三重积分

多变量函数的积分学

张瑞
中国科学技术大学数学科学学院

三重积分

三重积分的概念

在空间$\mathbb{R}^3$中的一个有界几何体$V$上分布着某种物质，已知这种物质的密度函数为$\rho(x,y,z)$，求这种物质在 $V$ 上的总质量!

\begin{tikzpicture}[small region/.style={shift={(0.5,0.4)}, dashed, blue}] % 画3维中的有界区域 \draw[] (0,0) .. controls (0,2.4) and (2,2.4) .. (2,0); \draw[] (0,0) .. controls (0,-2.26) and (2,-2.26) .. (2,0); \draw[] (0,0) .. controls (0.4,-0.23) and (1.6,-0.23) .. (2,0); \draw[dashed] (0,0) .. controls (0.4,0.23) and (1.6,0.23) .. (2,0); % 画区域中的小块 \draw[small region] (0,0)--(0.4,0)--(0.4,0.42)--(0,0.42)--cycle; \draw[small region] (0,0.42) -- ++(0.1,0.1)--++(0.4,0)--++(-0.1,-0.1); \draw[small region] (0.4,0) -- ++(0.1,0.1)--++(0,0.42); \node at (0.5,0.4) [anchor=south west, font=\tiny] {$V_i$}; \end{tikzpicture}

将几何体$V$分成互不重叠的小几何体$V_1$,$\cdots$,$V_n$，它们的体积为$\Delta V_1$, $\cdots$, $\Delta V_n$，其上的质量近似为$\rho(M_i)\Delta V_i$，其中$M_i$为$V_i$内任意一点。则总质量就是

\[\displaystyle\sum_{i=1}^n\rho(M_i)\Delta V_i \]

的极限。

定义 1.
设$f(x,y,z)$是定义在有界集$V$上的有界函数。令

\[f_V(x,y,z)=\begin{cases} f(x,y,z) , & (x,y,z)\in V, \\ 0 , & (x,y,z)\in V^c \end{cases} \]

取三维区间$R\supset V$。用平行于坐标平面的平面把区间$R$分割成若干三维区间$R_i$， $\lambda$是所有$R_i$ 直径的最大值。如果存在常数$A$，使得对于任意的分割和任意的取值$(\xi_i,\eta_i,\zeta_i)\in R_i$，都有

\[\lim_{\lambda\to0}\sum_if_V(\xi_i,\eta_i,\zeta_i)\Delta x_i\Delta y_i\Delta z_i=A \]

则称$f(x,y,z)$在$V$上可积，$A$称为$f(x,y,z)$在$V$上的积分

记为

\[\iiint\limits_Vf_V(x,y,z)dxdydx, \mbox{或} \int_V f \]
  • 三重积分的定义与二重积分类似。
  • 如果有界集$V$上的常值函数$f(x,y,z)=1$在$V$上可积，则称$V$是有体积的， 积分$\displaystyle\int_V 1$就是$V$的体积。
  • 可以证明：若$V$是由有限张光滑曲面围成的有界区域，则$V$是有体积的。

注. 今后，不作特殊说明，总假定积分域是由有限张光滑曲面围成的有界区域。

定理 1.
$V$是由有限张光滑曲面围成的有界区域，$f(x,y,z)$是$V$上的函数。

  1. 若$f(x,y,z)$在$V$上可积，则$f(x,y,z)$在$V$上有界；
  2. 若$f(x,y,z)$在$V$上有界，且$f(x,y,z)$的不连续点分布在有限张光滑曲面上，则$f(x,y,z)$在$V$上可积。
  • 三重积分与二重积分的性质完全一样。前面所列的关于二重积分的性质，对三重积分也成立。

三重积分的累次积分法

类似于二重积分的累次积分，三重积分的计算可以化为累次积分。对于三维区间$R=I_1\times I_2\times I_3$

\[\begin{aligned} \iiint\limits_V f(x,y,z)dxdydz &=\iint\limits_{I_1\times I_2}dxdy\int\limits_{I_3}f(x,y,z)dx \\ &=\int\limits_{I_1}dx\int\limits_{I_2}dy\int\limits_{I_3}f(x,y,z)dx \\ &=\int\limits_{I_1}dx\iint\limits_{I_2\times I_3}f(x,y,z)dx \\ \end{aligned} \]
  • 三重积分还有其他积分次序! 选择合适的累次积分次序是十分重要的!

