Multivariable Integral

多元积分 Multivariable Integral

概念与定义 Concepts and Definitions

二重积分 Double Integral

对于一个定义在闭区域 $D$ 上的连续函数 $f(x,y)$，二重积分表示为：

$${\iint }_{D}f(x,y)dA$$

其中 $dA$ 表示微小区域元。二重积分可以看作是在区域 $D$ 上对函数 $f(x,y)$ 进行累积求和

For a continuous function $f(x,y)$ defined over a closed region $D$, the double integral is denoted as:

$${\iint }_{D}f(x,y)dA$$

where $dA$ represents a small area element. The double integral can be considered as the accumulated sum of the function $f(x,y)$ over the region $D$

三重积分 Triple Integral

对于一个定义在闭区域 $V$ 上的连续函数 $f(x,y,z)$，三重积分表示为：

$${\iiint }_{V}f(x,y,z)dV$$

其中 $dV$ 表示微小体积元。三重积分可以看作是在体积 $V$ 上对函数 $f(x,y,z)$ 进行累积求和

For a continuous function $f(x,y,z)$ defined over a closed region $V$, the triple integral is denoted as:

$${\iiint }_{V}f(x,y,z)dV$$

where $dV$ represents a small volume element. The triple integral can be considered as the accumulated sum of the function $f(x,y,z)$ over the volume $V$

性质与定理 Properties and Theorems

迭代积分 Iterated Integrals

二重积分可以通过迭代积分来计算。假设区域 $D$ 由 $x$ 轴上的 $a$ 到 $b$ 和 $y$ 轴上的 $c$ 到 $d$ 界定，则二重积分可以表示为：

$${\iint }_{D}f(x,y)dA={\int }_a^b\left({\int }_c^{d}f(x,y)dy\right)dx$$

或者

$${\iint }_{D}f(x,y)dA={\int }_c^d\left({\int }_a^{b}f(x,y)dx\right)dy$$

The double integral can be computed using iterated integrals. Suppose the region $D$ is bounded by $a$ to $b$ on the $x$-axis and $c$ to $d$ on the $y$-axis, then the double integral can be expressed as:

$${\iint }_{D}f(x,y)dA={\int }_a^b\left({\int }_c^{d}f(x,y)dy\right)dx$$

or

$${\iint }_{D}f(x,y)dA={\int }_c^d\left({\int }_a^{b}f(x,y)dx\right)dy$$

同理，三重积分可以表示为：

$${\iiint }_{V}f(x,y,z)dV={\int }_a^b\left({\int }_c^d\left({\int }_e^{f}f(x,y,z)dz\right)dy\right)dx$$

Similarly, the triple integral can be expressed as:

$${\iiint }_{V}f(x,y,z)dV={\int }_a^b\left({\int }_c^d\left({\int }_e^{f}f(x,y,z)dz\right)dy\right)dx$$

变量替换法 Change of Variables

Jacobi 行列式 Jacobi Determinant

在多元积分中，当进行变量替换时，Jacobi 行列式（Jacobian Determinant）是非常重要的工具。设有一组变量 $(u_1,u_2,\dots ,u_n)$ 和 $(x_1,x_2,\dots ,x_n)$，变量之间的关系为：

$$x_i=x_i(u_1,u_2,\dots ,u_n)\text{for }i=1,2,\dots ,n$$

Jacobi 行列式定义为以下偏导数行列式：

$$J=\frac{\partial (x_1,x_2,\dots ,x_n)}{\partial (u_1,u_2,\dots ,u_n)}=\left|\begin{array}{cccc}\frac{\partial x_1}{\partial u_1}& \frac{\partial x_1}{\partial u_2}& \cdots & \frac{\partial x_1}{\partial u_n}\\ \frac{\partial x_2}{\partial u_1}& \frac{\partial x_2}{\partial u_2}& \cdots & \frac{\partial x_2}{\partial u_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial x_3}{\partial u_1}& \frac{\partial x_3}{\partial u_2}& \cdots & \frac{\partial x_3}{\partial u_n}\end{array}\right|$$

