From 58fb4ed53672a6596a8619401644288da951d05e Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 22 Jul 2021 23:15:07 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=9B=B2=E9=9D=A2=E7=A7=AF?= =?UTF-8?q?=E5=88=86=E4=B8=8E=E6=9B=B2=E7=BA=BF=E7=A7=AF=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ial-calculus-of-multivariate-functions.pdf | Bin 0 -> 52778 bytes ...ial-calculus-of-multivariate-functions.tex | 141 +++++++ ...or-algebra-and-space-analytic-geometry.pdf | Bin 91074 -> 108329 bytes ...or-algebra-and-space-analytic-geometry.tex | 38 +- ...ial-calculus-of-multivariate-functions.pdf | Bin 151649 -> 140099 bytes ...ial-calculus-of-multivariate-functions.tex | 82 ---- ...ral-calculus-of-multivariate-functions.pdf | Bin 106676 -> 159831 bytes ...ral-calculus-of-multivariate-functions.tex | 376 +++++++++++++++++- 8 files changed, 547 insertions(+), 90 deletions(-) create mode 100644 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b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -0,0 +1,141 @@ +\documentclass[UTF8, 12pt]{ctexart} +% UTF8编码,ctexart现实中文 +\usepackage{xcolor} +% 使用颜色 +\definecolor{orange}{RGB}{255,127,0} +\definecolor{violet}{RGB}{192,0,255} +\definecolor{aqua}{RGB}{0,255,255} +\usepackage{geometry} +\setcounter{tocdepth}{4} +\setcounter{secnumdepth}{4} +% 设置四级目录与标题 +\geometry{papersize={21cm,29.7cm}} +% 默认大小为A4 +\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} +% 默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\usepackage{pifont} +% 圆圈序号 +\usepackage{tikz} +% 绘图 +\author{Didnelpsun} +\title{多元函数微分学} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{多元函数微分应用} + +\subsection{空间曲线的切线与法平面} + +\subsubsection{参数方程} + +设空间曲线$\varGamma$由参数方程$\left\{\begin{array}{l} + x=\phi(t) \\ + y=\psi(t) \\ + z=\omega(t) +\end{array}\right.$给出,其中$\phi(t),\psi(t),\omega(t)$均可导,$P_0(x_0,y_0,z_0)$为$\varOmega$上的点,且当$t=t_0$时,$\phi'(t_0)$,$\psi'(t_0)$,$\omega'(t_0)$均不为0,则: + +\begin{itemize} + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切向量为$\vec{\tau}=(\phi'(t_0),\psi'(t_0),\omega'(t_0))$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切线方程为$\dfrac{x-x_0}{\phi'(t_0)}=\dfrac{y-y_0}{\psi'(t_0)}=\dfrac{z-z_0}{\omega'(t_0)}$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的法平面(过$P_0$且与切线垂直的平面)方程为$\phi'(t_0)(x-x_0)+\psi'(t_0)(y-y_0)+\omega'(t_0)(z-z_0)=0$。 +\end{itemize} + +\subsubsection{交面式方程} + +设空间曲线$\varGamma$由交面方程$\left\{\begin{array}{l} + F(x,y,z)=0 \\ + G(x,y,z)=0 +\end{array}\right.