定理 2.
设$V$为$\mathbb{R}^3$中的有界闭域，$f(x,y,z)$是$V$上的连续函数。

  1. (“切细条法”)设$V$在平面$Oxy$上的投影为平面区域$D$，而$V$是由曲面
    \[z=z_1(x,y), z=z_2(x,y) , z_1(x,y)\leq z_2(x,y), (x,y)\in D \]
    以及以$\partial D$为准线且平行于$z$轴的柱面围成的，则
\[\iiint_V f(x,y,z)dxdydz=\iint_D dxdy\int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z)dz \]
\begin{tikzpicture} %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,3,0) node[anchor=west]{$z$}; \draw[->] (0,0,0) -- (0,0,2) node[anchor=west]{$x$}; % 画oxy平面的区域D \fill[black!20, draw=red!50] (0.8,0,1) .. controls (1.1,0,2) and (3.0,0,2).. (2.8,0,1) .. controls (2.6,0,0) and (0.7,0,0) .. cycle; %\draw[blue] (0.8,0,1) -- (2.8,0,1); \node at (1.8,0,1) {$D$}; \draw[red] (0.8,1,1) .. controls (1.1,1,2) and (3.0,1,2).. (2.8,1,1); \draw[red, dashed] (0.8,1,1) .. controls (0.7,1,0) and (2.6,1,0).. (2.8,1,1); \draw[red] (0.8,0.91,1) .. controls (1.1,0.41,1) and (2.6,0.41,1).. (2.8,1,1); %\draw[red] (0.8,2,1) .. controls (1.1,2,2) and (3.0,2,2).. (2.8,2,1); %\draw[red, dashed] (0.8,2,1) .. controls (0.7,2,0) and (2.6,2,0).. (2.8,2,1); \draw[red] (0.8,3,1) .. controls (1.1,3,2) and (3.0,3,2).. (2.8,3,1); \draw[red, dashed] (0.8,3,1) .. controls (0.7,3,0) and (2.6,3,0).. (2.8,3,1); \draw[red] (0.8,3,1) .. controls (1.1,3.85,1) and (2.6,3.85,1).. (2.8,3,1); \draw[blue] (0.8,1,1) -- (0.8,3,1); \draw[blue] (2.8,1,1) -- (2.8,3,1); \draw[blue,dashed] (0.8,1,1) -- (0.8,0,1); \draw[blue,dashed] (2.8,1,1) -- (2.8,0,1); %\node at (1.8,3,1) [font=\tiny] {$z=z_2(x,y)$}; \draw[->] (-0.48,0.75,1) node (bfunc) [font=\tiny] {$z=z_1(x,y)$} (node cs:name=bfunc) -- (1.10,0.67,1); \draw[->] (-0.48,3,1) node (tfunc) [font=\tiny] {$z=z_2(x,y)$} (node cs:name=tfunc) -- (1.10,3,1); \draw[green] (1.1,0,1) -- (1.4,0,1) -- (1.4, 0, 0.7) -- (1.1,0,0.7) -- cycle; \draw[green] (1.1,0.68,1) -- (1.4,0.68,1) -- (1.4, 0.68, 0.7) -- (1.1,0.68,0.7) -- cycle; \draw[green] (1.1,3.3,1) -- (1.4,3.3,1) -- (1.4, 3.3, 0.7) -- (1.1,3.3,0.7) -- cycle; \draw[green,dashed] (1.1,0,1) -- (1.1,3.3,1); \draw[green,dashed] (1.4,0,1) -- (1.4,3.3,1); \draw[green,dashed] (1.4, 0, 0.7) -- (1.4, 3.3, 0.7); \draw[green,dashed] (1.1,0,0.7) -- (1.1,3.3,0.7); \end{tikzpicture}
\begin{tikzpicture} [top sur/.style={shift={(0.3,0,0)}}, mid sur/.style={shift={(0.5,0,0)}}, global scale=0.8 ] %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,3.5,0) node[anchor=west]{$z$}; \draw[->] (0,0,0) -- (0,0,2) node[anchor=west]{$x$}; \draw[red] (0.8,1,1) .. controls (1.1,1,2) and (3.0,1,2).. (2.8,1,1); \draw[red, dashed] (0.8,1,1) .. controls (0.7,1,0) and (2.4,1,0).. (2.8,1,1); \draw[red] (0.8,0.91,1) .. controls (1.1,0.41,1) and (2.6,0.41,1).. (2.8,1,1); \draw[red,top sur] (0.8,3,1) .. controls (1.1,3,2) and (3.0,3,2).. (2.8,3,1); \draw[red,top sur, dashed] (0.8,3,1) .. controls (0.7,3,0) and (2.4,3,0).. (2.8,3,1); \draw[red,top sur] (0.8,3,1) .. controls (1.1,3.5,1) and (2.6,3.5,1).. (2.8,3,1); \draw[green,mid sur] (0.8,2,1) .. controls (1.1,2,2) and (3.0,2,2).. (2.8,2,1); \draw[green,mid sur, dashed] (0.8,2,1) .. controls (0.7,2,0) and (2.4,2,0).. (2.8,2,1); \draw[green] (-1,1.5,1) -- (3,1.5,1) -- (3,1.5,-1.5)--(-1,1.5,-1.5) --cycle; \draw[green, dashed] (1.3,2,1) --(0,1.65,0) node [anchor=east] {$z$}; \node at (1.5,2,1) [anchor=west] {$D_z$} ; \draw[blue] plot[smooth] coordinates { (0.8,1,1) (1.3,2,1) (1.1, 3,1)}; \draw[blue] plot[smooth] coordinates { (2.8,1,1) (3.3,2,1) (3.1,3,1)}; \draw[blue,dashed] (0,0.15,0) node [anchor=east] {$a$} -- (1.8,0.15,0); \draw[blue,dashed] (0,3.0,0) node [anchor=east] {$b$} -- (1.8,3.0,0); \end{tikzpicture}
  1. (“切薄片法”)设$V$在$z$轴上投影为区间$I$，过$I$上一点$(0,0,z)$与$z$轴垂直的平面与$V$相交的平面图形在平面$Oxy$上的投影为区域$D_z$，则
    \[\iiint_V f(x,y,z)dxdydz=\int_I dz\iint_{D_z}f(x,y,z)dxdy \]

