Prerequisite 正交曲线坐标系
，梯度
，旋度
令eq. 1 中三个分母为 $f(u,v,w), g(u,v,w), h(u,v,w)$，则
\begin{equation}
\begin{aligned}
\,\mathrm{d}{ \boldsymbol{\mathbf{r}} } &= \frac{\partial \boldsymbol{\mathbf{r}} }{\partial u} \,\mathrm{d}{u} + \frac{\partial \boldsymbol{\mathbf{r}} }{\partial v} \,\mathrm{d}{v} + \frac{\partial \boldsymbol{\mathbf{r}} }{\partial w} \,\mathrm{d}{w} \\
&= f \hat{\boldsymbol{\mathbf{u}}} \,\mathrm{d}{u} + g \hat{\boldsymbol{\mathbf{v}}} \,\mathrm{d}{v} + h \hat{\boldsymbol{\mathbf{w}}} \,\mathrm{d}{w}
\end{aligned}
\end{equation}
令 $s(u, v, w)$ 为标量函数，$ \boldsymbol{\mathbf{A}} (u, v, w)$ 为矢量函数，且
\begin{equation}
\boldsymbol{\mathbf{A}} (u, v, w) = A_x(u, v, w) \hat{\boldsymbol{\mathbf{u}}} + A_y(u, v, w) \hat{\boldsymbol{\mathbf{v}}} + A_z(u, v, w) \hat{\boldsymbol{\mathbf{w}}}
\end{equation}
那么该坐标系中的梯度
，散度
，旋度
算符分别为
\begin{equation}
\boldsymbol\nabla s = \frac{1}{f} \frac{\partial s}{\partial u} \hat{\boldsymbol{\mathbf{u}}} + \frac{1}{g} \frac{\partial s}{\partial v} \hat{\boldsymbol{\mathbf{v}}} + \frac{1}{h} \frac{\partial s}{\partial w} \hat{\boldsymbol{\mathbf{w}}}
\end{equation}
\begin{equation}
\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathbf{A}} = \frac{1}{fgh} \left[ \frac{\partial}{\partial{u}} (ghA_u) + \frac{\partial}{\partial{v}} (fhA_v) + \frac{\partial}{\partial{w}} (fgA_w) \right]
\end{equation}
\begin{equation}
\begin{aligned}
& \boldsymbol{\nabla}\boldsymbol{\times} \boldsymbol{\mathbf{A}} = \frac{1}{gh} \left[ \frac{\partial}{\partial{v}} (hA_w) - \frac{\partial}{\partial{w}} (gA_v) \right] \hat{\boldsymbol{\mathbf{u}}} \\
&\quad + \frac{1}{fh} \left[ \frac{\partial}{\partial{w}} (fA_u) - \frac{\partial}{\partial{u}} (hA_w) \right] \hat{\boldsymbol{\mathbf{v}}}
+ \frac{1}{fg} \left[ \frac{\partial}{\partial{u}} (gA_v) - \frac{\partial}{\partial{v}} (fA_u) \right] \hat{\boldsymbol{\mathbf{w}}}
\end{aligned}
\end{equation}
\begin{equation}
\boldsymbol{\nabla}^2 s = \frac{1}{fgh} \left[ \frac{\partial}{\partial{u}} \left(\frac{gh}{f} \frac{\partial s}{\partial u} \right) + \frac{\partial}{\partial{v}} \left(\frac{fh}{g} \frac{\partial s}{\partial v} \right) + \frac{\partial}{\partial{w}} \left(\frac{fg}{h} \frac{\partial s}{\partial w} \right) \right]
\end{equation}