%\renewcommand{\arraystretch}{1.375}208%\begingroup\displaystyle209%\rule[0pt]{0pt}{11pt}#1\endgroup%210%\over\displaystyle\rule[-3pt]{0pt}{11pt}#2211%}}%212213\newenvironment{mfrac}[2]%214{\raise0.5ex\hbox{$#1$}\! \left/ \! \lower0.5ex\hbox{$#2$}\right.}215216%定义新数学符号217\DeclareMathOperator{\sgn}{sgn}218\DeclareMathOperator{\arccot}{arccot}219\DeclareMathOperator{\arccosh}{arccosh}220\DeclareMathOperator{\arcsinh}{arcsinh}221\DeclareMathOperator{\arctanh}{arctanh}222\DeclareMathOperator{\arccoth}{arccoth}223\DeclareMathOperator{\grad}{\bf{grad}}224225\title{\heiti 全国大学生数学竞赛习题总结}226\author{\kaishu 唐渊 (XuanYuan\_huan)}227\date{\today}228229\bibliographystyle{plain}230231%开始写文章232\begin{document}233234\maketitle235236\newpage237238\tableofcontents239240\newpage241\section{多元函数微分学}242243\subsection{函数与图形}244245\no1.平面方程246247$\dis\Sigma:A(x-x_0)+B(y-y_0)+C(z-z_0)=0,$ 法向量$\stackrel{\rightarrow}{n}=248(A,B,C),$249250方向余弦$\dis\cos\alpha=\dfrac{A}{\sqrt{A^2+B^2+C^2}},\ 251\cos\beta=\dfrac{B}{\sqrt{A^2+B^2+C^2}},\ 252\cos\gamma=\dfrac{C}{\sqrt{A^2+B^2+C^2}},$ 253254其中$\alpha=\left<\stackrel{\rightarrow}{n},\stackrel{\rightarrow}{i}\right>255=\left<\Sigma,yOz\right>,\ 256\beta=\left<\stackrel{\rightarrow}{n},\stackrel{\rightarrow}{j}\right>257=\left<\Sigma,zOx\right>,\ 258\gamma=\left<\stackrel{\rightarrow}{n},\stackrel{\rightarrow}{k}\right>259=\left<\Sigma,xOy\right>.$260261\no2.直线方程262263(1) 两面交线264265(2) $\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n};\ x=x_0+lt,\ 266y=y_0+mt,\ z=z_0+nt.$267268\no3.二次曲面269270单叶双曲面:$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1,$271 双叶双曲面:$\dfrac{z^2}{c^2}-\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1.$272(单双看负号个数)273274\no4.圆锥面铺平275276$\dis z=a\sqrt{x^2+y^2},\ a=\cot \alpha,\ \alpha$为半顶角, 铺平后的圆心角277$\beta=2\pi\sin\alpha.$278279\no5.旋转曲面的参数方程280281$L:x=x(t),y=y(t),z=z(t),a\ls t\ls b,$ 绕$z$轴形成的曲面为282283$\dis\Sigma: x=\sqrt{x^2(t)+y^2(t)}\cos\theta,\ 284y=\sqrt{x^2(t)+y^2(t)}\sin\theta,\ z=z(t),\ a\ls t\ls b,\ 0\ls\theta\ls 2\pi.$285286\no6.直纹面上过一点的直线方程287288参数方程代入法;因式分解比例法289290\no7.顶点在原点的锥面:$n$次齐次方程$F(x,y,z)=0,\ $291其中$F(tx,ty,tz)=t^nF(x,y,z).$292293\no8.投影方程:$xOy$面$\Longrightarrow$关于$z$的方程有解,得出$x,y$的限制方程.294295\no9.空间特殊曲线:曲线定义;投影回溯(借助方向余弦).296297\subsection{极限,连续与微分}298299\no1.二重极限定义:$\limit_{(x,y)\to(x_0,y_0)}f(x,y)=A,\ $  300 连续:$\limit_{(x,y)\to(x_0,y_0)}f(x,y)=f(x_0,y_0).$301302\begin{spacing}{2.5}303304\no 偏导数:$z=f(x,y)\longrightarrow\pzx,\ \pfx,\ z'_x,\ z_x,\ f'_x,\ f_x,\ f'_1,\ f_1.$305306\end{spacing}307308\no 混合二阶偏导(注意次序):$\dfrac{\partial}{\partial y}\left(\pzx\right)=\pzxy,309\ f_{12},\ f''_{xy};310\qquad\dfrac{\partial}{\partial x}\left(\pzy\right)=\pzyx,\ f_{21},\ f''_{yx}.$311312\no2.链式法则313314\no 全微分:$\dis\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)=A\Delta x+B\Delta y+o(\rho),\ 315\rho=\sqrt{\Delta x^2+\Delta y^2}\Rightarrow\Delta z\triangleq\dif z.$316317\no 全微分形式不变性318319\no 隐函数求导法:直接求导法,公式法(全微分推出),全微分法320321\no3.方向导数:$\stackrel{\rightarrow}{l}=(\cos\alpha,\cos\beta),$322323$\dfrac{\partial f}{\partial \stackrel{\rightarrow}{l}}\Bigg|_{(x_0,y_0)}=324\limit_{\rho\to0^+}\dfrac{f(x_0+\rho\cos\alpha,y_0+\rho\cos\beta)-f(x_0,y_0)}{\rho}325=f'_x(x_0,y_0)\cos\alpha+f'_y(x_0,y_0)\cos\beta.$326327\vspace{0.3cm}328329由两个正交偏导数决定了一圈偏导数;方向导数是单向的(射线型),330而偏导数是双向的(直线型)。