$V$由

\[x_1\leq x\leq x_2, y_1(x)\leq y \leq y_2(x), z_1(x,y)\leq z\leq z_2(x,y) \]

则

\[\begin{aligned} \iiint\limits_V f(x,y,z)dxdydz &=\int_{x_1}^{x_2}dx\int_{y_1(x)}^{y_2(x)}dy\int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z)dx \\ &=\int_{x_1}^{x_2}dx\iint\limits_{S(x)}f(x,y,z)dydz \end{aligned} \]

其中$S(X)$是平面$x=X$截$V$得到的截断面

例 1. $V$由$x+y+z=1$，$x=0$, $y=0$, $z=0$围成

\[\iiint\limits_Vf(x,y,z)dxdydz \]

例 2. $V$由$x^2+y^2+z^2=1$，$x\geq 0$, $y\geq 0$, $z\geq 0$围成

\[\iiint\limits_V xyzdxdydz \]

例 3. $V$由$x^2+y^2=z^2$，$z=1$围成

\[\iiint\limits_V \sqrt{x^2+y^2} dxdydz \]

int3d-ex-tetrahedron int3d-ex-tetrahedron-ii

chap7-2-ex2

chap7-2-ex3

变量代换

设变换

\[\vec r: x=x(u,v,w), y=y(u,v,w), z=z(u,v,w) \]
\begin{tikzpicture} [x=2cm, y=2cm, global scale=0.5] \coordinate (ox) at (-125:1); \coordinate (oy) at (-5:1); \coordinate (oz) at (90:1); \draw[->, ] (0,0) -- ($ 2*(oy) $) node[right] {$v$}; \draw[->, ] (0,0) -- ($ 1.5*(ox) $) node[below] {$u$}; \draw[->, ] (0,0) -- ($ 2*(oz) $) node[above] {$w$}; \coordinate (P) at (-65:0.75); \coordinate (Q) at ($ (P)+1.5*(oz) $); \draw[ thick] (P)-- node[left]{$dw$} (Q); \draw[ thick] (P) -- node[below]{$dv$} ++ (oy) -- node[right]{$du$} ++ ($ -0.5*(ox) $) -- ++ ($ 1.5*(oz) $) -- ++ ($ -1*(oy) $) -- ++ ($ 0.5*(ox) $) -- ++ ($ 1*(oy) $) -- ++ ($ -0.5*(ox) $); \draw[ thick] ($ (P)+(oy) $) --($ (Q)+(oy) $); \draw[ thick, dashed] (P) -- ++ ($ -0.5*(ox) $) -- + ($ 1.5*(oz) $) ($ (P)-0.5*(ox) $) -- +(oy); \begin{scope}[xshift=6cm] \coordinate (ox) at (-125:1); \coordinate (oy) at (-5:1); \coordinate (oz) at (90:1); \draw[->, ] (0,0) -- ($ 2*(oy) $) node[right] {$y$}; \draw[->, ] (0,0) -- ($ 1.5*(ox) $) node[below] {$x$}; \draw[->, ] (0,0) -- ($ 2*(oz) $) node[above] {$z$}; \coordinate (P) at (-65:0.75); \draw[-latex, red] (P) -- + (40:0.6) node[above ] {$\vec r'_u du$}; \draw[-latex, red] (P) -- + ($ 0.95*(oy) $) node[right] {$\vec r'_v dv$}; \draw[-latex, red] (P) -- + ($ 1.53*(oz) $) node[above] {$\vec r'_w dw$}; \coordinate (Q) at ($ (P)+(80:1.75) $); \coordinate (Q1) at ($ (Q)+(-10:1) $); \coordinate (Q2) at ($ (Q1)+(20:0.7) $); \coordinate (Q3) at ($ (Q)+(40:0.6) $); \coordinate (P1) at ($ (P)+(-20:1.05) $); \coordinate (P2) at ($ (P1)+(40:0.5) $); \coordinate (P3) at ($ (P)+(30:0.7) $); \draw[ thick] (P) to[out=90, in=-90] (Q) to[out=0, in=170] (Q1) to[out=-100, in=90] (P1) to[out=160, in=0] (P); \draw[ thick] (P1) to[out=30, in=210] (P2) to[out=90, in=-100] (Q2) to[out=170, in=-20] (Q3) to[out=-110, in=50] (Q) (Q1) -- (Q2); \draw[ thick, dashed] (P2) to[out=130, in=-20] (P3) to[out=-110, in=50] (P) (Q3) -- (P3); \end{scope} % 画一个映射曲线 \draw[->, ultra thick] (2,1) to[out=30,in=150] node[above] {$\vec r(u,v,w)$} (2.5,1); \end{tikzpicture}
\[dxdydz=|(r'_udu \times r'_v dv)\cdot r'_w dw| \]