In multiple integrals, when performing variable substitution, the Jacobi determinant (Jacobian Determinant) is a crucial tool. Given a set of variables $(u_1,u_2,\dots ,u_n)$ and $(x_1,x_2,\dots ,x_n)$, with the relationships:

$$x_i=x_i(u_1,u_2,\dots ,u_n)\text{for }i=1,2,\dots ,n$$

The Jacobi determinant is defined as the determinant of the following partial derivatives:

$$J=\frac{\partial (x_1,x_2,\dots ,x_n)}{\partial (u_1,u_2,\dots ,u_n)}=\left|\begin{array}{cccc}\frac{\partial x_1}{\partial u_1}& \frac{\partial x_1}{\partial u_2}& \cdots & \frac{\partial x_1}{\partial u_n}\\ \frac{\partial x_2}{\partial u_1}& \frac{\partial x_2}{\partial u_2}& \cdots & \frac{\partial x_2}{\partial u_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial x_3}{\partial u_1}& \frac{\partial x_3}{\partial u_2}& \cdots & \frac{\partial x_3}{\partial u_n}\end{array}\right|$$

Jacobi 行列式在多元积分中的应用 Applications of the Jacobi Determinant in Multiple Integrals

在进行变量替换时，积分区域和被积函数都会随之改变。设原积分变量为 $(x_1,x_2,\dots ,x_n)$，替换后的变量为 $(u_1,u_2,\dots ,u_n)$，原积分区域为 $D$，替换后的积分区域为 $D^{\prime }$。则有以下关系：

$$dV=\left|J\right|dU$$

其中 $\left|J\right|$ 表示 Jacobi 行列式的绝对值。对于多重积分，我们有：

$${\iiint }_{D}f(x_1,x_2,x_3)dV={\iiint }_{D^{\prime }}f(x_1(u_1,u_2,u_3),x_2(u_1,u_2,u_3),x_3(u_1,u_2,u_3))\left|J\right|dU$$

When performing variable substitution, both the integration region and the integrand change accordingly. Let the original integration variables be $(x_1,x_2,\dots ,x_n)$ and the substituted variables be $(u_1,u_2,\dots ,u_n)$. The original integration region is $D$ and the substituted integration region is $D^{\prime }$. The relationship is given by:

$$dV=\left|J\right|dU$$

where $\left|J\right|$ represents the absolute value of the Jacobi determinant. For multiple integrals, we have:

$${\iiint }_{D}f(x_1,x_2,x_3)dV={\iiint }_{D^{\prime }}f(x_1(u_1,u_2,u_3),x_2(u_1,u_2,u_3),x_3(u_1,u_2,u_3))\left|J\right|dU$$

变量替换及其 Jacobi 行列式 Variable Substitution and Its Jacobi Determinant

极坐标 Polar Coordinates

对于二重积分，如果积分区域 $D$ 是圆形或部分圆形，可以将直角坐标系 $(x,y)$ 换成极坐标系 $(r,\theta )$，转换关系为：

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ dA=rdrd\theta \end{gathered}$$

Jacobi 行列式为：

$$J=\left|\frac{\partial (x,y)}{\partial (r,\theta )}\right|=\left|\left|\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right|\right|=\left|\left|\begin{array}{cc}\cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \end{array}\right|\right|=r$$

二重积分表示为：

$${\iint }_{D}f(x,y)dA={\iint }_{D^{\prime }}f(r\cos \theta ,r\sin \theta )rdrd\theta$$

For double integrals, if the integration region $D$ is circular or partially circular, Cartesian coordinates $(x,y)$ can be changed to polar coordinates $(r,\theta )$. The transformation relations are:

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ dA=rdrd\theta \end{gathered}$$

The Jacobi determinant is:

$$J=\left|\frac{\partial (x,y)}{\partial (r,\theta )}\right|=\left|\left|\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right|\right|=\left|\left|\begin{array}{cc}\cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \end{array}\right|\right|=r$$

The double integral is expressed as:

$${\iint }_{D}f(x,y)dA={\iint }_{D^{\prime }}f(r\cos \theta ,r\sin \theta )rdrd\theta$$

柱坐标 Cylindrical Coordinates

对于三重积分，如果积分区域 $V$ 是圆柱形，可以将直角坐标系 $(x,y,z)$ 换成柱坐标系 $(r,\theta ,z)$，转换关系为：

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ z=z \\ dV=rdrd\theta dz \end{gathered}$$

