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zOk|^C|FJ|ZxEv(muf}>)19F$DN;zE^G)wIFH~WIyj78$XGYS(oi&gia;)SW2g+Fs= z?f(F9vmpZiLA?j#r=F2!Oo|O+v*`EDILz)`FZ-CA=~6vtj=G@DBz}lbOOe>_^hYpK zskpN5F|o)Ofdbgch_}N%E}%rq$14NR*qocg(VpW=(o_BTLURrph)%xFw>7n`*Idz@ z`t#Moe#R0%Cf7ZR4`V@0pd@WWNx)R1VjWQTDjss)%2sO{1MNIjl2){9_-&1R1)Xt9 zo9!Ih+gmnY5Q9Q%xQ<~f13I8^R6`j%la?aE>mE?|lR>#&#eoaBOf)ek2;SM*>eeDK zIY f(nLuawX4f94m|vbN{25XA|fG-$I7atrHuDK0$且$L_0$不超过$D$,$L_0$为逆时针。中间的环为$D_1$,最里侧的圆为$D_2$。 + +所以对中间的环$D_1$使用格林公式:$\oint_{L+L_0^-}=\iint_{D_1}\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\textrm{d}\sigma=0$。 + +$\therefore\oint\limits_L+\oint\limits_{L_0^-}=\oint\limits_L-\oint\limits_{L_0}=0$,$\oint_L=\oint_{L_0}$。 + +$I=\displaystyle{\oint\limits_{L_0}\dfrac{x\textrm{d}y-y\textrm{d}x}{x^2+y^2}=\dfrac{1}{r^2}\oint\limits_{L_0}x\,\textrm{d}y-y\,\textrm{d}x=\dfrac{2}{r^2}\iint\limits_{D_2}\textrm{d}\sigma=2\pi}$。 + +\textbf{例题:}计算曲线积分$\oint\limits_L\dfrac{x\textrm{d}y-y\textrm{d}x}{4^x2+y^2}$,其中$L$是以点$(1,0)$为圆心,$R\geqslant1$为半径的圆,取逆时针方向。 + +解:由于是逆时针在$L$上,所以是正向:$=\displaystyle{\oint\limits_{L^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}$。 + +又对于$L$所围成的圆面$D$,因为$4x^2+y^2\neq0$,所以$(0,0)$应该被挖去。 + +因为逆时针的方向下挖去这个点做的运动顺时针是负方向的,所以令其为$C^-$。 + +又因为格林公式$\oint\limits_{L^++C^-}P\,\textrm{d}x+Q\,\textrm{d}y=\displaystyle{\iint\limits_D\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\textrm{d}\sigma}=$\\$\displaystyle{\iint\limits_D\left(\dfrac{4x^2+y^2-3x^2}{(4x^2+y^2)^2}-\dfrac{-(4x^2+y^2)+2y^2}{(4x^2+y^2)^2}\right)\textrm{d}\sigma}=0$。旋度为0。 + +$=\oint\limits_{L^++C^-}-\oint\limits_{C^-}=0-\oint\limits_{C^-}=\oint\limits_{C^+}$。取$C:4x^2+y^2=\delta^2$,$\delta$为一个足够小的常数。(分母取$\delta^2$) + +$=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{\delta^2}\textrm{d}x+\dfrac{x}{\delta^2}\textrm{d}y\right)}$ + +$=\dfrac{1}{\delta^2}\oint\limits_{C^+}-y\,\textrm{d}x+x\,\textrm{d}y$,利用格林公式,$C^+$所成区域为$D'$:$\dfrac{1}{\delta^2}\oint\limits_{D'}(1-(-1))\,\textrm{d}\sigma=\dfrac{2}{\delta^2}D'=\dfrac{2}{\delta^2}\pi\dfrac{\delta}{2}\delta=\pi$。 + \end{document} diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf index ad1e30dc45e7fbd19d2869d5508230aa288e0b02..a5817c1e80fb60e8da5cd09dd583e7ed06a30f80 100644 GIT binary patch delta 169053 zcmY(pV{o8Nw5}c7wv&l%Yhrt1+xC-*?TPJ)ZQC{{PA0aKZ{D-dsoLMauI|;Vs=I2T zufE?m;lhg$YF!|J1C9HOkI&Tkh(ivuwG8EkwQ4empXeg2JntW21E{4hrF~HM 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b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -426,34 +426,28 @@ $\therefore I=\dfrac{1}{3}\iiint\limits_\Omega(x^2+y^2+z^2)\,\textrm{d}x\textrm{ 用于计算密度不均匀的不规则形状细线质量。 -\subsubsection{弧长} +\subsubsection{几何性质} -$L$是在$xOy$上的曲线段,$f(x,y)$在$L$上有界,将$L$分割为多个线段$\Delta S_1,\Delta S_2,\cdots,\Delta S_n$,假如取该线段某点$\forall(\xi_i,\eta_i)\in\Delta S_i$,则该线段的质量可以近似为$\Delta m_i\approx\rho(\xi_i,\eta_i)\Delta S_i$,所以整体线的质量$m\approx\sum\limits_{i=1}^n\rho(\xi_i,\eta_i)\Delta S_i$。$\lambda=\max\{\Delta S_1,\cdots,\Delta S_n\}$,若极限$m=\lim\limits_{\delta\to0}\sum\limits_{i=1}^n\rho(\xi_i,\eta_i)\Delta S_i$存在,称该极限为$f(x,y)$在$L$上对弧长的曲线积分$\int_Lf(x,y)\,\textrm{d}S$。 +$L$是在$xOy$上的曲线段,$f(x,y)$在$L$上有界,将$L$分割为多个线段$\Delta S_1,\Delta S_2,\cdots,\Delta S_n$,假如取该线段某点$\forall(\xi_i,\eta_i)\in\Delta S_i$,则该线段的质量可以近似为$\Delta m_i\approx\rho(\xi_i,\eta_i)\Delta S_i$,所以整体线的质量$m\approx\sum\limits_{i=1}^n\rho(\xi_i,\eta_i)\Delta S_i$。$\lambda=\max\{\Delta S_1,\cdots,\Delta S_n\}$,若极限$m=\lim\limits_{\delta\to0}\sum\limits_{i=1}^n\rho(\xi_i,\eta_i)\Delta S_i$存在,称该极限为$f(x,y)$在$L$上对弧长的曲线积分$\int_Lf(x,y)\,\textrm{d}s$。 \subsubsection{定义} -\begin{itemize} - \item $L:y=g(x)$($a\leqslant x\leqslant b$),$\int_Lf(x,y)\,\textrm{d}S=\int_a^bf[x,g(x)]\sqrt{1+g'^2(x)}\textrm{d}x$。 - \item $L:\left\{\begin{array}{c} - x=\phi(t) \\ - y=\psi(t) - \end{array}\right.$($\alpha\leqslant t\leqslant\beta$),$\int_Lf(x,y)\,\textrm{d}S=\int_\alpha^\beta f[\phi(t),\psi(t)]\\\sqrt{\phi'^2(t)+\psi'^2(t)}\textrm{d}t$。 -\end{itemize} +第一类曲线积分为$\int_Lf(x,y)\,\textrm{d}s$。 \subsubsection{性质} \begin{itemize} - \item $\int_L1\,\textrm{d}S=l$。 - \item $\int_Lf(x,y)\,\textrm{d}S=\int_{L_1}f(x,y)\,\textrm{d}S+\int_{L_2}f(x,y)\,\textrm{d}S$。 - \item $\int_L(k_1f(x,y))\textrm{d}S\pm\int_L(k_2f(x,y))\textrm{d}S=k_1\int_Lf(x,y)\,\textrm{d}S\pm k_2\int_Lf(x,y)\textrm{d}S$。 + \item $\int_L1\,\textrm{d}s=l$。 + \item $\int_Lf(x,y)\,\textrm{d}s=\int_{L_1}f(x,y)\,\textrm{d}s+\int_{L_2}f(x,y)\,\textrm{d}s$。 + \item $\int_L(k_1f(x,y))\textrm{d}s\pm\int_L(k_2f(x,y))\textrm{d}s=k_1\int_Lf(x,y)\,\textrm{d}s\pm k_2\int_Lf(x,y)\textrm{d}s$。 \end{itemize} \subsubsection{对称性} \begin{itemize} - \item $L$关于$y$轴对称,右边部分为$L_1$,若$f(-x,y)=-f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}S=0$,若$f(-x,y)=f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}S=2\int_{L_1}f(x,y)\,\textrm{d}S$。 - \item $L$关于$x$轴对称,上边部分为$L_1$,若$f(x,-y)=-f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}S=0$,若$f(x,-y)=f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}S=2\int_{L_1}f(x,y)\,\textrm{d}S$。 - \item $L$关于$y=x$对称,则$\int_Lf(x,y)\,\textrm{d}S=\int_Lf(x,y)\,\textrm{d}S$。 + \item $L$关于$y$轴对称,右边部分为$L_1$,若$f(-x,y)=-f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}s=0$,若$f(-x,y)=f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}s=2\int_{L_1}f(x,y)\,\textrm{d}s$。 + \item $L$关于$x$轴对称,上边部分为$L_1$,若$f(x,-y)=-f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}s=0$,若$f(x,-y)=f(x,y)$,则$\int_Lf(x,y)\,\textrm{d}s=2\int_{L_1}f(x,y)\,\textrm{d}s$。 + \item $L$关于$y=x$对称,则$\int_Lf(x,y)\,\textrm{d}s=\int_Lf(x,y)\,\textrm{d}s$。 \end{itemize} \subsection{计算} @@ -462,6 +456,14 @@ $L$是在$xOy$上的曲线段,$f(x,y)$在$L$上有界,将$L$分割为多个 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)。 +\begin{itemize} + \item $L:y=g(x)$($a\leqslant x\leqslant b$),$\int_Lf(x,y)\,\textrm{d}s=\int_a^bf[x,g(x)]\sqrt{1+g'^2(x)}\textrm{d}x$。 + \item $L:\left\{\begin{array}{c} + x=\phi(t) \\ + y=\psi(t) + \end{array}\right.$($\alpha\leqslant t\leqslant\beta$),$\int_Lf(x,y)\,\textrm{d}s=\int_\alpha^\beta f[\phi(t),\psi(t)]\\\sqrt{\phi'^2(t)+\psi'^2(t)}\textrm{d}t$。 +\end{itemize} + \paragraph{平面} \leavevmode \medskip \textbf{例题:}计算$\oint\limits_\Gamma\vert y\vert\,\textrm{d}s$,其中$\Gamma$为球面$x^2+y^2+z^2=2$与平面$x=y$的交线。 @@ -490,39 +492,69 @@ $=4\int_0^\frac{\pi}{2}\cos t\sqrt{2}\,\textrm{d}t=4\sqrt{2}$。 \section{第一型曲面积分} -$\textrm{d}S$为面微分。 +$\textrm{d}s$为面微分。即对面积曲面积分。 \subsection{概念} +\subsubsection{几何性质} + +若一块厚度不计的物体$\Sigma$在空间中,对$\Sigma$取面积微元$\forall\textrm{d}s\in\Sigma$,此时面积微元的面密度认定为均匀,其质量$\textrm{d}m=\rho(x,y,z)\textrm{d}s$。所以总质量$m=\iint_\Sigma\textrm{d}m=\iint_\Sigma\rho(x,y,z)\,\textrm{d}s$。 + \subsubsection{定义} +$\iint_\Sigma f(x,y,z)\,\textrm{d}s$为$f(x,y,z)$在曲面$\Sigma$上的面积的曲面积分。 + \subsubsection{性质} -\subsubsection{对称性} +\begin{itemize} + \item $\iint_\Sigma1\,\textrm{d}s=\sigma$。 + \item $\Sigma$关于$xOy$对称,上部为$\Sigma_1$,如果$f(x,y,-z)=-f(x,y,z)$,则$\iint_\Sigma=0$,如果$f(x,y,-z)=f(x,y,z)$,则$\iint_\Sigma=2\iint_{\Sigma_1}$。 +\end{itemize} \subsection{计算} +\subsubsection{二重积分法} + 还是一投二代三计算。 -\subsubsection{基础方法} +\begin{enumerate} + \item 对$\Sigma$作图。 + \item 查看图形的奇偶性和对称性,如果对称则消去对应$x$、$y$、$z$的奇数次项。如果图形关于$xOy$对称,则出现$z$的奇数次项就去掉;同理关于$yOz$存在$x$的奇数次项就去掉,$xOz$存在$y$的奇数次项就去掉。 + \item 查看是否可以将题目中的常数项式子对$I$化简。 + \item 投影,如将$\Sigma$投影到$xOy$平面,$z=\psi(x,y)$,$(x,y)\in D_{xy}$。 + \item 求$\dfrac{\partial z}{\partial x}$、$\dfrac{\partial z}{\partial y}$,$\textrm{d}s=\sqrt{1+\left(\dfrac{\partial z}{\partial x}\right)^2+\left(\dfrac{\partial z}{\partial y}\right)^2}\textrm{d}\sigma$。 + \item 代入$\displaystyle{\iint\limits_\Sigma f(x,y,z)\,\textrm{d}s=\iint\limits_{D_{xy}}f[x,y,\psi(x,y)]\sqrt{1+\left(\dfrac{\partial z}{\partial x}\right)^2+\left(\dfrac{\partial z}{\partial y}\right)^2}\textrm{d}\sigma}$。 +\end{enumerate} -即化为二重积分。 +\textbf{例题:}$I=\oiint_\Sigma(xy^2+z)\textrm{d}s$。$\Sigma:z=x^2+y^2,z\leqslant1$。 -\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:已知$\Sigma$是一个开口向$z$轴正方向的抛物面。不关于$z$轴对称,关于$x$轴和$y$轴对称。