在球坐标和柱坐标下散度旋度公式
柱坐标下的散度公式
$$
\nabla\cdot\overrightarrow{F}=\frac{\partial(\rho F_\rho)}{\rho\partial\rho}+\frac{\partial F_\phi}{\rho\partial\phi}+\frac{\partial F_z}{\partial z}=\frac{1}{\rho}[\frac{\partial}{\partial\rho},\frac{\partial}{\partial\phi},\frac{\partial}{\partial z}]
\begin{bmatrix}
\rho F_\rho\\
F_\phi\\
\rho F_z
\end{bmatrix}
$$
柱坐标下的旋度公式
$$
\nabla\times\overrightarrow{F}=\frac{1}{\rho}
\begin{bmatrix}
\overrightarrow{e_\rho}&\overrightarrow{\rho e_\phi}&\overrightarrow{e_z}\\
\frac{\partial}{\partial \rho}&\frac{\partial}{\partial\phi}&\frac{\partial}{\partial z}\\
F_\rho&\rho F_\phi&F_z
\end{bmatrix}
$$
球坐标系下的散度方程
$$
\nabla\cdot\overrightarrow{F}=\frac{\partial(r^2F_r)}{r^2\partial r}+\frac{\partial(sin\theta F_\theta)}{rsin\theta\partial\theta}+\frac{\partial F_\phi}{rsin\theta\partial\phi}=\frac{1}{r^2sin\theta}[\frac{\partial}{\partial r},\frac{\partial}{\partial\theta},\frac{\partial}{\partial\phi}]
\begin{bmatrix}
r^2sin\theta F_r\\
rsin\theta F_\theta\\
rF_\phi
\end{bmatrix}
$$
球坐标系下的旋度方程
$$
\nabla\times\overrightarrow{F}=\frac{1}{r^2sin\theta}
\begin{bmatrix}
\overrightarrow{e_r}&\rho \overrightarrow{e_\theta}&rsin\theta \overrightarrow{e_\phi}\\
\frac{\partial}{\partial r}&\frac{\partial}{\partial\theta}&\frac{\partial}{\partial\phi}\\
F_r&rF_\theta&rsin\theta F_\phi
\end{bmatrix}
$$
