重积分

XantC

二重积分

定义

\[ \iint_{D}f(x,y)\mathrm{d}\sigma = \lim_{\lambda \to 0}\sum_{i=1}^{n}f(\xi_i, \eta_i)\Delta \sigma_i \]

计算

直角坐标

\[ \begin{align*} \iint_{d}f(x,y)\mathrm{d}\sigma&=\int^b_a\mathrm{d}x\int^{\varphi_1(x)}_{\varphi_2(x)}f(x,y)\mathrm{d}y\\ &=\int^d_c\mathrm{d}y\int^{\psi_1(y)}_{\psi_2(y)}f(x,y)\mathrm{d}x \end{align*} \]

极坐标

\[ \rho_1\leq\rho\leq\rho_2,\;\alpha\leq\theta\leq\beta \]

\[ \begin{align*} \iint_Df(\rho\cos\theta, \rho\sin\theta)\rho\mathrm{d}\rho\mathrm{d}\theta=\int^\beta_\alpha\mathrm{d}\theta\int^{\rho_2}_{\rho_1}f(\rho\cos\theta, \rho\sin\theta)\rho\mathrm{d}\rho \end{align*} \]

求高斯函数

\[ 对于D=\{(x,y)|x^2+y^2\leq a^2\}\colon \]

\[ \iint_{D}e^{-x^2-y^2}=\iint_{D^{'}}e^{-\rho^2}\rho\mathrm{d}\rho\mathrm{d}\theta=\pi(1-e^{-a^2}) \]

\[ 令D_1=\{(x,y)|x^2+y^2\leq R^2,x\geq 0,y\geq 0\} \]

\[ D_2=\{(x,y)|x^2+y^2\leq(\sqrt{2}R)^2,x\geq0,y\geq0\} \]

\[ S=\{(x,y)|0\leq x\leq R,0\leq y\leq R\} \]

\[ \Rightarrow \iint_{D_1}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y<\iint_{S}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y<\iint_{D_2}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y \]

\[ \begin{align*} \because &\iint_{D_1}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y=\frac{\pi}{4}(1-e^{-R^2})\\ &\iint_{S}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y=\int^R_0 e^{-x^2}\mathrm{d}x\cdot\int^R_0 e^{-y^2}\mathrm{d}y=(\int^R_0 e^{-x^2}\mathrm{d}x)^2\\ &\iint_{D_2}e^{-x^2-y^2}\mathrm{d}x\mathrm{d}y=\frac{\pi}{4}(1-e^{-2R^2}) \end{align*} \]

\[ \therefore\frac{\pi}{4}(1-e^{-R^2})<(\int^R_0 e^{-x^2}\mathrm{d}x)^2<\frac{\pi}{4}(1-e^{-2R^2}) \]

\[ R\to +\infty\colon \int^{+\infty}_0e^{-x^2}=\frac{\sqrt{\pi}}{2}\]

一般换元

\[ x=x(u,v),\;y=y(u,v),\;J(u,v)=\frac{\partial(x,y)}{\partial(u,v)} \]

\[ \iint_{D}f(x,y)\mathrm{d}x\mathrm{d}y=\iint_{D^{'}} f(x(u,v),y(u,v))|J(u,v)|\mathrm{d}u\mathrm{d}v \]

证明

\[ 证明的关键是表示面积元素 \]

\[ 在u-v坐标系取一小正方形M_1^{'}M_2^{'}M_3^{'}M_4^{'},则在x-y坐标系中的对应曲变形为M_1M_2M_3M_4\]

\[ 令M_1^{'}(u,v), M_2^{'}(u+h,v),M_3^{'}(u+h,v+h),M_4^{'}(u,v+h) \]

\[ M_1\colon x_1=x(u,v), y_1=y(u,v) \]

\[ \begin{align*} M_2\colon&x_2=x(u+h,v)=x(u,v)+x_u(u,v)h+\mathcal{o}(h)\\ &y_2=y(u+h,v)=y(u,v)+y_u(u,v)h+\mathcal{o}(h) \end{align*} \]

\[ \begin{align*} M_3\colon&x_3=x(u+h,v+h)=x(u,v)+x_u(u,v)h+x_v(u,v)h+\mathcal{o}(h)\\ &y_3=y(u+h,v+h)=y(u,v)+y_u(u,v)h+y_v(u,v)h+\mathcal{o}(h) \end{align*} \]

\[ \begin{align*} M_4\colon&x_4=x(u,v+h)=x(u,v)+x_v(u,v)h+\mathcal{o}(h)\\ &y_4=y(u,v+h)=y(u,v)+y_v(u,v)h+\mathcal{o}(h) \end{align*} \]

\[ \because x_1+x_3=x_2+x_4,\;y_1+y_3=y_2+y_4 \]

\[ \therefore M_1M_2M_3M_4为平行四边形 \]

\[ \Rightarrow \Delta \sigma=\begin{vmatrix} x_2-x_1&y_2-y_1\\ x_4-x_1&y_4-y_1 \end{vmatrix}=\begin{vmatrix} x_u(u,v)h&y_u(u,v)h\\ x_v(u,v)h&y_v(u,v)h \end{vmatrix}=h^2\bigg|\frac{\partial(x,y)}{\partial(u,v)}\bigg| \]

\[ \Rightarrow \Delta\sigma=\bigg|\frac{\partial (x,y)}{\partial(u,v)}\bigg|\Delta \sigma^{'}+\mathcal{o}(\Delta \sigma^{'}) \]

\[ \therefore \iint_{D}f(x,y)\mathrm{d}\sigma=\iint_{D'}f(x(u,v),y(u,v))\bigg|\frac{\partial(x,y)}{\partial(u,v)}\bigg|\mathrm{d}u\mathrm{d}v \]

