引言

本文介绍了散度、梯度和旋度在直角坐标系、柱坐标系和球坐标系三种常见坐标系下的表示。记录一下，具体可以利用梅拉系数进行推导。

谨记：
梯度：标量求梯度得到矢量。
散度：矢量求散度得到标量。
旋度：矢量求旋度得到矢量。

1.直角坐标系

标量表示
$$f=f(x,y,z)$$

矢量表示
$$\overrightarrow{\mathbf{f}}=f_x\overrightarrow{{\mathbf{e}}_{\mathbf{x}}}+f_y\overrightarrow{{\mathbf{e}}_{\mathbf{y}}}+f_z\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}$$

梯度：
$$▽f=\frac{\partial f}{\partial x}\overrightarrow{{\mathbf{e}}_{\mathbf{x}}}+\frac{\partial f}{\partial y}\overrightarrow{{\mathbf{e}}_{\mathbf{y}}}+\frac{\partial f}{\partial z}\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}$$

散度：
$$▽\cdot \overrightarrow{\mathbf{f}}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}$$

旋度：
$$\begin{array}{rl}▽\times \overrightarrow{\mathbf{f}}& =\left[\begin{array}{ccc}\overrightarrow{{\mathbf{e}}_{\mathbf{x}}}& \overrightarrow{{\mathbf{e}}_{\mathbf{y}}}& \overrightarrow{{\mathbf{e}}_{\mathbf{z}}}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ f_x& f_y& f_z\end{array}\right]\\ & =(\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z})\overrightarrow{{\mathbf{e}}_{\mathbf{x}}}+(\frac{\partial f_x}{\partial z}-\frac{\partial f_z}{\partial x})\overrightarrow{{\mathbf{e}}_{\mathbf{y}}}+(\frac{\partial f_y}{\partial x}-\frac{\partial f_x}{\partial y})\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}\end{array}$$

2.柱坐标系

标量表示
$$f=f(\rho ,\theta ,z)$$

矢量表示
$$\overrightarrow{\mathbf{f}}=f_{\rho }\overrightarrow{{\mathbf{e}}_{\rho }}+f_{\theta }\overrightarrow{{\mathbf{e}}_{\theta }}+f_z\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}$$

梯度：
$$▽f=\frac{\partial f}{\partial \rho }\overrightarrow{{\mathbf{e}}_{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \theta }\overrightarrow{{\mathbf{e}}_{\theta }}+\frac{\partial f}{\partial z}\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}$$

散度：
$$▽\cdot \overrightarrow{\mathbf{f}}=\frac{1}{\rho }\frac{\partial (\rho f_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial f_{\theta }}{\partial \theta }+\frac{\partial f_z}{\partial z}$$

旋度：
$$\begin{array}{rl}▽\times \overrightarrow{\mathbf{f}}& =\frac{1}{\rho }\left[\begin{array}{ccc}\overrightarrow{{\mathbf{e}}_{\rho }}& \rho \overrightarrow{{\mathbf{e}}_{\theta }}& \overrightarrow{{\mathbf{e}}_{\mathbf{z}}}\\ \frac{\partial }{\partial \rho }& \frac{\partial }{\partial \theta }& \frac{\partial }{\partial z}\\ f_{\rho }& \rho f_{\theta }& f_z\end{array}\right]\\ & =\frac{1}{\rho }\left[(\frac{\partial f_z}{\partial \theta }-\frac{\partial (\rho f_{\theta })}{\partial z})\overrightarrow{{\mathbf{e}}_{\rho }}+(\frac{\partial f_{\rho }}{\partial z}-\frac{\partial f_z}{\partial \rho })\rho \overrightarrow{{\mathbf{e}}_{\theta }}+(\frac{\partial (\rho f_{\theta })}{\partial \rho }-\frac{\partial f_{\rho }}{\partial \theta })\overrightarrow{{\mathbf{e}}_{\mathbf{z}}}\right]\end{array}$$

3.球坐标系

(请注意这里的$\theta$和柱坐标系中的$\theta$的定义不同，详细见图)

标量表示
$$f=f(\rho ,\theta ,\varphi )$$

矢量表示
$$\overrightarrow{\mathbf{f}}=f_{\rho }\overrightarrow{{\mathbf{e}}_{\rho }}+f_{\theta }\overrightarrow{{\mathbf{e}}_{\theta }}+f_{\varphi }\overrightarrow{{\mathbf{e}}_{\varphi }}$$

梯度：
$$▽f=\frac{\partial f}{\partial \rho }\overrightarrow{{\mathbf{e}}_{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \theta }\overrightarrow{{\mathbf{e}}_{\theta }}+\frac{1}{\rho sin\theta }\frac{\partial f}{\partial \varphi }\overrightarrow{{\mathbf{e}}_{\varphi }}$$