$I$中存在$x$的奇数次项$xy^2$则去掉,由于不存在$y$的奇数次项则跳过。 -解:曲面$\Sigma$是一个正八面体。又普通对称性得$\oiint\limits_\Sigma x\,\textrm{d}S=0$。 +$\therefore I=\iint_\Sigma z\,\textrm{d}s$。所以$\Sigma$对$xOy$投影最好,为一个单位圆。 -令第一卦限为$\Sigma_1$,所以根据普通对称性$\oiint\limits_\Sigma\vert y\vert\,\textrm{d}S=8\iint\limits_{\Sigma_1}y\,\textrm{d}S$ +所以投影为单位圆$D_{xy}:x^2+y^2\leqslant1$。 -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。 +$\dfrac{\partial z}{\partial x}=2x$,$\dfrac{\partial z}{\partial y}=2y$,$\textrm{d}s=\sqrt{1+4x^2+4y^2}\textrm{d}\sigma$。 -根据轮换对称性$\iint\limits_{\Sigma_1}y\,\textrm{d}S=\iint\limits_{\Sigma_1}x\,\textrm{d}S$。 +$I=\iint_{D_{xy}}(x^2+y^2)\sqrt{1+4(x^2+y^2)}\,\textrm{d}\sigma=\int_0^2\pi\textrm{d}\theta\int_0^1r^3\sqrt{1+4r^2}\,\textrm{d}r=\\\pi\int_0^1r^2\sqrt{1+4r^2}\,\textrm{d}(r^2)$。 -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。 +令$r^2=t$,$I=\pi\int_0^1t\sqrt{1+4t}\,\textrm{d}t$。令$\sqrt{1+4t}=u$,$=\pi\int_1^{\sqrt{5}}\dfrac{u^2-1}{4}u\dfrac{u}{2}\,\textrm{d}u=\dfrac{\pi}{8}\int_1^{\sqrt{5}}(u^4-u^2)\textrm{d}u=\dfrac{\pi}{8}(\dfrac{u^5}{5}-\dfrac{u^3}{3})\bigg\vert_1^{\sqrt{5}}=\dfrac{10}{3}\sqrt{5}+\dfrac{2}{15}$。 -根据轮换对称性$\iint\limits_{\Sigma_1}y\,\textrm{d}S=\iint\limits_{\Sigma_1}x\,\textrm{d}S=\iint\limits_{\Sigma_1}z\,\textrm{d}S$。 +\textbf{例题:}设曲面$\Sigma:\vert x\vert+\vert y\vert+\vert z\vert=1$,求$\oiint_\Sigma(x+\vert y\vert)\,\textrm{d}s$。 -$\therefore8\iint\limits_{\Sigma_1}y\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sigma_1}(x+y+z)\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sigma_1}\textrm{d}S=\dfrac{8}{3}S_{\Sigma_1}=\dfrac{8}{3}\dfrac{\sqrt{3}}{2}=\dfrac{4}{3}\sqrt{3}$。 +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} @@ -602,15 +634,15 @@ $\therefore8\iint\limits_{\Sigma_1}y\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sig 数量场就是对应数量没有方向。向量场就是有数量也有方向。 -\subsubsection{变力沿曲线做功} +\subsubsection{几何性质} 对于双理想状态,对一个物体沿直线且均匀力道,则其功为$\vec{F}\cdot\overrightarrow{AB}$($\vec{F}$为力向量,$\overrightarrow{AB}$为物体移动向量)。 而对于双不理想状态,对一个物体沿曲线且变动力道做功,则无法得出结论。 -令曲线为$L$,对$L$进行切分为微元$\forall\overrightarrow{\textrm{d}S}\in L$,则$\overrightarrow{\textrm{d}S}=\{\textrm{d}x,\textrm{d}y\}$。设变力$\vec{F}(x,y)=\{P(x,y),Q(x,y)\}$,则功的微元为$\textrm{d}\omega=\vec{F}\cdot\overrightarrow{\textrm{d}S}=P(x,y)\textrm{d}x+Q(x,y)\textrm{d}y$,所以对整体功进行积分$\omega=\int_L\textrm{d}\omega=\int_LP(x,y)\textrm{d}x+Q(x,y)\textrm{d}y$。 +令曲线为$L$,对$L$进行切分为微元$\forall\overrightarrow{\textrm{d}s}\in L$,则$\overrightarrow{\textrm{d}s}=\{\textrm{d}x,\textrm{d}y\}$。设变力$\vec{F}(x,y)=\{P(x,y),Q(x,y)\}$,则功的微元为$\textrm{d}\omega=\vec{F}\cdot\overrightarrow{\textrm{d}s}=P(x,y)\textrm{d}x+Q(x,y)\textrm{d}y$,所以对整体功进行积分$\omega=\int_L\textrm{d}\omega=\int_LP(x,y)\textrm{d}x+Q(x,y)\textrm{d}y$。 -令曲线为$L$,对$L$进行切分为微元$\forall\overrightarrow{\textrm{d}S}\in L$,则$\overrightarrow{\textrm{d}S}=\{\textrm{d}x,\textrm{d}y,\textrm{d}z\}$。设变力$\vec{F}(x,y,z)=\{P(x,y,z),Q(x,y,z),R(x,y,z)\}$,则功的微元为$\textrm{d}\omega=\vec{F}\cdot\overrightarrow{\textrm{d}S}=P(x,y,z)\textrm{d}x+Q(x,y,z)\textrm{d}y+R(x,y,z)\textrm{d}z$,所以对整体功进行积分$\omega=\int_L\textrm{d}\omega=\int_LP(x,y,z)\textrm{d}x+Q(x,y,z)\textrm{d}y+R(x,y,z)\textrm{d}z$。 +令曲线为$L$,对$L$进行切分为微元$\forall\overrightarrow{\textrm{d}s}\in L$,则$\overrightarrow{\textrm{d}s}=\{\textrm{d}x,\textrm{d}y,\textrm{d}z\}$。设变力$\vec{F}(x,y,z)=\{P(x,y,z),Q(x,y,z),R(x,y,z)\}$,则功的微元为$\textrm{d}\omega=\vec{F}\cdot\overrightarrow{\textrm{d}s}=P(x,y,z)\textrm{d}x+Q(x,y,z)\textrm{d}y+R(x,y,z)\textrm{d}z$,所以对整体功进行积分$\omega=\int_L\textrm{d}\omega=\int_LP(x,y,z)\textrm{d}x+Q(x,y,z)\textrm{d}y+R(x,y,z)\textrm{d}z$。 \subsubsection{定义} @@ -622,56 +654,158 @@ $\therefore8\iint\limits_{\Sigma_1}y\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sig \begin{itemize} \item $\int_{L^-}P(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=-\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y$。 - \item $\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_L(P(x,y)\cos\alpha+Q(x,y)\cos\beta)\textrm{d}S$。(利用方向余弦将第二类曲线积分化成第一类曲线积分) + \item $\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_L(P(x,y)\cos\alpha+Q(x,y)\cos\beta)\textrm{d}s$。(利用方向余弦将第二类曲线积分化成第一类曲线积分) \end{itemize} \subsection{计算} -\subsubsection{基础方法} +对于不封闭曲线使用定积分法,对于封闭曲线或封闭曲线部分使用二重积分法。 -即化为定积分。 +\subsubsection{定积分法} -\subsubsection{格林公式} +由于第二类曲线积分积分值可正可负,所以只需要关心其起点$x=a$,其终点$x=b$。 -是考试的重点,基本上都会考到。非常重要。 +\begin{itemize} + \item $L:y=g(x)$($x$起于$x=a$终于$x=b$),$\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_a^bP[x,g(x)]\,\textrm{d}x+Q[x,g(x)]g'(x)\,\textrm{d}x$。 + \item $L:\left\{\begin{array}{c} + x=\phi(t) \\ + y=\psi(t) + \end{array}\right.$($t$起于$\alpha$终于$\beta$),$\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\\\int_\alpha^\beta P[\phi(t),\psi(t)]\phi'(t)\,\textrm{d}t+Q[\phi(t),\psi(t)]\psi'(t)\textrm{d}t$。 +\end{itemize} -\textbf{例题:}计算曲线积分$\oint\limits_L\dfrac{x\textrm{d}y-y\textrm{d}x}{4^x2+y^2}$,其中$L$是以点$(1,0)$为圆心,$R\geqslant1$为半径的圆,取逆时针方向。 +\subsubsection{二重积分法} -解:由于是逆时针在$L$上,所以是正向:$=\displaystyle{\oint\limits_{L^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}$。 +是考试的重点,基本上都会考到,非常重要。 -又对于$L$所围成的圆面$D$,因为$4x^2+y^2\neq0$,所以$(0,0)$应该被挖去。 +\paragraph{背景} \leavevmode \medskip -因为逆时针的方向下挖去这个点做的运动顺时针是负方向的,所以令其为$C^-$。 +复平面上的一个区域$G$,如果在其中任做一条简单闭曲线,而闭曲线的内部总属于$G$,就称$G$为单连通区域。一个区域如果不是单连通区域,就称为多连通区域。