定理 3.
变换$r$

\[\begin{aligned} r:V'\to V , \begin{cases} x=x(u,v,w) \\ y=y(u,v,w) \\ z=z(u,v,w) \end{cases} \end{aligned} \]

将$O'uvw$空间中的区域$V'$一一映射为$Oxyz$空间中的区域$V$， 且$\left|{\frac{\partial (x,y,z)}{\partial (u,v,w)}}\right|\neq0$，则有

\[\iiint\limits_Vf(x,y,z)dxdydz=\iiint\limits_{V'}f(x,y,z)\left|{\frac{\partial (x,y,z)}{\partial (u,v,w)}}\right| dudvdw \]

球坐标变换

\[\begin{cases} x=r\sin\theta\cos\phi \\ y=r\sin\theta\sin\phi \\ z=r\cos\theta \end{cases} \]

得到

\[\left|{\frac{\partial (x,y,z)}{\partial (r,\phi,\theta)}}\right|=r^2\sin\theta \]
\tdplotsetmaincoords{70}{120} \begin{tikzpicture}[scale=3,every node/.append style={scale=0.6}, tdplot_main_coords] %TODO: look into using 3d spherical coordinate system \pgfmathsetmacro{\rvec}{.8} \pgfmathsetmacro{\thetavec}{30} \pgfmathsetmacro{\phivec}{70} \pgfmathsetmacro{\dphivec}{20} \pgfmathsetmacro{\dthetavec}{20} \pgfmathsetmacro{\drvec}{0.2} %set up some coordinates %----------------------- \coordinate (O) at (0,0,0); %determine a coordinate (P) using (r,\theta,\phi) coordinates. This command %also determines (Pxy), (Pxz), and (Pyz): the xy-, xz-, and yz-projections %of the point (P). %syntax: \tdplotsetcoord{Coordinate name without parentheses}{r}{\theta}{\phi} \tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec} \tdplotsetcoord{P1}{\rvec}{\thetavec}{\phivec+\dphivec} \tdplotsetcoord{P2}{\rvec}{\thetavec+\dthetavec}{\phivec+\dphivec} \tdplotsetcoord{P3}{\rvec}{\thetavec+\dthetavec}{\phivec} \tdplotsetcoord{Q}{\rvec+\drvec}{\thetavec}{\phivec} \tdplotsetcoord{Q1}{\rvec+\drvec}{\thetavec}{\phivec+\dphivec} \tdplotsetcoord{Q2}{\rvec+\drvec}{\thetavec+\dthetavec}{\phivec+\dphivec} \tdplotsetcoord{Q3}{\rvec+\drvec}{\thetavec+\dthetavec}{\phivec} %draw figure contents %-------------------- %draw the main coordinate system axes \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$}; \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$}; \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$}; %draw a vector from origin to point (P) \draw[-stealth,color=red, dashed] (O) -- (P); \draw[color=red, dashed] (O) -- (P3); \draw[color=red, dashed] (O) -- (P1); \draw[color=red, dashed] (O) -- (P2); \draw[blue, thick] (P) -- (P1) (P2)--(P3); \draw[blue, thick] (Q) -- (Q1) (Q2)--(Q3); %draw projection on xy plane, and a connecting line \draw[dashed, color=red] (O) -- (Pxy); \draw[dashed, color=red] (O) -- (P1xy); \draw[dashed, color=red] (P) -- (Pxy); \draw[dashed, color=red] (P1) -- (P1xy); \draw[dashed, color=red] (Pxy) -- node[below, black, sloped]{$r\sin\theta d\phi$}(P1xy); %draw the angle \phi, and label it %syntax: \tdplotdrawarc[coordinate