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a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex index f153f91..dba5ff3 100644 --- a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex +++ b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -22,6 +22,8 @@ % 数学公式 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 +\usepackage{tikz} +% 绘图 \author{Didnelpsun} \title{多元函数积分学} \date{} @@ -48,6 +50,45 @@ \subsubsection{极坐标系} +\textbf{例题:}对$\int_{-\frac{\pi}{4}}^{\frac{\pi}{2}}\textrm{d}\theta\int_0^{2\cos\theta}f(r\cos\theta,r\sin\theta)r\,\textrm{d}r$交换积分次序。 + +解:对于极坐标的积分次序交换需要利用直角坐标系来画图了解,特别是对于$r$的上下限。 + +对$\theta=\dfrac{\pi}{2}$变为$y$轴,$y=-\dfrac{\pi}{4}$变为$y=-x$。 + +对$r=2\cos\theta$变为$xy$的表达式,$r^2=2\cos\theta$,即$x^2+y^2=2x$,$(x-1)^2+y^2=1$。 + +\begin{minipage}{0.625\linewidth} + 所以所得到的$\sigma$为一个圆割去一个扇形。 + + 交换积分次序后就需要以一个长度以极点为圆心做圆,切割$\sigma$。 + + 由$\sigma$可知取长度$\sqrt{2}$可以切分。 +\end{minipage} +\hfill +\begin{minipage}{0.25\linewidth} + \begin{tikzpicture}[scale=1] + \draw[-latex](-0.25,0) -- (2.5,0) node[below]{$x$}; + \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$}; + \filldraw[black] (0,0) node[below]{$O$}; + \draw (1,-1) arc (-90:180:1); + \draw (0,0) -- (1,-1); + \draw (0,-1.4) arc (-90:90:1.4); + \end{tikzpicture} +\end{minipage} + +所以$\sigma$可以分为左边的$\sigma_1$和右边的$\sigma_2$。 + +$\sigma_1$的$r\in[0,\sqrt{2}]$,$\sigma_2$的$r\in[\sqrt{2},2]$。 + +$\sigma_1$的$\theta$下限是$y=-x$这条边,即$\theta=-\dfrac{\pi}{4}$,上限是$r=2\cos\theta$这个圆,则$\theta=\arccos\dfrac{r}{2}$。 + +$\sigma_2$的$\theta$界限都是是$r=2\cos\theta$这个圆,但是上限是上半部分,此时$y>0$,而下限是下半部分,此时$y<0$,即上限$r\cos$,所以下限为$\theta=-\arccos\dfrac{r}{2}$。 + +综上交换积分次序结果为: + +$\int_0^{\sqrt{2}}r\,\textrm{d}r\int_{-\frac{\pi}{4}}^{\arccos\frac{r}{2}}f(r\cos\theta,r\sin\theta)\textrm{d}\theta+\int_{\sqrt{2}}^2r\,\textrm{d}r\int_{-\arccos\frac{r}{2}}^{\arccos\frac{r}{2}}f(r\cos\theta,r\sin\theta)\textrm{d}\theta$。 + \subsection{极直互化} \textbf{例题:}将$I=\int_0^{\frac{\sqrt{2}}{2}R}e^{-y^2}\textrm{d}y\int_0^ye^{-x^2}\,\textrm{d}x+\int_{\frac{\sqrt{2}}{2}R}^Re^{-y^2}\,\textrm{d}y\int_0^{\sqrt{R^2-y^2}}e^{-x^2}\,\textrm{d}x$转换为极坐标系并计算结果。 @@ -60,4 +101,12 @@ $\therefore I=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\textrm{d}\theta\int_0^Re^{-r^2}r\,\textrm{d}r$。 +\subsection{累次积分计算} + +二重积分若是累次积分形式出现,则计算可以使用上面两种方法简便运算。 + +\subsubsection{交换积分次序} + +\textbf{例题:} + \end{document}