即形象地说一块完整的凸图形即单连通,中间存在洞的或凹图形为多连通。 -又因为格林公式$\oint\limits_{L^++C^-}P\,\textrm{d}x+Q\,\textrm{d}y=\displaystyle{\iint\limits_D\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\textrm{d}\sigma}=$\\$\displaystyle{\iint\limits_D\left(\dfrac{4x^2+y^2-3x^2}{(4x^2+y^2)^2}-\dfrac{-(4x^2+y^2)+2y^2}{(4x^2+y^2)^2}\right)\textrm{d}\sigma}=0$。旋度为0。 +对于单连通区域,边界的正方向为逆时针方向,对于多连通区域,外边界的正方向是逆时针,内边界的正方向是顺时针。 -$=\oint\limits_{L^++C^-}-\oint\limits_{C^-}=0-\oint\limits_{C^-}=\oint\limits_{C^+}$。取$C:4x^2+y^2=\delta^2$,$\delta$为一个足够小的常数。(分母取$\delta^2$) +在定积分中,使用牛顿-莱布尼茨公式,将$\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)$,即将代表$[a,b]$范围积分域的积分值变为代表两点边界差值的函数值。在二重积分中,积分区域为一个连通域,其面积的积分就是二重积分,其周长的积分就是曲线积分,那么是否有这么一种方法,将域的关系转换为边界的关系,即是否存在一种方法将二重积分和曲线积分联系起来。 -$=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{\delta^2}\textrm{d}x+\dfrac{x}{\delta^2}\textrm{d}y\right)}$ +所以这个方法就是将曲线积分转换为二重积分,这个工具就是格林公式。 -$=\dfrac{1}{\delta^2}\oint\limits_{C^+}-y\,\textrm{d}x+x\,\textrm{d}y$,利用格林公式,$C^+$所成区域为$D'$:$\dfrac{1}{\delta^2}\oint\limits_{D'}(1-(-1))\,\textrm{d}\sigma=\dfrac{2}{\delta^2}D'=\dfrac{2}{\delta^2}\pi\dfrac{\delta}{2}\delta=\pi$。 +\paragraph{格林公式} \leavevmode \medskip +条件: + +\begin{itemize} + \item $D$为连通区域,$L$为$D$的正向边界。(如果$D$不是个完整的区域则添加线段,如果$L$为反向边界则添加负号) + \item $P(x,y)$、$Q(x,y)$在$D$上连续可偏导。 +\end{itemize} + +$\displaystyle{\oint\limits_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\iint\limits_D\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\textrm{d}\sigma}$。 + +\subsubsection{路径无关积分} + +对于第二类曲线积分$\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y$,其积分值可能与路径$L$相关,也可能与路径$L$无关。 + +若$D$为单连通区域,$P(x,y)$、$Q(x,y)$在$D$上连续可偏导,以下命题等价: + +\begin{enumerate} + \item $\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y$与路径无关。 + \item $\forall C\in D$($C$为闭区域),$\oint_CP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=0$。 + \item 柯西黎曼条件:$\dfrac{\partial Q}{\partial x}\equiv\dfrac{\partial P}{\partial y}$。(最合适) + \item $\exists u(x,y)$,使得$\textrm{d}u=P(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y$。即$\dfrac{\partial u}{\partial x}=P(x,y)$,$\dfrac{\partial u}{\partial y}=Q(x,y)$。 +\end{enumerate} + +则根据这些条件简化计算,若满足柯西黎曼条件: + +\begin{enumerate} + \item 求曲线积分:$\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_{(x_0,y_0)}^{(x_1,y_1)}P(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_{x_0}^{x_1}P(x,y_0)\,\textrm{d}x+\int_{y_0}^{y_1}Q(x_1,y)\,\textrm{d}y$(先水平后垂直)。 + \item 能找到$u$,求曲线积分:有$\textrm{d}u=P(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y$,则$\int_LP(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_{(x_0,y_0)}^{(x_1,y_1)}\textrm{d}u=u(x_1,y_1)-u(x_0,y_0)$。 + \item 求全微分对应$u$表达式:则$u(x,y)=\int_{(x_0,y_0)}^{(x,y)}P(x,y)\,\textrm{d}x+Q(x,y)\,\textrm{d}y=\int_{x_0}^xP(x,y_0)\,\textrm{d}x+\int_{y_0}^yQ(x_1,y)\,\textrm{d}y$(先水平后垂直)。 +\end{enumerate} + +\textbf{例题:}已知$\psi(x)$可导,$\psi(0)=2$,$\int_Lxy^2\,\textrm{d}x+\psi(x)y\,\textrm{d}y$与路径无关,求$\psi(x)$,并计算$I=\int_{(1,2)}^{(2,3)}xy^2\,\textrm{d}x+\psi(x)y\,\textrm{d}y$。 + +解:已知$P=xy^2$,$y=\psi(x)y$。由于曲线积分与路径无关,所以$\dfrac{\partial Q}{\partial x}\equiv\dfrac{\partial P}{\partial y}$,所以$\psi'(x)y\equiv2xy$,$\psi'(x)=2x$,$\psi(x)=x^2+C$,又$\psi(0)=2$,$C=2$,$\psi(x)=x^2+2$。 + +所以$I=\int_{(1,2)}^{(2,3)}xy^2\,\textrm{d}x+(x^2+2)y\,\textrm{d}y$。 + +方法一:$I=\int_1^2x\cdot2^2\,\textrm{d}x+\int_2^3(2^2+2)y\,\textrm{d}y=2x^2\vert_1^2+3y^2\vert_2^3=6+15=21$。 + +方法二:$I=\int_{(1,2)}^{(2,3)}(xy^2\,\textrm{d}x+x^2y\,\textrm{d}y)+2y\,\textrm{d}y=\int_{(1,2)}^{(2,3)}\textrm{d}(\dfrac{1}{2}x^2y^2+y^2)=(\dfrac{1}{2}x^2y^2+y^2)|_{(1,2)}^{(2,3)}=27-6=21$。 \section{第二型曲面积分} +与第二型曲线积分一样,是有方向的。即对坐标曲面积分。 + \subsection{概念} \subsubsection{向量场的通量} +令一个不可压缩的向量场$\vec{v}=\{P,Q,R\}$,向量场中有一个面$\Sigma$,有不规则方向的流体流入或流出这个面,现在要求单位时间内流入指定侧的流量。 + +取一块有侧的面$\forall\overrightarrow{\textrm{d}s}\in\Sigma$,从$x$轴正方向部分将$\overrightarrow{\textrm{d}s}$向$yOz$面的投影记为$\textrm{d}y\textrm{d}z$。 + +当$\cos\alpha\geqslant0$即方向余弦非负的情况下,$\textrm{d}y\textrm{d}z=\textrm{d}s_{yOz}$即就等于投影面积;当$\cos\alpha<0$即方向余弦为负的情况下,$\textrm{d}y\textrm{d}z=-\textrm{d}s_{yOz}$即就等于负投影面积。实际上$\textrm{d}y\textrm{d}z=\textrm{d}s\cdot\cos\alpha$。同理其他两个方向也是如此。 + +所以对$\overrightarrow{\textrm{d}s}$进行三轴投影,得到$\overrightarrow{\textrm{d}s}=\{\textrm{d}y\textrm{d}z,\textrm{d}z\textrm{d}x,\textrm{d}x\textrm{d}y\}$ + +所以$\overrightarrow{\textrm{d}s}$单位时间内流入指定侧的流量$\textrm{d}\phi=\vec{v}\cdot\overrightarrow{\textrm{d}s}=P(x,y,z)\textrm{d}y\textrm{d}z+Q(x,y,z)\textrm{d}z\textrm{d}x+R(x,y,z)\textrm{d}x\textrm{d}y$。 + +从而总流量为$\iint_\Sigma\textrm{d}\phi=\iint_\Sigma P(x,y,z)\textrm{d}y\textrm{d}z+Q(x,y,z)\textrm{d}z\textrm{d}x+R(x,y,z)\textrm{d}x\textrm{d}y$。 + \subsubsection{定义} +$\iint_\Sigma P(x,y,z)\textrm{d}y\textrm{d}z+Q(x,y,z)\textrm{d}z\textrm{d}x+R(x,y,z)\textrm{d}x\textrm{d}y$为在$\Sigma$上的曲面积分。其中$\iint_\Sigma P(x,y,z)\textrm{d}y\textrm{d}z$为$P(x,y,z)$在有侧曲面$\Sigma$上对坐标$y$、$z$的曲面积分。 + \subsubsection{性质} +\begin{itemize} + \item $\iint_{\Sigma^-}=-\iint_\Sigma$。 + \item $\iint_\Sigma P(x,y,z)\textrm{d}y\textrm{d}z+Q(x,y,z)\textrm{d}z\textrm{d}x+R(x,y,z)\textrm{d}x\textrm{d}y=\iint_\Sigma P(x,y,z)\cos\alpha+Q(x,y,z)\cos\beta+R(x,y,z)\cos\gamma$。(曲面积分转为曲线积分) + \item 若积分曲面对称:被积函数关于相应变量为奇函数,积分为半区间的2倍;若为偶函数,则积分等于0。(与一般奇偶性正好相反) +\end{itemize} + +对于奇偶性的解释:因为是第二型的曲面积分,会分前后左右上下,分别代表正负,所以被积函数为偶函数时如果是相反方向,就正好被减去了(两个积的结果相同,方向相反,可以考虑磁通量一边进,一边出),奇函数两边想减因为方向不同,所以--为正相加,即为两倍。