frame, draw options]{center point}{r}{angle}{label options}{label} \tdplotdrawarc{(O)}{0.2}{0}{\phivec}{anchor=north}{$\phi$} \draw[dashed] (\rvec,0,0) arc (0:90:\rvec); %set the rotated coordinate system so the x'-y' plane lies within the %"theta plane" of the main coordinate system %syntax: \tdplotsetthetaplanecoords{\phi} \tdplotsetthetaplanecoords{\phivec} \draw[dashed,tdplot_rotated_coords] (\rvec,0,0) arc (0:90:\rvec); %draw theta arc and label, using rotated coordinate system \tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.5}{0}{\thetavec}{anchor=south west}{$\theta$} \tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.25}{\thetavec}{\thetavec+\dthetavec}{anchor=south west}{$d\theta$} %draw some dashed arcs, demonstrating direct arc drawing %\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{\rvec}{\thetavec}{\thetavec+\dthetavec}{}{} %\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{\rvec+\drvec}{\thetavec}{\thetavec+\dthetavec}{}{} \draw[blue, fill=blue, fill opacity=0.5, thick, tdplot_rotated_coords] (P) arc (\thetavec:\thetavec+\dthetavec:\rvec) node[pos=0.5, below, sloped]{$rd\theta$} -- (Q3) arc (\thetavec+\dthetavec:\thetavec:\rvec+\drvec) node[left]{$dr$} -- cycle; \tdplotsetthetaplanecoords{\phivec+\dphivec} \draw[dashed,tdplot_rotated_coords] (\rvec,0,0) arc (0:90:\rvec); %\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{\rvec}{\thetavec}{\thetavec+\dthetavec}{}{} %\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{\rvec+\drvec}{\thetavec}{\thetavec+\dthetavec}{}{} %\draw[red,tdplot_rotated_coords] (\rvec,0,0) arc (0:\dthetavec:\rvec); \draw[blue, thick,tdplot_rotated_coords] (P1) arc (\thetavec:\thetavec+\dthetavec:\rvec) -- (Q2) arc (\thetavec+\dthetavec:\thetavec:\rvec+\drvec) -- cycle; %draw theta arc and label, using rotated coordinate system %\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.5}{0}{\thetavec}{anchor=south west}{} \end{tikzpicture}

体积微元为$r^2\sin\theta dr d\theta d\phi$

\[\begin{aligned} &\left|{\frac{\partial (x,y,z)}{\partial (r,\phi,\theta)}}\right| \\ =&\left|{ \begin{matrix} \sin\theta\cos\phi & r\sin\theta(-\sin\phi) & r\cos\theta\cos\phi \\ \sin\theta\sin\phi & r\sin\theta\cos\phi & r\cos\theta\sin\phi \\ \cos\theta & 0 & -r\sin\theta \end{matrix} }\right| \\ =&\cos\theta(-r^2\sin\theta\cos\theta\sin^2\phi-r^2\cos\theta\sin\theta\cos^2\phi) \\ &-r\sin\theta(r\sin^2\theta\cos^2\phi+r\sin^2\theta\sin^2\phi) \\ =&-r^2\sin\theta\cos^2\theta-\sin\theta\sin^2\theta =-r\sin\theta \end{aligned} \]

柱坐标变换(cylindrical coordinates)

\[\begin{cases} x=r\cos\theta \\ y=r\sin\theta \\ z=z \end{cases} \]