Jacobi 行列式为：

$$J=\left|\frac{\partial (x,y,z)}{\partial (r,\theta ,z)}\right|=\left|\left|\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial z}\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial z}\\ \frac{\partial z}{\partial r}& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial z}\end{array}\right|\right|=\left|\left|\begin{array}{ccc}\cos \theta & -r\sin \theta & 0\\ \sin \theta & r\cos \theta & 0\\ 0& 0& 1\end{array}\right|\right|=r$$

三重积分表示为：

$${\iiint }_{V}f(x,y,z)dV={\iiint }_{V^{\prime }}f(r\cos \theta ,r\sin \theta ,z)rdrd\theta dz$$

For triple integrals, if the integration region $V$ is cylindrical, Cartesian coordinates $(x,y,z)$ can be changed to cylindrical coordinates $(r,\theta ,z)$. The transformation relations are:

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ z=z \\ dV=rdrd\theta dz \end{gathered}$$

The Jacobi determinant is:

$$J=\left|\frac{\partial (x,y,z)}{\partial (r,\theta ,z)}\right|=\left|\left|\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial z}\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial z}\\ \frac{\partial z}{\partial r}& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial z}\end{array}\right|\right|=\left|\left|\begin{array}{ccc}\cos \theta & -r\sin \theta & 0\\ \sin \theta & r\cos \theta & 0\\ 0& 0& 1\end{array}\right|\right|=r$$

The triple integral is expressed as:

$${\iiint }_{V}f(x,y,z)dV={\iiint }_{V^{\prime }}f(r\cos \theta ,r\sin \theta ,z)rdrd\theta dz$$

球坐标 Spherical Coordinates

对于三重积分，如果积分区域 $V$ 是球形，可以将直角坐标系 $(x,y,z)$ 换成球坐标系 $(\rho ,\theta ,\varphi )$，转换关系为：

$$\begin{gathered} x=\rho \sin \varphi \cos \theta \\ y=\rho \sin \varphi \sin \theta \\ z=\rho \cos \varphi \\ dV={\rho }^2\sin \varphi d\rho d\varphi d\theta \end{gathered}$$

Jacobi 行列式为：

$$J=\left|\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}\right|=\left|\left|\begin{array}{ccc}\frac{\partial x}{\partial \rho }& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial \varphi }\\ \frac{\partial y}{\partial \rho }& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial \varphi }\\ \frac{\partial z}{\partial \rho }& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial \varphi }\end{array}\right|\right|=\left|\left|\begin{array}{ccc}\sin \varphi \cos \theta & -\rho \sin \varphi \sin \theta & \rho \cos \varphi \cos \theta \\ \sin \varphi \sin \theta & \rho \sin \varphi \cos \theta & \rho \cos \varphi \sin \theta \\ \cos \varphi & 0& -\rho \sin \varphi \end{array}\right|\right|={\rho }^2\sin \varphi$$

三重积分表示为：

$${\iiint }_{V}f(x,y,z)dV={\iiint }_{V^{\prime }}f(\rho \sin \varphi \cos \theta ,\rho \sin \varphi \sin \theta ,\rho \cos \varphi ){\rho }^2\sin \varphi d\rho d\varphi d\theta$$

For triple integrals, if the integration region $V$ is spherical, Cartesian coordinates $(x,y,z)$ can be changed to spherical coordinates $(\rho ,\theta ,\varphi )$. The transformation relations are:

$$\begin{gathered} x=\rho \sin \varphi \cos \theta \\ y=\rho \sin \varphi \sin \theta \\ z=\rho \cos \varphi \\ dV={\rho }^2\sin \varphi d\rho d\varphi d\theta \end{gathered}$$

The Jacobi determinant is:

$$J=\left|\frac{\partial (x,y,z)}{\partial (\rho ,\theta ,\varphi )}\right|=\left|\left|\begin{array}{ccc}\frac{\partial x}{\partial \rho }& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial \varphi }\\ \frac{\partial y}{\partial \rho }& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial \varphi }\\ \frac{\partial z}{\partial \rho }& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial \varphi }\end{array}\right|\right|=\left|\left|\begin{array}{ccc}\sin \varphi \cos \theta & -\rho \sin \varphi \sin \theta & \rho \cos \varphi \cos \theta \\ \sin \varphi \sin \theta & \rho \sin \varphi \cos \theta & \rho \cos \varphi \sin \theta \\ \cos \varphi & 0& -\rho \sin \varphi \end{array}\right|\right|={\rho }^2\sin \varphi$$