$给出,则: + +\begin{itemize} + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切向量为\\$\vec{\tau}=\left(\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0},\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0},\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}\right)$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切线方程为\\$\dfrac{x-x_0}{\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0}},\dfrac{y-y_0}{\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0}},\dfrac{z-z_0}{\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}}$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的法平面方程为\\$\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0}(x-x_0)+\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0}(y-y_0)+\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}(z-z_0)=0$。 +\end{itemize} + +\subsection{空间曲面的切平面与法线} + +\subsubsection{隐式} + +设空间曲面$\varSigma$由方程$F(x,y,z)=0$给出,$P_0(x_0,y_0,z_0)$是$\varSigma$上的点,则: + +\begin{itemize} + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的法向量为$\vec{n}=(F_x'(x_0,y_0,z_0),F_y'(x_0,y_0,z_0),$\\$F_z'(x_0,y_0,z_0))$且法线方程为$\dfrac{x-x_0}{F_x'(x_0,y_0,z_0)}=\dfrac{y-y_0}{F_y'(x_0,y_0,z_0)}=\dfrac{z-z_0}{F_z'(x_0,y_0,z_0}$。 + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的切平面方程为$F_x'(x_0,y_0,z_0)(x-x_0)+F_y'$\\$(x_0,y_0,z_0)(y-y_0)+F_z'(x_0,y_0,z_0)(z-z_0)=0$。 +\end{itemize} + +\subsubsection{显式} + +设空间曲面$\varSigma$由方程$z=f(x,y)$给出,令$F(x,y,z)=f(x,y)-z$,假定法向量的方向向下,即其余$z$轴正向所成的角为钝角,即$z$为-1,则: + +\begin{itemize} + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的法向量为$\vec{n}=(f_x'(x_0,y_0),f_y'(x_0,y_0),-1)$,且法线方程为$\dfrac{x-x_0}{f_x'(x_0,y_0)}=\dfrac{y-y_0}{f_y'(x_0,y_0)}=\dfrac{z-z_0}{-1}$。 + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的切平面方程为$f_x'(x_0,y_0)(x-x_0)+f_y'(x_0,y_0)$\\$(y-y_0)-(z-z_0)=0$。 +\end{itemize} + +若是反之成锐角,则将里面所有的-1都换成1。 + +若用$\alpha$,$\beta$,$\gamma$表示曲面$z=f(x,y)$在点$(x_0,y_0,z_0)$处的法向量的方向角,并这里假定法向量的方向是向上的,即其余$z$轴正向所成的角$\gamma$为锐角,则法向量\textbf{方向余弦}为$\cos\alpha=\dfrac{-f_x}{\sqrt{1+f_x^2+f_y^2}}$,$\cos\beta=\dfrac{-f_y}{\sqrt{1+f_x^2+f_y^2}}$,$\cos\gamma=\dfrac{1}{\sqrt{1+f_x^2+f_y^2}}$,其中$f_x=f_x'(x_0,y_0)$,$f_y=f_y'(x_0,y_0)$。 + +\textbf{例题:}设直线$L\left\{\begin{array}{l} + x+y+b=0 \\ + x+ay-z-3=0 +\end{array}\right.$在平面$\pi$上,而平面$\pi$与曲面$z=x^2+y^2$相切于$(1,-2,5)$,求$ab$的值。 + +解:$L$在$\pi$上且与曲面相切,则$\pi$为$L$的切平面。设曲面方程$F(x,y,z)=x^2+y^2-z$。 + +曲面法向量为$\vec{n}=\{F_x',F_y',F_z'\}=\{2x,2y,-1\}$,代入$(1,-2,5)$,则法向量为$\{2,-4,-1\}$。 + +又点法式:$\pi:2(x-1)-4(y+2)-(z-5)=0$,即$2x-4y-z-5=0$。 + +联立直线方程,得到:$(5+a)x+4b+ab-2=0$,又$x$是任意的。 + +解得$a=-5,b=-2$。 + +\end{document} diff --git 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a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex index b895f74..35f6ba7 100644 --- a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex +++ b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex @@ -270,12 +270,46 @@ \subsection{方向导数} +偏导数就是一个函数在坐标轴方向上的变化率,而方向导数就是函数在某点沿其他特定方向上的变化率。 + +\textcolor{violet}{\textbf{定义:}}设三元函数$u=u(x,y,z)$在点$P_0(x_0,y_0,z_0)$的某空间领域$U\in R^3$内有定义,$l$从点$P_0$出发的射线,$P(x,y,z)$为$l$上切在$U$内的任一点,则$\left\{\begin{array}{l} + x-x_0=\Delta x=t\cos\alpha \\ + y-y_0=\Delta y=t\cos\beta \\ + z-z_0=\Delta z=t\cos\gamma +\end{array}\right.