第一型曲面积分物理意义来源于对给定密度函数的空间曲面,计算该曲面的质量。第二型曲面积分物理意义来源对于给定的空间曲面和流体的流速,计算单位时间流经曲面的总流量。 + \subsection{计算} -\subsubsection{基础方法} +\subsubsection{二重积分法} -即化为二重积分。 +曲面积分的二重积分法较复杂。将积分化为不同面的投影。为什么会带一个正负号?因为前面的$\textrm{d}x\textrm{d}y\textrm{d}z$表示投影,其值可正可负,而后面的$\textrm{d}x\textrm{d}y\textrm{d}z$表示面积,必然为正。 -\subsubsection{高斯公式} +\begin{itemize} + \item $\oiint_\Sigma P(x,y,z)\,\textrm{d}y\textrm{d}z$:用$yz$表示$x$:$\Sigma:x=\phi(y,z)$,$(y,z)\in D_{yz}$,\\$\iint_\Sigma P(x,y,z)\,\textrm{d}y\textrm{d}z=\pm\iint_{D_{yz}}P[\phi(y,z),y,z]\textrm{d}y\textrm{d}z$。前正后负。 + \item $\oiint_\Sigma Q(x,y,z)\,\textrm{d}z\textrm{d}x$:用$xz$表示$y$:$\Sigma:y=\phi(z,x)$,$(z,x)\in D_{zx}$,\\$\iint_\Sigma Q(x,y,z)\,\textrm{d}z\textrm{d}x=\pm\iint_{D_{zx}}Q[x,\phi(z,x),z]\textrm{d}z\textrm{d}x$。右正左负。 + \item $\oiint_\Sigma R(x,y,z)\,\textrm{d}x\textrm{d}y$:用$xy$表示$z$:$\Sigma:z=\phi(x,y)$,$(x,y)\in D_{xy}$,\\$\iint_\Sigma R(x,y,z)\,\textrm{d}x\textrm{d}y=\pm\iint_{D_{xy}}R[x,y,\phi(x,y)]\textrm{d}x\textrm{d}y$。上正下负。 +\end{itemize} + +\subsubsection{三重积分法} 是考试的重点,基本上都会考到。非常重要。 +\paragraph{背景} \leavevmode \medskip + +牛顿莱布尼兹公式将定积分转换为函数值差,格林公式将曲线积分转换为二重积分,高斯公式就是将曲面积分转换为三重积分。 + +\paragraph{高斯公式} \leavevmode \medskip + +条件: + +\begin{itemize} + \item $\Omega$为几何体,$\Sigma$为$\Omega$的外表面。(如果不封闭则补全,如果是内表面就添加负号) + \item $P(x,y,z)$、$Q(x,y,z)$、$R(x,y,z)$在$\Omega$上连续可偏导。 +\end{itemize} + +$\displaystyle{\oiint_\Sigma P\,\textrm{d}y\textrm{d}z+Q\,\textrm{d}z\textrm{d}x+R\,\textrm{d}x\textrm{d}y=\iiint\limits_\Omega\left(\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}\right)\textrm{d}v}$。 + +\textbf{例题:}$I=\iint_\Sigma yz\,\textrm{d}z\textrm{d}x+2\,\textrm{d}x\textrm{d}y$,其中$\Sigma$为$x^2+y^2+z^2=4$的$xOy$面的上半部分。 + +解:由$I$可得$P=0$,$Q=yz$,$R=2$,$\dfrac{\partial P}{\partial x}=0$,$\dfrac{\partial Q}{\partial y}=z$,$\dfrac{\partial R}{\partial z}=0$。 + +由于$\Sigma$只有球面的上半部分,所以需要补充底面$\Sigma_0:y=0$($x^2+y^2\leqslant4$下侧),此时才是一个半球体的完整封闭表面积,$\Sigma_0$的法向量为$z$轴的反方向。所以$I=\oiint_{\Sigma+\Sigma_0}-\iint_{\Sigma_0}$。 + +又根据高斯公式$\oiint_{\Sigma+\Sigma_0}=\iiint_\Omega z\,\textrm{d}v=\int_0^{2\pi}\textrm{d}\theta\int_0^{\frac{\pi}{2}}\textrm{d}\psi\int_0^2r\cos\psi\cdot r^2\sin\psi\,\textrm{d}r=2\pi\int_0^{\frac{\pi}{2}}\sin\psi\,\textrm{d}(\sin\psi)\int_0^2r^3\,\textrm{d}r=2\pi\times\dfrac{1}{2}\sin^2\psi\vert_0^{\frac{\pi}{2}}\times\dfrac{r^4}{4}\vert_0^2=4\pi$。 + +而底面$\iint_{\Sigma_0}yz\,\textrm{d}z\textrm{d}x+2\,\textrm{d}x\textrm{d}y=\iint_{\Sigma_0}2\,\textrm{d}x\textrm{d}y=-2\iint_{D_{xy}}\textrm{d}x\textrm{d}y=-8\pi$。 + +$I=4\pi+8\pi=12\pi$。 + \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。