The triple integral is expressed as:

$${\iiint }_{V}f(x,y,z)dV={\iiint }_{V^{\prime }}f(\rho \sin \varphi \cos \theta ,\rho \sin \varphi \sin \theta ,\rho \cos \varphi ){\rho }^2\sin \varphi d\rho d\varphi d\theta$$

计算技巧 Calculation Techniques

判断积分区域 Determining the Integration Region

二重积分区域 Double Integral Region

常见的二重积分区域有矩形区域和一般区域。对于一般区域，可以根据边界曲线确定积分区域：

$$D=\{(x,y)\mid a\le x\le b,g_1(x)\le y\le g_2(x)\}$$

此时，二重积分表示为：

$${\iint }_{D}f(x,y)dA={\int }_a^b\left({\int }_{g_1(x)}^{g_2(x)}f(x,y)dy\right)dx$$

Common double integral regions include rectangular regions and general regions. For a general region, the integration region can be determined by boundary curves:

$$D=\{(x,y)\mid a\le x\le b,g_1(x)\le y\le g_2(x)\}$$

At this time, the double integral is expressed as:

$${\iint }_{D}f(x,y)dA={\int }_a^b\left({\int }_{g_1(x)}^{g_2(x)}f(x,y)dy\right)dx$$

三重积分区域 Triple Integral Region

常见的三重积分区域有矩形区域和一般区域。对于一般区域，可以根据边界曲面确定积分区域：

$$V=\{(x,y,z)\mid a\le x\le b,g_1(x)\le y\le g_2(x),h_1(x,y)\le z\le h_2(x,y)\}$$

此时，三重积分表示为：

$${\iiint }_{V}f(x,y,z)dV={\int }_a^b\left({\int }_{g_1(x)}^{g_2(x)}\left({\int }_{h_1(x,y)}^{h_2(x,y)}f(x,y,z)dz\right)dy\right)dx$$

Common triple integral regions include rectangular regions and general regions. For a general region, the integration region can be determined by boundary surfaces:

$$V=\{(x,y,z)\mid a\le x\le b,g_1(x)\le y\le g_2(x),h_1(x,y)\le z\le h_2(x,y)\}$$

At this time, the triple integral is expressed as:

$${\iiint }_{V}f(x,y,z)dV={\int }_a^b\left({\int }_{g_1(x)}^{g_2(x)}\left({\int }_{h_1(x,y)}^{h_2(x,y)}f(x,y,z)dz\right)dy\right)dx$$

常见积分区域换元技巧 Common Transformation Techniques for Integration Regions

圆区域 Circular Region

当积分区域为圆或部分圆时，使用极坐标变换，将积分区域从 $(x,y)$ 变换到 $(r,\theta )$，并使用 $dA=rdrd\theta$

When the integration region is a circle or part of a circle, use the polar coordinate transformation to change the integration region from $(x,y)$ to $(r,\theta )$, and use $dA=rdrd\theta$

柱形区域 Cylindrical Region

当积分区域为柱形时，使用柱坐标变换，将积分区域从 $(x,y,z)$ 变换到 $(r,\theta ,z)$，并使用 $dV=rdrd\theta dz$

When the integration region is cylindrical, use the cylindrical coordinate transformation to change the integration region from $(x,y,z)$ to $(r,\theta ,z)$, and use $dV=rdrd\theta dz$

球形区域 Spherical Region

当积分区域为球形时，使用球坐标变换，将积分区域从 $(x,y,z)$ 变换到 $(\rho ,\theta ,\varphi )$，并使用 $dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$

When the integration region is spherical, use the spherical coordinate transformation to change the integration region from $(x,y,z)$ to $(\rho ,\theta ,\varphi )$, and use $dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$

例题 Examples

例 1 Example 1

计算二重积分 ${\iint }_D(x^2+y^2)dA$，其中区域 $D$ 是单位圆 $x^2+y^2\le 1$

Calculate the double integral ${\iint }_D(x^2+y^2)dA$ where the region $D$ is the unit circle $x^2+y^2\le 1$

解：

使用极坐标变换：

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ dA=rdrd\theta \end{gathered}$$