$进行在坐标轴上投影。 + +以$t=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$表示$P$与$P_0$之间的距离。若极限$\lim\limits_{t\to0^+}$\\$\dfrac{u(P)-u(P_0)}{t}=\lim\limits_{t\to0^+}\dfrac{u(x_0+t\cos\alpha,y_0+t\cos\beta,z_0+t\cos\gamma)-u(x_0,y_0,z_0)}{t}$存在,则称此极限为函数$u=u(x,y,z)$在点$P_0$沿方向$l$的\textbf{方向导数},记为$\dfrac{\partial u}{\partial l}\bigg|_{P_0}$。 + +方向导数计算公式\textcolor{aqua}{\textbf{定理:}}设三元函数$u=u(x,y,z)$在点$P_0(x_0,y_0,z_0)$处可微分,则$u=u(x,y,z)$在点$P_0$处沿任一方向$l$的方向导数都存在,且$\dfrac{\partial u}{\partial l}\bigg|_{P_0}=u_x'(P_0)\cos\alpha+u_y'(P_0)\cos\beta+u_z'(P_0)\cos\gamma$,其中$\cos\alpha$,$\cos\beta$,$\cos\gamma$为方向$l$的方向余弦。 + +\textbf{例题:}求函数$z=xe^{2y}$在点$P(1,0)$处沿点$P(1,0)$指向$Q(2,-1)$方向的方向导数。 + +解:这是一个隐式的三元函数,所以基本上解决方法类似。不过需要将$z$对$xy$求偏导。 + +$\dfrac{\partial z}{\partial x}=e^{2y}$,$\dfrac{\partial z}{\partial y}=2xe^{2y}$,代入$P(1,0)$,得到$\{1,2\}$。 + +然后求方向余弦,对于$\overrightarrow{PQ}={1,-1}$方向余弦就是除它的模$\left\{\dfrac{1}{\sqrt{2}},-\dfrac{1}{\sqrt{2}}\right\}$。 + +方向导数就是相乘:$\dfrac{1}{\sqrt{2}}-\dfrac{2}{\sqrt{2}}=-\dfrac{\sqrt{2}}{2}$。 + \subsection{梯度} +在一个数量场中中,函数在给定点处沿不同的方向,其方向导数一般是不相同的。为研究哪个方向的方向导数最大,最大值为多少,增加速度最快,就引入了梯度。 + +\textcolor{violet}{\textbf{定义:}}设三元函数$u=u(x,y,z)$在点$P_0(x_0,y_0,z_0)$处具有一阶偏导数,定义$\text{grad}\,u|_{P_0}=(u'_x(P_0),u_y'(P_0),u_z'(P_0))$为函数$u=u(x,y,z)$在点$P_0$处的\textbf{梯度}。 + \subsection{方向导数与梯度关系} -\subsection{散度} +方向导数为梯度×梯度方向余弦。 -\subsection{旋度} +函数在某点的梯度是一个向量,其方向与取得最大方向导数的方向是一致的,其模就是方向导数最大值。 + +\subsection{散度与旋度} + +\textcolor{violet}{\textbf{定义:}}设向量场$\vec{A}(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z))$,则\textbf{散度}$\textrm{div}\,\vec{A}=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$,\textbf{旋度}$\overrightarrow{\textrm{rot}}\,\vec{A}=\left\vert\begin{array}{ccc} + \vec{i} & \vec{j} & \vec{k} \\ + \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ + P & Q & R +\end{array}\right\vert$。 \end{document} diff --git 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zL@Ar_#EA!>`U6d^S~L4*!(fu}r18UKJePyq{#pzIC#f*J7k72{JBtL@T7t}F$b5uy z6MZ9&E#8h)ut&3r@dOuQ5r_A@w~kZc@3osxeD2A!(~WInHD0a|=7yljpIRF3v diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex index 9b4ca61..ce7f988 100644 --- a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex +++ b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -6,9 +6,9 @@ \definecolor{violet}{RGB}{192,0,255} \definecolor{aqua}{RGB}{0,255,255} \usepackage{geometry} -\setcounter{tocdepth}{4} -\setcounter{secnumdepth}{4} -% 设置四级目录与标题 +\setcounter{tocdepth}{5} +\setcounter{secnumdepth}{5} +% 设置五级目录与标题 \geometry{papersize={21cm,29.7cm}} % 默认大小为A4 \geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} @@ -27,6 +27,8 @@ % 超链接 \usepackage{tikz} % 绘图 +\usepackage{unicode-math} +% 二重闭合积分 \author{Didnelpsun} \title{多元函数积分学} \date{} @@ -84,7 +86,7 @@ $A.I_3>I_2>I_1$\qquad$B.I_1>I_2>I_3$\qquad$C.I_2>I_1>I_3$\qquad$D.I_3>I_1>I_2$ $I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqrt{f(x)}+\sqrt{f(y)}}\textrm{d}\sigma=\iint\limits_D\dfrac{a\sqrt{f(y)}+b\sqrt{f(x)}}{\sqrt{f(y)}+\sqrt{f(x)}}\textrm{d}\sigma}$。 -$\therefore2I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqrt{f(x)}+\sqrt{f(y)}}\textrm{d}\sigma+\iint\limits_D\dfrac{a\sqrt{f(y)}+b\sqrt{f(x)}}{\sqrt{f(y)}+\sqrt{f(x)}}\textrm{d}\sigma}=\iint\limits_D(a+b)\,\textrm{d}\sigma=(a+b)\dfrac{\pi}{4}$。 +$\therefore2I=\displaystyle{\iint\limits_D\dfrac{a\sqrt{f(x)}+b\sqrt{f(y)}}{\sqrt{f(x)}+\sqrt{f(y)}}\textrm{d}\sigma+\iint\limits_D\dfrac{a\sqrt{f(y)}+b\sqrt{f(x)}}{\sqrt{f(y)}+\sqrt{f(x)}}\textrm{d}\sigma}=\iint\limits_D(a+b)\,\textrm{d}\sigma$\\$=(a+b)\dfrac{\pi}{4}$。 解得$I=\dfrac{a+b}{8}\pi$。 @@ -228,7 +230,7 @@ $\int_0^1\textrm{d}y\int_0^u\sin(\pi u^2)\,\textrm{d}x=\int_0^1\sin(\pi u^2)u\,\ 解:根据积分值与积分变量无关的性质: -$I^2=(\int_0^{+\infty}e^{-x^2}\,\textrm{d}x)^2=\int_0^{+\infty}e^{-x^2}\,\textrm{d}x\cdot\int_0^{+\infty}e^{-y^2}\,\textrm{d}y=\int_0^{+\infty}\int_0^{+\infty}e^{-x^2-y^2}\,\textrm{d}x\textrm{d}y$。 +$I^2=(\int_0^{+\infty}e^{-x^2}\,\textrm{d}x)^2=\int_0^{+\infty}e^{-x^2}\,\textrm{d}x\cdot\int_0^{+\infty}e^{-y^2}\,\textrm{d}y=\int_0^{+\infty}\int_0^{+\infty}e^{-x^2-y^2}\,\textrm{d}x\textrm{d}y$ $\textrm{d}\sigma$是第一象限,可以看作一个广义的圆,半径无限大,转换为极坐标系。 @@ -236,6 +238,368 @@ $=\int_0^\frac{\pi}{2}\textrm{d}\theta\int_0^{+\infty}e^{-r^2}r\,\textrm{d}r=\di \section{三重积分} -\section{曲线曲面积分} +\subsection{概念} + +三重积分的被积函数$f(x,y,z)$定义在三维空间$\Omega$上,是四维空间图形体积,非常抽象。 + +所以利用质量描述,设一质量非均匀的物体,体积密度为$f(x,y,z)$,则三重积分就是以此为点密度的空间物体的质量。 + +\subsubsection{定义} + +\textcolor{violet}{\textbf{定义:}}设三元函数$z=f(x,y,z)$定义在有界闭区域$\Omega$上将区域$\Omega$任意分成$n$个子域$\Delta v_i$($i=1,2,3,\cdots,n$)并以$\Delta v_i$表示第$i$个子域的体积。在$\Delta v_i$上任取一点$(\alpha_i,\beta_i,\gamma_i)$作和$\sum\limits_{i=1}^n\alpha_i\beta_i\gamma_i\Delta v_i$。如果当各个子域的直径中的最大值$\lambda$趋于零时,此和式的极限存在,则称此极限为函数$f(x,y,z)$在区域$\Omega$上的三重积分,记为$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v$,即$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=\lim\limits_{\lambda\to0}\sum\limits_{i=1}^nf(x_i,y_i,z_i)\Delta y_i$,其中$\textrm{d}v$叫做体积元素。 + +其中$\iiint$称为三重积分号,$f(x,y,z)$为被积函数,$f(x,y,z)\textrm{d}v$称为被积表达式,$\textrm{d}v$称为体积元,$x$、$y$、$z$为积分变量,$\Omega$为积分区域,$\sum f(\alpha_i,\beta_i,\gamma_i)\Delta v_i$为积分和。 + +\subsubsection{性质} + +假设$\Omega$为空间有界闭区域。 + +\begin{itemize} + \item 空间区域体积:$\iiint\limits_\Omega1\,\textrm{d}v=\iiint\limits_\Omega\textrm{d}v=V$,其中$V$为$\Omega$的体积。 + \item 可积函数必有界:设$f(x,y,z)$在$\Omega$上可积,则在$\Omega$上必有界。 + \item 积分线性:设$k_1,k_2$为常数,则$\iiint\limits_\Omega[k_1f(x,y,z)\pm k_2g(x,y,z)]\textrm{d}v=k_1\iiint\limits_\Omega$\\$f(x,y,z)\,\textrm{d}v\pm k_2\iiint\limits_\Omega g(x,y,z)\,\textrm{d}v$。 + \item 积分可加性:设$f(x,y,z)$在$\Omega$上可积,且$\Omega_1\cup\Omega_2=\Omega$,$\Omega_1\cap\Omega_2=\varnothing$,则$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=\iiint\limits_{\Omega_1}f(x,y,z)\,\textrm{d}v+\iiint\limits_{\Omega_2}f(x,y,z)\,\textrm{d}v$。 + \item 积分保号性:设$f(x,y,z)$,$g(x,y,z)$在$\Omega$上可积,且在$\Omega$上$f(x,y,z)\leqslant g(x,y,z)$,则有$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v\leqslant\iiint\limits_\Omega g(x,y,z)\,\textrm{d}v$。且利用不等式性质:$\vert\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v\vert\leqslant\iiint\limits_\Omega\vert f(x,y,z)\vert\,\textrm{d}v$。 + \item 三重积分估值定理:设$M,m$分别为$f(x,y,z)$在$\Omega$上的最大值和最小值,$V$为$\Omega$的体积,则$mV\leqslant\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v\leqslant MV$。 + \item 三种积分中值定理:设$f(x,y,z)$在$\Omega$上连续,$V$为$\Omega$的体积,则$\Omega$上至少存在一点$(\xi,\eta,\zeta)$使得$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=f(\xi,\eta,\zeta)V$。 +\end{itemize} + +\subsubsection{对称性} + +分析方法与二重积分完全一样。 + +\paragraph{普通对称性} \leavevmode \medskip + +假设$\Omega$关于$yOz$面对称,则$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=$\\$\left\{\begin{array}{ll} + 2\iiint\limits_{\Omega_1}f(x,y,z)\,\textrm{d}v, & f(x,y,z)=f(-x,y,z) \\ + 0, & f(x,y,z)=-f(-x,y,z) +\end{array}\right.$,其中$\Omega_1$为$\Omega$在$yOz$面前面的部分。 + +\paragraph{轮换对称性} \leavevmode \medskip + +若把$x$与$y$对调后,$\Omega$不变,则$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=\iiint\limits_\Omega f(y,x,z)\,\textrm{d}v$,这就是\textbf{轮换对称性}。其他情况类似。 + +在使用轮换对称性的时候需要根据题目进行轮换,特别是根据所要求的被积函数$f(x)$,若$f(x)$中存在某些变量,则要将没有出现的变量换去。 + +\subsection{计算} + +\subsubsection{基础方法} + +\paragraph{直角坐标系} \leavevmode \medskip + +即$\textrm{d}v=\textrm{d}x\textrm{d}y\textrm{d}z$,微元是一个长方体。 + +\subparagraph{先一后二法} \leavevmode \medskip + +先$z$后$xy$,也称为投影穿线法。先做定积分后做二重积分。 + +适用场合:$\Omega$有下曲面$z=z_1(x,y)$、上曲面$z=z_2(x,y)$,无侧面或侧面为柱面。 + +如二重积分:后积先定限,限内画条线,先交写下限,后交写上限。 + +$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=\iint\limits_{D_{xy}}\textrm{d}\sigma\int_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)\,\textrm{d}z$。 + +\textbf{例题:}计算三重积分$I=\displaystyle{\iiint\limits_D\dfrac{\textrm{d}x\textrm{d}y\textrm{d}z}{(1+x+y+z)^3}}$,其中$\Omega$是由平面$x=0,y=0,z=0$及$x+y+z=1$所围成的四面体。 + +解:根据图形,已知是一个四面体,所以下底面是一个1×1的等腰直角三角形$D_{xy}$,上曲面为一个等边三角形$z=-1-x-y$,有两个侧柱面。 + +则先将$I$消去$z$,再计算$xy$:$I=\displaystyle{\iint\limits_{D_{xy}}\textrm{d}\sigma\int_0^{1-x-y}\dfrac{1}{(1+x+y+z)^3}\textrm{d}z}$ + +$=\displaystyle{\iint\limits_{D_{xy}}\textrm{d}\sigma\left(-\dfrac{1}{2}\dfrac{1}{(1+x+y+z)^2}\bigg|_{z=0}^{z=1-x-y}\right)}=\displaystyle{\iint\limits_{D_{xy}}\dfrac{1}{2}\left(\dfrac{1}{(1+x+y)^2}-\dfrac{1}{4}\right)\,\textrm{d}\sigma}$\\$=\dfrac{1}{2}\int_0^1\textrm{d}x\int_0^{1-x}\left(\dfrac{1}{(1+x+y)^2}-\dfrac{1}{4}\right)\,\textrm{d}y=\dfrac{1}{2}\displaystyle{\int_0^1\left(-\dfrac{1}{4}x+\dfrac{1}{1+x}-\dfrac{1}{4}\right)\,\textrm{d}x}$\\$=\left(-\dfrac{1}{8}x^2+\ln(1+x)-\dfrac{1}{4}x\right)\bigg|_0^1=\dfrac{1}{2}\left(\ln2-\dfrac{5}{8}\right)$。 + +\subparagraph{先二后一法} \leavevmode \medskip + +先$xy$后$z$,也称为定限截面法。先做二重积分后做定积分。 + +适用场合:$\Omega$是旋转体,上面和下面都是平面,中间为曲面,旋转曲面方程为$\Sigma:z=z(x,y)$。 + +后积先定限,限内截个面限。 + +$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}v=\int\limits_a^b\textrm{d}z\iint\limits_{D_z}f(x,y,z)\,\textrm{d}\sigma$。 + +\paragraph{柱面坐标系} \leavevmode \medskip + +若二重积分部分$\iint\limits_{D_{xy}}\textrm{d}\sigma$适用于极坐标系(即与圆相关),使用极坐标系表示,令$x=r\cos\theta$,$y=r\sin\theta$,便有$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}x\textrm{d}y\textrm{d}z=\iiint\limits_\Omega f(r\cos\theta,r\sin\theta,z)r\,$\\$\textrm{d}r\textrm{d}\theta\textrm{d}z$。这就是柱面坐标系下三重积分的计算。 + +适用场合:被积函数含有$x^2+y^2$,积分区域为圆或部分圆。 + +即一个定积分加上一个极坐标系下的二重积分。 + +% 在直接坐标系的先二后一法,因为$\textrm{d}\sigma$是在与$z$相关的$D_z$区域,需要用$z$来表示函数,所以无法用两个变量来表示函数。 + +\textbf{例题:}计算$\iiint\limits_\Omega(x^2+y^2)\,\textrm{d}v$,其中$\Omega$是$\left\{\begin{array}{ll} + y^2=2z \\ + x=0 +\end{array}\right.$绕$z$轴旋转一周形成的曲面与平面$z=8$所围成的区域。 + +解:已知平面曲线绕$z$轴旋转,首先求这个旋转曲面。 + +首先令$P_1(x_1,y_1,z_1)$在该曲线上,即得到两个方程:$y_1^2=2z_1$,$x_1=0$。 + +取在旋转轴$z$轴上一点$P_0(0,0,0)$,对于纬圆上任一点$P(x,y,z)$,其中$\vert P_0P_1\vert=\vert PP_0\vert$,即$x_1^2+y_1^2+z_1^2=x^2+y^2+z^2$。 + +且向量$\overrightarrow{PP_1}$垂直于旋转轴$z$轴,所以$(x_1-x,y_1-y,z_1-z)\bot(0,0,1)$,$z_1-z=0$,$z_1=z$。 + +代入方程所以$x_1^2+y_1^2=x^2+y^2$,再代入$y_1^2=2z$,$x_1=0$,得到$2z=x^2+y^2$。 + +旋转曲面为$z=\dfrac{x^2+y^2}{2}$。且于$z=8$所得到一个旋转体。 + +因为选择体上下都是平面,侧面是曲面,所以使用先二后一法。其中$D:x^2+y^2\leqslant2z$。 + +$I=\int_0^8\textrm{d}z\iint\limits_D(x^2+y^2)\,\textrm{d}\sigma=\int_0^8\textrm{d}z\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2z}}r^2r\,\textrm{d}r=\dfrac{1024}{3}\pi$。 + +\paragraph{球面坐标系} \leavevmode \medskip + +\subparagraph{适用场合} \leavevmode \medskip + +被积函数含有$x^2+y^2+z^2$或$x^2+y^2$,积分区域为球或球的部分,锥或锥的部分。 + +\subparagraph{计算方法} \leavevmode \medskip + +利用三族面对$\Omega$进行切割: + +\begin{enumerate} + \item 首先用$r=r_0$从原点开始向外做球体进行切割,求半径为$r_0$,增量为$\textrm{d}r$。$r_0\in[0,+\infty)$。 + \item 然后用$\phi=\phi_0$从$z$轴为中心,原点为定点,做半顶角为$\phi_0$的圆锥面进行切割,增量为$\textrm{d}\phi$。$\phi_0\in[0,\pi]$。 + \item 最后用$\theta=\theta_0$以$z$为轴做半平面,与$xOz$夹角为$\theta_0$,增量为$\textrm{d}\theta$。$\theta\in[0,2\pi]$。 +\end{enumerate} + +首先在极坐标系中,弧长等于弧度乘半径,所以微元的由$\phi$确定的一边为$r\textrm{d}\phi$,对于$\theta$确定的一边,首先需要根据勾股定理得到弧长$r\sin\phi$,然后乘$\textrm{d}\theta$得到微元边长$r\sin\phi\textrm{d}\theta$,最后乘上$\textrm{d}r$,从而得到微元就是三边相乘:$r\textrm{d}\phi r\sin\phi\textrm{d}\theta\textrm{d}r$。 + +对$xyz$,由$\phi$推出的一个直角三角形的斜边为$r$,半顶角为$\phi$,所以$z$轴的直角边为$z=r\cos\phi$,又$x^2+y^2+z^2=r^2$,所以$x^2+y^2=r^2-z^2=r^2\sin^2\phi$,又$xOy$夹角为$\theta$,所以$x=r\sin\phi\cos\theta$,$y=r\sin\phi\sin\theta$。 + +$\iiint\limits_\Omega f(x,y,z)\,\textrm{d}x\textrm{d}y\textrm{d}z=\iiint\limits_\Omega f(r\sin\phi\cos\theta,r\sin\phi\sin\theta,r\cos\phi)r^2\sin\phi\,\textrm{d}\theta\textrm{d}\phi\textrm{d}r$ + +\textbf{例题:}计算三重积分$\iiint\limits_\Omega(x^2+y^2)\,\textrm{d}v$,其中$\Omega$是半球面$x^2+y^2+z^2=a^2$($y\geqslant0$)与$xOz$面所围成的区域。 + +解:根据图形是一个右半球,所以$\theta$是$x$正轴到负轴一共$\pi$,$\phi$到正轴到负轴一共$\pi$,$r$从原点到最外面一共$a$。$f(x)=x^2+y^2=r^2\sin^2\phi$。 + +$\therefore I=\int_0^\pi\textrm{d}\theta\int_0^\pi\textrm{d}\phi\int_0^a(r^2\sin^2\phi)r^2\sin\phi\textrm{d}r$ + +\paragraph{对称性} \leavevmode \medskip + +\subparagraph{普通对称性} \leavevmode \medskip + +\textbf{例题:}计算$\iiint\limits_\Omega e^{\vert z\vert}\,\textrm{d}v$,其中$\Omega:x^2+y^2+z^2\leqslant1$。 + +解:已知$\Omega$为一个半径为1的球体。且球体球心在原点,利用普通对称性代入:$f(x,y,z)\,\textrm{d}v=e^{\vert z\vert}\,\textrm{d}v=\textrm{d}m=f(x,y,-z)\,\textrm{d}v=e^{\vert z\vert}\,\textrm{d}v=\textrm{d}m$,所以对于$f(x,y,z)$,在球体上下积分相同。 + +$\therefore\iiint\limits_\Omega e^{\vert z\vert}\,\textrm{d}v=2\iiint\limits_{\Omega_1}e^z\,\textrm{d}v$,其中$\Omega_1$为上半球体,即$\Omega_1:x^2+y^2+z^2\leqslant1,z\geqslant0$。 + +由于$f(x)=e^z$,只包含$z$的变量,所以使用先二后一法更简单。 + +由于截面是一个圆,所以令$z=z$,代入方程得到面积:$D:x^2+y^2\leqslant1-z^2$。所以这个圆的半径的平方就是$r^2=1-z^2$,面积为$\pi r^2=\pi(1-z^2)$。 + +$=2\int_0^1\textrm{d}z\iint\limits_De^z\,\textrm{d}\sigma=2\int_0^1e^z\cdot\pi(1-z^2)\,\textrm{d}z=2\pi$。 + +\subparagraph{轮换对称性} \leavevmode \medskip + +\textbf{例题:}设$\Omega=\{(x,y,z)|x^2+y^2+z^2\leqslant1\}$,求$\iiint\limits_\Omega z^2\,\textrm{d}x\textrm{d}y\textrm{d}z$。 + +解:利用轮换对称性可知$\iiint\limits_\Omega x^2\,\textrm{d}x\textrm{d}y\textrm{d}z=\iiint\limits_\Omega y^2\,\textrm{d}x\textrm{d}y\textrm{d}z=\iiint\limits_\Omega z^2\,\textrm{d}x\textrm{d}y\textrm{d}z$。 + +$\therefore I=\dfrac{1}{3}\iiint\limits_\Omega(x^2+y^2+z^2)\,\textrm{d}x\textrm{d}y\textrm{d}z$。 + +\paragraph{形心公式逆用} \leavevmode \medskip + +由$\overline{x}=\dfrac{\iiint\limits_\Omega x\,\textrm{d}v}{\iiint\limits_\Omega\textrm{d}v}$推出$\iiint\limits_\Omega x\,\textrm{d}v=\overline{x}\cdot V$,其中$V$是$\Omega$的体积。 + +\section{第一型曲线积分} + +是由定积分推广而来。 + +\subsection{概念} + +\subsubsection{定义} + +\subsubsection{性质} + +\subsubsection{对称性} + +\subsection{计算} + +\subsubsection{基础方法} + +即化为定积分。一投(投影)二代(代入关系方程)三计算($\textrm{d}s$转换为$\textrm{d}x$等)。 + +\paragraph{平面} \leavevmode \medskip + +\textbf{例题:}计算$\oint_\Gamma\vert y\vert\,\textrm{d}s$,其中$Gamma$为球面$x^2+y^2+z^2=2$与平面$x=y$的交线。 + +解:根据普通对称性,对于$\vert y\vert$而言,其他象限的函数值都与第一象限区域的函数相等。令第一象限区域为$\Gamma_1$: + +$\therefore\oint_\Gamma\vert y\vert\,\textrm{d}s=4\oint_{\Gamma_1}y\,\textrm{d}s$ + +根据交线方程进行联立,得到$x=\cos t$,$y=\cos t$,$z=\sqrt{2}\sin t$。 + +对于$\Gamma_1$,其角度为$t\in\left(0,\dfrac{\pi}{2}\right)$,$y=\cos t$。 + +又$\textrm{d}s=\sqrt{x_t'^2+y_t'^2+z_t'^2}$,所以$\textrm{d}t=\sqrt{2}\,\textrm{d}t$ + +$=4\int_0^\frac{\pi}{2}\cos t\sqrt{2}\,\textrm{d}t=4\sqrt{2}$。 + +\paragraph{空间} \leavevmode \medskip + +\subsubsection{技术方法} + +\paragraph{边界方程代入被积函数} \leavevmode \medskip + +\paragraph{对称性} \leavevmode \medskip + +\paragraph{形心公式逆用} \leavevmode \medskip + +\section{第一型曲面积分} + +$\textrm{d}S$为面微分。 + +\subsection{概念} + +\subsubsection{定义} + +\subsubsection{性质} + +\subsubsection{对称性} + +\subsection{计算} + +还是一投二代三计算。 + +\subsubsection{基础方法} + +即化为二重积分。 + +\textbf{例题:}设曲面$\Sigma:\vert x\vert+\vert y\vert+\vert z\vert=1$,求$\oiint\limits_\Sigma(x+\vert y\vert)\,\textrm{d}S$。 + +解:曲面$\Sigma$是一个正八面体。又普通对称性得$\oiint\limits_\Sigma x\,\textrm{d}S=0$。 + +令第一卦限为$\Sigma_1$,所以根据普通对称性$\oiint\limits_\Sigma\vert y\vert\,\textrm{d}S=8\iint\limits_{\Sigma_1}y\,\textrm{d}S$ + +因为$x+y+z=1$和$\textrm{d}S=\sqrt{1+z_x'^2+z_y'^2}\,\textrm{d}x\textrm{d}y$交换$xy$保持不变。 + +根据轮换对称性$\iint\limits_{\Sigma_1}y\,\textrm{d}S=\iint\limits_{\Sigma_1}x\,\textrm{d}S$。 + +且对于$x+y+z=1$和用$xz$替换$y$(把微元曲面投到不同的坐标轴平面):$\textrm{d}S=\sqrt{1+y_x'^2+y_z'^2}\,\textrm{d}x\textrm{d}z$交换$xz$保持不变。 + +根据轮换对称性$\iint\limits_{\Sigma_1}y\,\textrm{d}S=\iint\limits_{\Sigma_1}x\,\textrm{d}S=\iint\limits_{\Sigma_1}z\,\textrm{d}S$。 + +$\therefore8\iint\limits_{\Sigma_1}y\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sigma_1}(x+y+z)\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sigma_1}\textrm{d}S=\dfrac{8}{3}S_{\Sigma_1}=\dfrac{8}{3}\dfrac{\sqrt{3}}{2}=\dfrac{4}{3}\sqrt{3}$。 + +\subsubsection{技术方法} + +\paragraph{边界方程代入被积函数} \leavevmode \medskip + +\paragraph{对称性} \leavevmode \medskip + +\paragraph{形心公式逆用} \leavevmode \medskip + +\section{多元积分应用} + +包括二重积分、三重积分、一型曲线积分、一型曲面积分四个部分。 + +\subsection{几何量} + +\subsubsection{平面区域} + +\subsubsection{空间区域} + +\subsubsection{空间曲线} + +\subsubsection{空间曲面} + +是考试的重点。 + +\subsection{重心与形心} + +当密度$\rho$为一个固定常数时重心就是形心。 + +\subsubsection{平面薄片} + +\subsubsection{空间物体} + +是考试的重点。 + +\textbf{例题:}设空间物体$\Omega=\{(x,y,z)|x^2+y^2\leqslant z\leqslant1\}$,求$\Omega$的形心的竖坐标$\overline{z}$。 + +% 解:形心公式$\overline{z}=\dfrac{}{}$ + +\subsubsection{空间曲线} + +\subsubsection{空间曲面} + +\subsection{转动惯量} + +可能会考到。 + +\subsubsection{平面薄片} + +\subsubsection{空间物体} + +\subsubsection{空间曲线} + +\subsubsection{空间曲面} + +\subsection{引力} + +考的可能性很小。 + +\subsubsection{平面薄片} + +\subsubsection{空间物体} + +\subsubsection{空间曲线} + +\subsubsection{空间曲面} + +\section{第二型曲线积分} + +第二型与第一型的差别就是第二型具有物理意义是有向的,而第一型具有几何意义是无向的。 + +\subsection{概念} + +\subsubsection{场的概念} + +\textcolor{violet}{\textbf{定义:}}就是空间区域$\Omega$上的一种对应法则。 + +数量场就是对应数量没有方向。向量场就是有数量也有方向。 + +\subsubsection{变力沿曲线做功} + +\subsubsection{定义} + +\subsubsection{性质} + +\subsection{计算} + +\subsubsection{基础方法} + +即化为定积分。 + +\subsubsection{格林公式} + +是考试的重点,基本上都会考到。非常重要。 + +\section{第二型曲面积分} + +\subsection{概念} + +\subsubsection{向量场的通量} + +\subsubsection{定义} + +\subsubsection{性质} + +\subsection{计算} + +\subsubsection{基础方法} + +即化为二重积分。 + +\subsubsection{高斯公式} + +是考试的重点,基本上都会考到。非常重要。 + +\section{空间第二型曲线积分计算} + +是第二型曲线积分的应用。使用的是斯托克斯公式。 \end{document}