得到

\[\begin{aligned} &\left|{\frac{\partial (x,y,z)}{\partial (r,\theta,z)}}\right| \\ =& \begin{vmatrix} \cos\theta & -r\sin\theta & 0 \\ \sin\theta & r\cos\theta & 0 \\ 0 & 0 & 1 \end{vmatrix} =r \end{aligned} \]
\begin{tikzpicture} [global scale=0.5] \coordinate (O) at (0,0); \coordinate (Ox) at (-3,-3); \coordinate (Oy) at (4.243,0); % sqrt{18} \coordinate (Oz) at (0, 6); % draw axis \draw[-latex, line width=1] (O)-- (Ox) node[below] {$x$}; \draw[-latex, line width=1] (O)-- (Oy) node[right] {$y$}; \draw[-latex, line width=1] (O)-- (Oz) node[above] {$z$}; % draw arcs \draw[thick] ($(0, 0) + (236:3cm and 2cm)$(P) arc (236:360:3cm and 2cm); \draw[thick] ($(0, 0) + (236:3cm and 2cm)$(P) arc (236:360:3cm and 2cm); \draw[thick] ($(0, 5) + (236:3cm and 2cm)$(P) arc (236:360:3cm and 2cm); \draw[thick, -latex] ($(0, 0) + (236:1.5cm and 1cm)$(P) arc (236:310:1.5cm and 1cm); \coordinate (Phi) at (0,-1) ; \node[below] at (Phi) {$\theta$}; \coordinate (A1) at (0, 5); \coordinate (B) at (3, 5); \coordinate (C) at (-1.7, 3.3); \draw[thick] (A1)--(B); \draw[thick] (A1)--(C); % radius \coordinate (D) at (1.9,-1.5); \coordinate (P) at (1.9,3.5); \draw[thick] (O)--(D); \draw[thick, dashed] (A1)--(P) node[right, yshift=-1mm] {$P$}; \draw[thick] (D)--(P); \fill[black] (P) circle (3pt); \coordinate (A) at (2.6, 4.0); \draw[thick, dashed] (A1)--(A) node[right, yshift=-1mm, xshift=-1mm] {$A$}; % arcs \draw[thick] ($(0, 5) + (310:1.8cm and 1.2cm)$(P) arc (310:330:1.8cm and 1.2cm); \draw[thick] ($(0, 3.5) + (310:1.8cm and 1.2cm)$(P) arc (310:330:1.8cm and 1.2cm); \draw[thick] ($(0, 3.5) + (310:3cm and 2cm)$(P) arc (310:330:3cm and 2cm); \coordinate (Q) at (1.9,1.97); \node[below,xshift=2mm] at (Q) {$Q$}; % \fill[black] (Q) circle (3pt); \coordinate (B) at (2.6, 2.5); \node[below,xshift=1mm] at (B) {$B$}; % \fill[black] (B) circle (3pt); \draw[thick] (A) --(B); \coordinate (S) at (1.15, 4.1); \node[below, xshift=-2mm] at (S) {$S$}; % \fill[black] (S) circle (3pt); \coordinate (R) at (1.15, 2.6); \node[below, xshift=-2mm] at (R) {$R$}; %\fill[black] (R) circle (3pt); \coordinate (D) at (1.52, 4.42); \node[above] at (D) {$D$}; % \fill[black] (D) circle (3pt); \coordinate (C) at (1.54, 2.86); \node[below] at (C) {$C$}; %\fill[black] (C) circle (3pt); \draw[thick] (S) --(R); \draw[thick] (D) --(C); \draw[thick] (R) --(Q); \draw[thick] (C) --(B); % verticals on the planes \coordinate (H) at (-1.65,-1.65); %\fill[black] (H) circle (3pt); % \coordinate (I) at (-1.65,3.35); %\fill[black] (I) circle (3pt); \draw[thick] (H) --(I); \coordinate (J) at (3,0); %\fill[black] (J) circle (3pt); \coordinate (K) at (3,5); %\fill[black] (K) circle (3pt); \draw[thick] (J) --(K); % filling \filldraw[opacity=0.2] (D)--(A) arc (325:306:3cm and 2.2cm)--(S) arc (305:325:1.8cm and 1.2cm)--cycle; \filldraw[opacity=0.2] (P) arc (306:325:3cm and 2.2cm)--(B) arc (325:306:3.0cm and 2.2cm)--cycle; \filldraw[opacity=0.2] (P)--(Q)--(R)--(S)--cycle; % differential labels \node[right, yshift=1mm,xshift=2mm, rotate=-20] at (Q) {$r d \theta$}; \node[right, yshift=6mm, xshift=-1mm ] at (B) {$dz$}; \node[right,xshift=3mm, yshift=2mm, rotate=-20] at (D) {$d r$}; \end{tikzpicture}