积分区域 $D$ 变为 $0\le r\le 1,0\le \theta \le 2\pi$

$$\begin{gathered} {\iint }_D(x^2+y^2)dA={\int }_0^{2\pi }{\int }_0^1(r^2)rdrd\theta \\ ={\int }_0^{2\pi }{\int }_0^1{r}^{3}drd\theta \\ ={\int }_0^{2\pi }{\left[\frac{r^4}{4}\right]}_0^{1}d\theta \\ ={\int }_0^{2\pi }\frac{1}{4}d\theta \\ =\frac{1}{4}\cdot 2\pi \\ =\frac{\pi }{2} \end{gathered}$$

Solution:

Using the polar coordinate transformation:

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \\ dA=rdrd\theta \end{gathered}$$

The integration region $D$ becomes $0\le r\le 1,0\le \theta \le 2\pi$

$$\begin{gathered} {\iint }_D(x^2+y^2)dA={\int }_0^{2\pi }{\int }_0^1(r^2)rdrd\theta \\ ={\int }_0^{2\pi }{\int }_0^1{r}^{3}drd\theta \\ ={\int }_0^{2\pi }{\left[\frac{r^4}{4}\right]}_0^{1}d\theta \\ ={\int }_0^{2\pi }\frac{1}{4}d\theta \\ =\frac{1}{4}\cdot 2\pi \\ =\frac{\pi }{2} \end{gathered}$$

例 2 Example 2

计算三重积分 ${\iiint }_V(x^2+y^2+z^2)dV$，其中区域 $V$ 是单位球 $x^2+y^2+z^2\le 1$

Calculate the triple integral ${\iiint }_V(x^2+y^2+z^2)dV$ where the region $V$ is the unit sphere $x^2+y^2+z^2\le 1$

解：

使用球坐标变换：

$$\begin{gathered} x=\rho \sin \varphi \cos \theta \\ y=\rho \sin \varphi \sin \theta \\ z=\rho \cos \varphi \\ dV={\rho }^2\sin \varphi d\rho d\varphi d\theta \end{gathered}$$

积分区域 $V$ 变为 $0\le \rho \le 1,0\le \varphi \le \pi ,0\le \theta \le 2\pi$

$$\begin{gathered} {\iiint }_V(x^2+y^2+z^2)dV={\iiint }_{V^{\prime }}({\rho }^2){\rho }^2\sin \varphi d\rho d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }{\int }_0^1{\rho }^4\sin \varphi d\rho d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }{\left[\frac{{\rho }^5}{5}\right]}_0^1\sin \varphi d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }\frac{1}{5}\sin \varphi d\varphi d\theta \\ =\frac{1}{5}{\int }_0^{2\pi }{\left[-\cos \varphi \right]}_0^{\pi }d\theta \\ =\frac{1}{5}{\int }_0^{2\pi }(2)d\theta \\ =\frac{2}{5}\cdot 2\pi \\ =\frac{4\pi }{5} \end{gathered}$$

Solution:

Using the spherical coordinate transformation:

$$\begin{gathered} x=\rho \sin \varphi \cos \theta \\ y=\rho \sin \varphi \sin \theta \\ z=\rho \cos \varphi \\ dV={\rho }^2\sin \varphi d\rho d\varphi d\theta \end{gathered}$$

The integration region $V$ becomes $0\le \rho \le 1,0\le \varphi \le \pi ,0\le \theta \le 2\pi$

$$\begin{gathered} {\iiint }_V(x^2+y^2+z^2)dV={\iiint }_{V^{\prime }}({\rho }^2){\rho }^2\sin \varphi d\rho d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }{\int }_0^1{\rho }^4\sin \varphi d\rho d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }{\left[\frac{{\rho }^5}{5}\right]}_0^1\sin \varphi d\varphi d\theta \\ ={\int }_0^{2\pi }{\int }_0^{\pi }\frac{1}{5}\sin \varphi d\varphi d\theta \\ =\frac{1}{5}{\int }_0^{2\pi }{\left[-\cos \varphi \right]}_0^{\pi }d\theta \\ =\frac{1}{5}{\int }_0^{2\pi }(2)d\theta \\ =\frac{2}{5}\cdot 2\pi \\ =\frac{4\pi }{5} \end{gathered}$$