常见坐标系下的梯度、散度、旋度和拉普拉斯算子

本文给出常见的正交曲线坐标系下梯度、散度、旋度和拉普拉斯算子的形式，以供快速查阅。

正交曲线坐标系

拉梅系数

$$\mathrm{d} \boldsymbol{r}=\frac{\partial \boldsymbol{r}}{\partial q_{1}} \mathrm{d} q_{1}+\frac{\partial \boldsymbol{r}}{\partial q_{2}} \mathrm{d} q_{2}+\frac{\partial \boldsymbol{r}}{\partial q_{3}} \mathrm{d} q_{3}$$

$$\left\{\begin{array}{l} \left|\dfrac{\partial \bm{r}}{\partial q_{1}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{1}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{1}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{1}}\right)^{2}}=H_{1} \\ \left|\dfrac{\partial \bm{r}}{\partial q_{2}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{2}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{2}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{2}}\right)^{2}}=H_{2} \\ \left|\dfrac{\partial \bm{r}}{\partial q_{1}}\right|=\sqrt{\left(\dfrac{\partial x}{\partial q_{3}}\right)^{2}+\left(\dfrac{\partial y}{\partial q_{3}}\right)^{2}+\left(\dfrac{\partial z}{\partial q_{3}}\right)^{2}}=H_{3} \end{array}\right.$$

弧长

$$|\mathrm{d} \boldsymbol{r}|=\mathrm{d} s=\sqrt{H_{1}^{2} \mathrm{d} q_{1}^{2}+H_{2}^{2} \mathrm{d} q_{2}^{2}+H_{3}^{2} \mathrm{d} q_{3}^{2}}$$

面积元

$$\left\{\begin{array}{l} \mathrm{d} \sigma_{1}=H_{2} H_{3} \mathrm{d} q_{2} \mathrm{d} q_{3} \\ \mathrm{d} \sigma_{2}=H_{1} H_{3} \mathrm{d} q_{1} \mathrm{d} q_{3} \\ \mathrm{d} \sigma_{3}=H_{2} H_{1} \mathrm{d} q_{2} \mathrm{d} q_{1} \end{array}\right.$$

体积元

$$\mathrm{d} V=\mathrm{d} s_{1} \mathrm{d} s_{2} \mathrm{d} s_{3}=H_{1} H_{2} H_{3} \mathrm{d} q_{1} \mathrm{d} q_{2} \mathrm{d} q_{3}$$

梯度

$$\nabla f=\frac{1}{H_{1}} \frac{\partial f}{\partial q_{1}} \bm{e}_{1}+\frac{1}{H_{2}} \frac{\partial f}{\partial q_{2}} \bm{e}_{2}+\frac{1}{H_{3}} \frac{\partial f}{\partial q_{3}} \bm{e}_{3}$$

散度

$$\nabla \cdot \boldsymbol{A}=\frac{1}{H_{1} H_{2} H_{3}}\left(\frac{\partial\left(A_{1} H_{2} H_{3}\right)}{\partial q_{1}}+\frac{\partial\left(A_{2} H_{1} H_{3}\right)}{\partial q_{2}}+\frac{\partial\left(A_{3} H_{2} H_{1}\right)}{\partial q_{3}}\right)$$

旋度

$$\nabla \times \boldsymbol{A}=\frac{1}{H_{1} H_{2} H_{3}}\begin{vmatrix} H_{1} \boldsymbol{e}_{1} & H_{2} \boldsymbol{e}_{2} & H_{3} \boldsymbol{e}_{3} \\ \frac{\partial}{\partial q_{1}} & \frac{\partial}{\partial q_{2}} & \frac{\partial}{\partial q_{3}} \\ H_{1} A_{1} & H_{2} A_{2} & H_{3} A_{3} \end{vmatrix}$$

拉普拉斯算子

$$\Delta f=\frac{1}{H_{1} H_{2} H_{3}}\left[\frac{\partial}{\partial q_{1}}\left(\frac{H_{2} H_{3}}{H_{1}} \frac{\partial f}{\partial q_{1}}\right)+\frac{\partial}{\partial q_{2}}\left(\frac{H_{1} H_{3}}{H_{2}} \frac{\partial f}{\partial q_{2}}\right)+\frac{\partial}{\partial q_{3}}\left(\frac{H_{2} H_{1}}{H_{3}} \frac{\partial f}{\partial q_{3}}\right)\right]$$

直角坐标系

梯度

$$\nabla f(x, y, z)=\frac{\partial f}{\partial x} \boldsymbol{i}+\frac{\partial f}{\partial y} \boldsymbol{j}+\frac{\partial f}{\partial z} \boldsymbol{k}$$

散度

$$\nabla \cdot \bm{A}=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}$$

旋度

$$\nabla \times \bm{A}=\begin{vmatrix} \bm{i} & \bm{j} & \bm{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_{x} & A_{y} & A_{z} \end{vmatrix}$$

拉普拉斯算子

$$\Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}$$

柱坐标系

梯度

$$\nabla f(\rho, \theta, z)=\frac{\partial f}{\partial \rho} \bm{e}_{\rho}+\frac{1}{\rho} \frac{\partial f}{\partial \theta} \bm{e}_{\theta}+\frac{\partial f}{\partial z} \bm{e}_{z}$$

散度

$$\nabla \cdot \bm{A}=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho A_{\rho}\right)+\frac{1}{\rho} \frac{\partial A_{\varphi}}{\partial \varphi}+\frac{\partial A_{z}}{\partial z}$$

旋度

$$\nabla \times \bm{A}=\begin{vmatrix} \frac{1}{\rho} \bm{e}_{\rho} & \bm{e}_{\theta} & \frac{1}{\rho} \bm{e}_{z} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\ A_{\rho} & \rho A_{\theta} & A_{z} \end{vmatrix}$$

拉普拉斯算子

$$\Delta f=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial f}{\partial \rho}\right)+\frac{1}{\rho^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}$$

球坐标系

梯度

$$\nabla f(r, \theta, \phi)=\frac{\partial f}{\partial r} \bm{e}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \bm{e}_{\theta}+\frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \bm{e}_{\phi}$$

散度

$$\nabla \cdot \bm{A}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} A_{r}\right)+\frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta A_{\theta}\right)+\frac{1}{r \sin \theta} \frac{\partial A_{\phi}}{\partial \phi}$$

旋度

$$\nabla \times \bm{A}=\frac{1}{r^{2} \sin \theta}\begin{vmatrix} \bm{e}_{r} & r \bm{e}_{\theta} & r \sin \theta \bm{e}_{\phi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ A_{r} & r A_{\theta} & r \sin \theta A_{\phi} \end{vmatrix}$$

拉普拉斯算子

$$\Delta f=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}}$$