散度：
$$▽\cdot \overrightarrow{\mathbf{f}}=\frac{1}{{\rho }^2}\frac{\partial ({\rho }^2{f}_{\rho })}{\partial \rho }+\frac{1}{\rho sin\theta }\frac{\partial (sin\theta f_{\theta })}{\partial \theta }+\frac{1}{\rho sin\theta }\frac{\partial f_{\varphi }}{\partial \varphi }$$

旋度：
$$\begin{array}{rl}▽\times \overrightarrow{\mathbf{f}}& =\frac{1}{{\rho }^{2}sin\theta }\left[\begin{array}{ccc}\overrightarrow{{\mathbf{e}}_{\rho }}& \rho \overrightarrow{{\mathbf{e}}_{\theta }}& \rho sin\theta \overrightarrow{{\mathbf{e}}_{\varphi }}\\ \frac{\partial }{\partial \rho }& \frac{\partial }{\partial \theta }& \frac{\partial }{\partial \varphi }\\ f_{\rho }& \rho f_{\theta }& \rho sin\theta f_{\varphi }\end{array}\right]\\ & =\frac{1}{{\rho }^{2}sin\theta }\left[(\frac{\partial (\rho sin\theta f_{\varphi })}{\partial \theta }-\frac{\partial (\rho f_{\theta })}{\partial \varphi })\overrightarrow{{\mathbf{e}}_{\rho }}+(\frac{\partial f_{\rho }}{\partial \varphi }-\frac{\partial (\rho sin\theta f_{\varphi })}{\partial \rho })\rho \overrightarrow{{\mathbf{e}}_{\theta }}+(\frac{\partial (\rho f_{\theta })}{\partial \rho }-\frac{\partial f_{\rho }}{\partial \theta })\rho sin\theta \overrightarrow{{\mathbf{e}}_{\varphi }}\right]\end{array}$$

由于笔者之前一直想不清楚柱坐标系和球坐标系下三度的变换，于是查阅资料，其中发现[1]是极好的，推荐一看！！！详细推导见下面的参考资料[1]，利用梅拉系数很容易地可以进行推导，利用拉梅系数还可以很直接地推导了斯托克斯公式和高斯公式，非常地简单易懂！！感谢知乎@弧长长长长长。另外[2]写得也比较详细，可以用作其他的参考。

[1] 浅谈：拉梅系数那些事儿
[2] 柱面及球面坐标系中散度、旋度的应用

附：

1.常用的梅拉系数

直角坐标系：
$$H_x=1,H_y=1,H_z=1$$

柱坐标系：
$$H_r=1,H_{\theta }=r,H_z=1$$

球坐标系：
$$H_r=1,H_{\theta }=r,H_{\varphi }=rsin\theta$$

2.通用公式

梯度：

$$▽f=\frac{1}{H_1}\frac{\partial f}{\partial q_1}\overrightarrow{{\mathbf{e}}_1}+\frac{1}{H_2}\frac{\partial f}{\partial q_2}\overrightarrow{{\mathbf{e}}_2}+\frac{1}{H_3}\frac{\partial f}{\partial q_3}\overrightarrow{{\mathbf{e}}_3}$$

散度：
$$\mathrm{div}\mathbf{r}=\underset{V\to 0}{\lim }\frac{{\oint }_S{r}_{i}dS}{V}=\frac{1}{H_1{H}_2{H}_3}\left(\frac{\partial \left(r_1{H}_2{H}_3\right)}{\partial q_1}+\frac{\partial \left(r_2{H}_1{H}_3\right)}{\partial q_2}+\frac{\partial \left(r_3{H}_2{H}_1\right)}{\partial q_3}\right)$$

旋度：

$$\{\begin{array}{l}{\mathrm{rot}}_{q_1}\mathbf{r}=\frac{1}{H_2{H}_3}\left[\frac{\partial \left(r_3{H}_3\right)}{\partial q_2}-\frac{\partial \left(r_2{H}_2\right)}{\partial q_3}\right]\\ {\mathrm{rot}}_{q_2}\mathbf{r}=\frac{1}{H_1{H}_3}\left[\frac{\partial \left(r_1{H}_1\right)}{\partial q_3}-\frac{\partial \left(r_3{H}_3\right)}{\partial q_1}\right]\\ {\mathrm{rot}}_{q_3}\mathbf{r}=\frac{1}{H_2{H}_1}\left[\frac{\partial \left(r_2{H}_2\right)}{\partial q_1}-\frac{\partial \left(r_1{H}_1\right)}{\partial q_2}\right]\end{array}$$
或
$$\mathrm{rot}\mathbf{r}=\frac{1}{H_1{H}_2{H}_3}\left|\begin{array}{ccc}{H}_1{\mathbf{e}}_1& H_2{\mathbf{e}}_2& H_3{\mathbf{e}}_3\\ \frac{\partial }{\partial q_1}& \frac{\partial }{\partial q_2}& \frac{\partial }{\partial q_3}\\ H_1{r}_1& H_2{r}_2& H_3{r}_3\end{array}\right|$$