则体积微分为$r dr d\theta dz$

例 4. $V$为$x^2+y^2+z^2=z$， 求积分

\[\iiint\limits_V\sqrt{x^2+y^2+z^2}dxdydz \]
\begin{tikzpicture} \begin{axis} [ view={110}{30}, axis lines=middle, width=7cm, %xlabel=$x$, zlabel={$z$} ] \addplot3[ domain=0:360, samples = 60, samples y=0, thick, red, ] ({0.5*cos(x)}, {0.5*sin(x)}, 0.5); \addplot3[ domain=0:360, samples = 60, samples y=0, thick, red, ] ({0.5*cos(x)*cos(120)}, {0.5*cos(x)*sin(120)}, {0.5*sin(x)+0.5}); \addplot3[line legend] coordinates { (-0.1,0,0) (1.3,0,0) } node[left] {$x$}; \addplot3[->] coordinates { (0,-0.1,0) (0,1.1,0) } node[right] {$y$}; \addplot3[->] coordinates { (0,0,-0.1) (0,0,1.3) } node[above] {$z$}; \addplot3[->, blue] coordinates { (0,0,0) ({0.5*cos(60)*cos(120)}, {0.5*cos(60)*sin(120)}, {0.5*sin(60)+0.5}) } node[right] {$(r,\theta,\phi)$}; \addplot3[blue] coordinates { ({0.5*cos(60)*cos(120)}, {0.5*cos(60)*sin(120)}, {0.5*sin(60)+0.5}) ({0.5*cos(60)*cos(120)}, {0.5*cos(60)*sin(120)}, 0) (0,0,0) } node[above right] {$\theta$}; \end{axis} \end{tikzpicture}

例 5. $V$为$x^2+y^2=2z$与$z=2$所围成的区域，求

\[\iiint\limits_V(x^2+y^2)dxdydz \]
\begin{tikzpicture}[scale=2, tdplot_main_coords] %draw the main coordinate system axes \draw[thick,->] (0,0,0) -- (1.3,0,0) node[anchor=north east]{$x$}; \draw[thick,->] (0,0,0) -- (0,1.3,0) node[anchor=north west]{$y$}; \draw[thick,->] (0,0,0) -- (0,0,1.5) node[anchor=south]{$z$}; \coordinate (O) at (0,0,0); \draw [thick, fill=blue, fill opacity=0.5](0,0,1) circle (1); \draw [dashed, red] (O) circle (1); %determine a coordinate (P) using (r,\theta,\phi) coordinates. This command %also determines (Pxy), (Pxz), and (Pyz): the xy-, xz-, and yz-projections %of the point (P). %syntax: \tdplotsetcoord{Coordinate name without parentheses}{r}{\theta}{\phi} \tdplotsetcoord{P}{{sqrt(2)}}{45}{100} \tdplotsetcoord{Q}{{sqrt(5)}}{atan(2)}{-40} \draw[thick] (0,0,1) node[left]{$2$}--(P); \draw[dashed] (O) -- (Pxy) (P)--(Pxy) node[right] {$2$}; % 画曲线 \tdplotsetrotatedcoords{-50}{0}{0} \draw[thick, domain=0:1, tdplot_rotated_coords] plot (\x,0,{\x*\x}); \tdplotsetrotatedcoords{110}{0}{0} \draw[thick, domain=0:1, tdplot_rotated_coords] plot (\x,0,{\x*\x}); \end{tikzpicture}

5.

chap7-ex-2-5

例 6. $V$为$z=ay^2$, $z=by^2$, $y>0$, $(0<a<b)$，$z=\alpha x$, $z=\beta x$, $(0<\alpha<\beta)$, $z=h$所围成，求

\[\iiint\limits_V x^2 dxdydz \]

例 7. $V$为$x^2+y^2+z^2\leq 1$，求

\[\iiint_V\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)dxdydz \]

6.

chap7-ex-2-6

例 8. (例7.2.5) $V$由球面$x^2+y^2+z^2=4$和抛物面$x^2+y^2=3z$所围成，求

\[\iiint_V z dxdydz \]
\tdplotsetmaincoords{80}{60} \begin{tikzpicture}[scale=1.5,tdplot_main_coords] %\begin{tikzpicture}[x={(215:2cm/sqrt 2)},y={(2cm,0cm)},z={(0cm,2cm)},samples=40,scale=1.5] \draw[densely dashed] (0,0,0)--(1,0,0); %,(2,0,0)}; \draw[->] (1,0,0) -- (2,0,0) node[left,below] {$x$}; \draw[densely dashed] (0,0,0)--(0,1,0); %,(2,0,0)}; \draw[->] (0,1,0) -- (0,1.5,0) node[right] {$y$}; \draw[densely dashed] (0,0,0)--(0,0,1); %,(2,0,0)}; \draw[->] (0,0,1) -- (0,0,1.5) node[above] {$z$}; % 柱面 \draw[color=cyan, densely dashed] plot[domain=0:2*pi] ({cos(\x r)}, {sin(\x r)}, 0) node {$D$}; \draw[ color=cyan] plot[domain=0:2*pi] ({cos(\x r)}, {sin(\x r)}, 1); \draw[densely dashed, color=cyan] (0,-1,0) -- (0,-1,1) (0,1,0) -- (0,1,1) (-1,0,0) -- (-1,0,1) (1,0,0) -- (1,0,1); % 两个曲面的交线 \draw[color=red] plot[domain=-1:1, samples=50] ({\x*sin(30)}, {\x*cos(30)}, {sqrt(2-\x*\x)}); \draw[color=red] plot[domain=-1:1, samples=50] ({\x*sin(30)}, {\x*cos(30)}, {\x*\x}); \end{tikzpicture}