331332\vspace{0.3cm}333334偏导连续$\Longrightarrow$可微$\Longrightarrow\left\{\begin{aligned}335&\text{连续}\Longrightarrow\text{极限存在}\\336&\text{偏导存在}\\337\end{aligned}\right.$338(未画出均不可推)339340\vspace{0.3cm}341342其中,偏导连续指的是$f'_x(x,y),\ f'_y(x,y)$两个偏导函数在全平面范围的意义下在该点连续。343344\no4. \textcolor[rgb]{1,0,0}{二元函数Taylor公式}345346$\dis f(x_0+\Delta x,y_0+\Delta y)\triangleq f(x,y)=347f(x_0.y_0)+\big(f'_x\ f'_y\big)348\binom{\Delta x}{\Delta y}349+\dfrac{1}{2!}\big(\Delta x\ \Delta y\big)350\left[          351  \begin{array}{cc}   352    f''_{xx} & f''_{xy} \\  353    f''_{yx} &f''_{yy} \\  354  \end{array}355\right]               356\binom{\Delta x}{\Delta y}357+\cdots+358\dfrac{1}{n!}\left(\Delta x\dfrac{\partial}{\partial x}359+\Delta y\dfrac{\partial}{\partial y}\right)^nf(x_0,y_0)+\cdots.$360361其中,整体记号(二项展开):362$\left(\Delta x\dfrac{\partial}{\partial x}363+\Delta y\dfrac{\partial}{\partial y}\right)^nf(x_0,y_0)364=\sumkn C_n^k\Delta x^k\Delta y^{n-k}\dfrac{\partial^nf}{\partial x^k\partial y^{n-k}}365\Bigg|_{(x_0,y_0)}.$366367\no5.偏导等式问题368369\vspace{0.3cm}370371$\left\{\begin{aligned}372yf'_x-xf'_y=0\Longrightarrow f(x,y)=f(\cos\theta,\sin\theta),\\373xf'_x+kyf'_y=0\Longrightarrow f(x,y)=f\left(t,\dfrac{y_0}{x_0^k}t^k\right).\\374\end{aligned}\right.$375376\vspace{0.3cm}377378若有参数或借助参数,则对参数求导得函数为常数,取特殊参数值,可证。379380\subsection{几何意义,极值与最值}381382\no1.切线,法线,切平面,法平面383384对参数求导得切向量,对方程求导得法向量。385386\no2.极值387388$\nabla f(x_0,y_0)=0$,称驻点。389390Hessian矩阵391$\left[          392  \begin{array}{cc}   393    f''_{xx} & f''_{xy} \\  394    f''_{yx} &f''_{yy} \\  395  \end{array}396\right]     397=398\left[          399  \begin{array}{cc}   400    A &B \\  401    B &C \\  402  \end{array}403\right]     404\longrightarrow405\left[          406  \begin{array}{cc}  407\lambda_1&0\\4080&\lambda_2409\end{array}410\right]  411,\ $412特征值$\lambda_1,\ \lambda_2$:413414$\left\{\begin{aligned}415&++\Longrightarrow\text{正定}\Longrightarrow\text{极小值}\\416&--\Longrightarrow\text{负定}\Longrightarrow\text{极大值}\\417&+-,\ -+\Longrightarrow\text{不定}\Longrightarrow\text{非极值}\\418&+0,\ -0\Longrightarrow\text{半正定,半负定}\Longrightarrow419\text{定义法,观察可疑极值点周围函数值大小(一般是正负)}\\420\end{aligned}\right.$421422\vspace{0.3cm}423424三元及以上的函数$f(x_1,x_2,\cdots,x_n)$,仍由Hessian矩阵$H(f)=\{f''_{x_ix_j}\}$425的正定,负定,不定,半正定,半负定决定是否取得极值,且与二元时的判断规则相同。426427\no3.条件极值428429$z=f(x,y)$在$\varphi(x,y)=0$下的极值:430$ \textcolor[rgb]{1,0,0}{\nabla f//\nabla \varphi}\Longrightarrow431\left\{\begin{aligned}432&\text{最远、近点的垂线原理}\\433&\text{Lagrange乘数法}434\end{aligned}\right.$435436\vspace{0.3cm}437438$\varphi(x,y)$若能化为参数方程或能显式解出$y=g(x)$则直接代入,不使用439Lagrange乘数法。440441\vspace{0.3cm}442443\no4.曲线簇$F(x,y,c)=0$的包络线:444$\left\{\begin{aligned}445&F(x,y,c)=0\\446&F'_c(x,y,c)=0447\end{aligned}\right.$448449\no5.曲线切向量450451$\stackrel{\rightarrow}{t}452=\left(\limit_{\Delta t\to0}\dfrac{x(t+\Delta t)-x(t)}{\Delta t^n},\ 453\limit_{\Delta t\to0}\dfrac{y(t+\Delta t)-y(t)}{\Delta t^n},\ 454\limit_{\Delta t\to0}\dfrac{z(t+\Delta t)-z(t)}{\Delta t^n}455\right),\ n=1,2,\cdots,\ $456457\vspace{0.2cm}458459直到$\stackrel{\rightarrow}{t}\neq\stackrel{\rightarrow}{0}$为止.460461\no6.最小距离和462463距平面上$n$条\textcolor[rgb]{1,0,0}{互不平行}的直线的距离和最小的点必为某个交点。464465\vspace{0.3cm}466467\no7.过直线$L:468\left\{\begin{aligned}469&F(x,y,z)=0\\470&G(x,y,z)=0471\end{aligned}\right.$472的平面簇方程$\Sigma:\ F(x,y,z)+\lambda G(x,y,z)=0.$473\end{document}