有用的定理

\[ \iint_{D}f(x,y)\mathrm{d}\sigma=\iint_{D}f_1(x)f_2(y)\mathrm{d}x\mathrm{d}y=\int^b_af_1(x)\mathrm{d}x\int^d_cf_2(y)\mathrm{d}y \]

三重积分

定义

\[ \iiint_{\Omega}f(x,y,z)\mathrm{d}V = \lim_{\lambda \to 0}\sum_{i=1}^{n}f(\xi_i, \eta_i,\zeta_i)\Delta V_i \]

计算

直角坐标

\[ \begin{align*} \iiint_{\Omega}f(x,y,z)\mathrm{d}V&=\int^{x_1}_{x_2}\mathrm{d}x\int^{y_1(x)}_{y_2(x)}\mathrm{d}y\int^{z_1(x,y)}_{z_2(x,y)}f(x,y,z)\mathrm{d}z\\ &=\int^{z_1}_{z_2}\mathrm{d}z\iint_{D_z}f(x,y,z)\mathrm{d}x\mathrm{d}y \end{align*} \]

柱面坐标

\[ \iiint_{\Omega}f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z=\iiint_{\Omega}f(\rho\cos\theta, \rho\sin\theta, z)\rho\mathrm{d}\rho\mathrm{d}\theta\mathrm{d}z \]

球面坐标

\[ \iiint_{\Omega}f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z=\iiint_{\Omega}f(r\sin\varphi\cos\theta, r\sin\varphi\sin\theta, r\cos\varphi)r^2\sin\varphi\mathrm{d}r\mathrm{d}\varphi\mathrm{d}\theta \]

有用的定理

\[ \iiint_{\Omega}f_1(x)f_2(y)f_3(z)\mathrm{d}x\mathrm{d}y\mathrm{d}z=\int^b_af_1(x)\mathrm{d}x\int^d_cf_2(y)\mathrm{d}y\int^f_ef_3(z)\mathrm{d}z \]

重积分应用

曲面面积

\[ \begin{align*} A&=\iint_{D_{xy}}\sqrt{1+(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2}\mathrm{d}x\mathrm{d}y\\ &=\iint_{D_{xz}}\sqrt{1+(\frac{\partial y}{\partial x})^2+(\frac{\partial y}{\partial z})^2}\mathrm{d}x\mathrm{d}z\\ &=\iint_{D_{yz}}\sqrt{1+(\frac{\partial x}{\partial y})^2+(\frac{\partial x}{\partial z})^2}\mathrm{d}y\mathrm{d}z \end{align*} \]

质心

\[ 一般情况\colon\bar{x}=\frac{\iint_{D}x\mu(x,y)\mathrm{d}\sigma}{\iint_{D}\mu(x,y)\mathrm{d}\sigma},\;\bar{y}=\frac{\iint_{D}y\mu(x,y)\mathrm{d}\sigma}{\iint_{D}\mu(x,y)\mathrm{d}\sigma} \]

\[ 质量均匀\colon \bar{x}=\frac{1}{A}\iint_{D}x\mathrm{d}\sigma,\;\bar{y}=\frac{1}{A}\iint_{D}y\mathrm{d}\sigma \]

万有引力

\[ 三维\colon\begin{align*} \boldsymbol{F}&=(F_x,F_y,F_z)\\ &=(\iiint_{\Omega}G\frac{m\rho(x_0,y_0,z_0)(x-x_0)}{r^3}\mathrm{d}V,\iiint_{\Omega}G\frac{m\rho(x_0,y_0,z_0)(y-y_0)}{r^3}\mathrm{d}V,\iiint_{\Omega}G\frac{m\rho(x_0,y_0,z_0)(z-z_0)}{r^3}\mathrm{d}V) \end{align*} \]

\[ 二维\colon\begin{align*} \boldsymbol{F}&=(F_x, F_y, F_z)\\ &=(\iint_{D}G\frac{m\mu(x_0,y_0)(x-x_0)}{r^3}\mathrm{d}\sigma, \iint_{D}G\frac{m\mu(x_0,y_0)(y-y_0)}{r^3}\mathrm{d}\sigma, \iint_{D}G\frac{m\mu(x_0,y_0)(z-z_0)}{r^3}\mathrm{d}\sigma) \end{align*} \]