例 9. 求$V$的体积

(1)(习题) $V: (x^2+y^2+z^2)^3=3xyz$

(2) $x^2+y^2+z^2=a^2$, $x^2+y^2+z^2=b^2$, $x^2+y^2=z^2$ , $z\geq 0$所围成，其中 $b>a>0$

(3)(习题) $\displaystyle V: (\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}$

(4) $z=x^2+y^2$, $z=2x^2+2y^2$, $xy=a^2$, $xy=2a^2$, $x=2y$, $2x=y$, 所围成

9. (1)

chap7-2-ex8-1

\begin{tikzpicture} \begin{axis} [ view={120}{30}, axis lines=middle, width=7cm, %ztick=\empty,ytick=\empty,xtick=\empty, %xlabel=$x$, zlabel={$z$} ] % 内层球壳 \addplot3[ domain=0:360, samples = 60, samples y=0, thick, red, dashed, ] ({cos(x)}, {sin(x)}, 0); % node[pos=0] {$x$}; \addplot3[ domain=0:180, samples = 60, samples y=0, thick, red, dashed, ] ({cos(x)*cos(120)}, {cos(x)*sin(120)}, {sin(x)}); % 外层球壳 \addplot3[ domain=0:360, samples = 60, samples y=0, thick, red, ] ({2*cos(x)}, {2*sin(x)}, 0); % node[pos=0] {$x$}; \addplot3[ domain=0:180, samples = 60, samples y=0, thick, red, ] ({2*cos(x)*cos(120)}, {2*cos(x)*sin(120)}, {2*sin(x)}); %\addplot3[line legend] coordinates { (-0.1,0,0) (1.3,0,0) } node[left] {$x$}; %\addplot3[->] coordinates { (0,-0.1,0) (0,1.3,0) } node[right] {$y$}; %\addplot3[->] coordinates { (0,0,-0.1) (0,0,1.3) } node[above] {$z$}; \addplot3[blue,thick] coordinates { (0,0,0) ({2.5*cos(45)*cos(120)}, {2.5*cos(45)*sin(120)}, {2.5*sin(45)}) };% node[right] {$(r,\theta,\phi)$}; \addplot3[blue,thick] coordinates { (0,0,0) ({2.5*cos(45)*cos(300)}, {2.5*cos(45)*sin(300)}, {2.5*sin(45)}) };% node[right] {$(r,\theta,\phi)$}; \addplot3[ domain=0:360, samples = 60, samples y=0, thick, blue, ] ({2*sin(45)*cos(x)}, {2*sin(45)*sin(x)}, {2*sin(45)}); \addplot3[ domain=0:360, samples = 60, samples y=0, thick, blue, dashed, ] ({sin(45)*cos(x)}, {sin(45)*sin(x)}, {sin(45)}); % node[pos=0] {$x$}; \end{axis} \end{tikzpicture}

(3)

chap7-2-ex8-3

(4)

chap7-2-ex8-4

例 10. 求$F'(t)$，

(1)(习题) $\displaystyle F(t)=\iiint\limits_{x^2+y^2+z^2\leq t^2}f(x^2+y^2+z^2)dxdydz$

(2) $\displaystyle F(t)=\iiint\limits_{0\leq x\leq t, 0\leq y\leq t, 0\leq z\leq t}f(xyz)dxdydz$

目录

谢谢

例 11. 本节读完

11.

椭圆坐标

\[\begin{cases} x=ar\cos^{\beta}\psi\cos^{\alpha}\phi \\ y=br\cos^{\beta}\psi\sin^{\alpha}\phi \\ z=cr\sin^{\beta}\psi \end{cases} \]

其中$a$, $b$, $c$, $\alpha$, $\beta$为常数。则

\[\left|{\frac{\partial (x,y,z)}{\partial (r,\psi,\phi)}}\right| =abcr^2\alpha\beta\cos^{\alpha-1}\phi\sin^{\alpha-1}\phi\cos^{2\beta-1}\psi\sin^{\beta-1}\psi \]
\[\begin{aligned} \left|{\frac{\partial (x,y,z)}{\partial (r,\psi,\phi)}}\right| =&\begin{vmatrix} a \cos^\beta\psi \cos\alpha\phi & ar \beta\cos^{\beta-1}\psi (-\sin\psi)\\ b \cos^\beta\psi \sin\alpha\phi & \\ c \sin^\beta\psi & \\ \end{vmatrix} \\ =& \end{aligned} \]