抛物线坐标系
贡献者： addis
\begin{equation}
r = \frac{\eta}{1 - \cos \theta }~.
\end{equation}
\begin{equation}
r = \frac{\xi}{1 + \cos \theta }~.
\end{equation}
图 1：抛物线坐标系，极轴向上（来自 Wikipedia）
若把这些曲线绕极轴旋转一周变为一系列抛物面，那么我们只需要再指定一个方位角 $\phi$ 就可以用坐标 $(\xi, \eta, \phi)$ 确定空间中的任意一点。
1. 与直角坐标和球坐标的转换
一般令极轴与直角坐标的 $z$ 轴重合，则根据定义有
\begin{equation}
\xi = r(1 + \cos\theta) = \sqrt{x^2 + y^2 + z^2} + z~,
\end{equation}
\begin{equation}
\eta = r(1 - \cos\theta) = \sqrt{x^2 + y^2 + z^2} - z~,
\end{equation}
\begin{equation}
\phi = \operatorname{Arctan} (y, x)~,
\end{equation}
\begin{equation}
z = \frac{\xi - \eta}{2}~,
\end{equation}
\begin{equation}
x = \sqrt{\xi\eta}\cos\phi ~,\qquad
y = \sqrt{\xi\eta}\sin\phi~.
\end{equation}
未完成：另开文章
2. 正交曲线坐标系
可以证明抛物线坐标系是一个正交曲线坐标系，即图 1 中过任意一点的两条坐标曲线都垂直：
在某点 $ \boldsymbol{\mathbf{r}} $，延 $\eta$ 曲线的切线方向为 $ \partial \boldsymbol{\mathbf{r}} /\partial \xi $，延 $\xi$ 曲线的切线方向为 $ \partial \boldsymbol{\mathbf{r}} /\partial \eta $
\begin{equation}
\,\mathrm{d}{x} = \sqrt{\frac{\eta}{\xi}}\frac{\cos\phi}{2} \,\mathrm{d}{\xi} + \sqrt{\frac{\xi}{\eta}}\frac{\cos\phi}{2\sqrt{\xi}} \,\mathrm{d}{\eta} - \sqrt{\xi\eta}\sin\phi \,\mathrm{d}{\phi} ~,
\end{equation}
\begin{equation}
\,\mathrm{d}{y} = \sqrt{\frac{\eta}{\xi}}\frac{\sin\phi}{2} \,\mathrm{d}{\xi} + \sqrt{\frac{\xi}{\eta}}\frac{\sin\phi}{2} \,\mathrm{d}{\eta} + \sqrt{\xi\eta}\cos\phi \,\mathrm{d}{\phi} ~,
\end{equation}
\begin{equation}
\,\mathrm{d}{z} = \frac{1}{2} \,\mathrm{d}{\xi} - \frac{1}{2} \,\mathrm{d}{\eta} ~.
\end{equation}
\begin{equation}
\hat{\boldsymbol{\mathbf{\xi}}} = \sqrt{\frac{\eta}{\xi+\eta}}\cos\phi\, \hat{\boldsymbol{\mathbf{x}}} + \sqrt{\frac{\eta}{\xi+\eta}}\sin\phi\, \hat{\boldsymbol{\mathbf{y}}} + \sqrt{\frac{\xi}{\xi+\eta}}\, \hat{\boldsymbol{\mathbf{z}}} ~,
\end{equation}
\begin{equation}
\hat{\boldsymbol{\mathbf{\eta}}} = \sqrt{\frac{\xi}{\xi+\eta}}\cos\phi\, \hat{\boldsymbol{\mathbf{x}}} + \sqrt{\frac{\xi}{\xi+\eta}}\sin\phi\, \hat{\boldsymbol{\mathbf{y}}} + \sqrt{\frac{\eta}{\xi+\eta}}\, \hat{\boldsymbol{\mathbf{z}}} ~,
\end{equation}
\begin{equation}
\hat{\boldsymbol{\mathbf{\phi}}} = -\sin\phi \, \hat{\boldsymbol{\mathbf{x}}} + \cos\phi \, \hat{\boldsymbol{\mathbf{y}}} ~,
\end{equation}
\begin{equation}
\,\mathrm{d}{ \boldsymbol{\mathbf{r}} } = \frac{1}{2}\sqrt{\frac{\xi + \eta}{\xi}} \hat{\boldsymbol{\mathbf{\xi}}} \,\mathrm{d}{\xi}
+ \frac{1}{2}\sqrt{\frac{\xi + \eta}{\eta}} \hat{\boldsymbol{\mathbf{\eta}}} \,\mathrm{d}{\eta}
+ \sqrt{\xi\eta}\, \hat{\boldsymbol{\mathbf{\phi}}} \,\mathrm{d}{\phi} ~,
\end{equation}
\begin{equation}
\,\mathrm{d}{V} = \frac{1}{4} (\xi + \eta) \,\mathrm{d}{\xi} \,\mathrm{d}{\eta} \,\mathrm{d}{\phi} ~.
\end{equation}
3. 矢量算符
\begin{equation}
\boldsymbol\nabla u = 2\sqrt{\frac{\xi}{\xi + \eta}} \frac{\partial u}{\partial \xi} + 2\sqrt{\frac{\eta}{\xi + \eta}} \frac{\partial u}{\partial \eta} + \frac{1}{\sqrt{\xi\eta}} \frac{\partial u}{\partial \phi} ~,
\end{equation}
\begin{equation}
\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathbf{A}} = \frac{2}{\xi + \eta} \left[ \frac{\partial}{\partial{\xi}} \left(\sqrt{\xi(\xi+\eta)}A_\xi \right) + \frac{\partial}{\partial{\eta}} \left(\sqrt{\eta(\xi+\eta)}A_\eta \right) \right] + \frac{1}{\sqrt{\xi\eta}} \frac{\partial A_\phi}{\partial \phi} ~,
\end{equation}
\begin{equation}
\begin{aligned}
\boldsymbol{\nabla}\boldsymbol{\times} \boldsymbol{\mathbf{A}} = & \left[\frac{2}{\sqrt{\xi+\eta}} \frac{\partial}{\partial{\eta}} (\sqrt{\eta}A_\phi) - \frac{1}{\sqrt{\xi\eta}} \frac{\partial A_\eta}{\partial \phi} \right] \hat{\boldsymbol{\mathbf{\xi}}} \\
&+ \left[\frac{1}{\sqrt{\xi\eta}} \frac{\partial A_\xi}{\partial \phi} - \frac{2}{\sqrt{\xi+\eta}} \frac{\partial}{\partial{\xi}} (\sqrt{\xi}A_\phi) \right] \hat{\boldsymbol{\mathbf{\eta}}} \\
&+ \left[\frac{2\sqrt{\xi}}{\xi+\eta} \frac{\partial}{\partial{\xi}} (\sqrt{\xi+\eta}A_\eta) - \frac{2\sqrt{\eta}}{\xi+\eta} \frac{\partial}{\partial{\eta}} (\sqrt{\xi+\eta}A_\xi) \right] \hat{\boldsymbol{\mathbf{\phi}}} ~,
\end{aligned}
\end{equation}
\begin{equation} \boldsymbol{\nabla}^2 u = \frac{4}{\xi + \eta} \left[ \frac{\partial u}{\partial \xi} \left(\xi \frac{\partial}{\partial{\xi}} \right) + \frac{\partial u}{\partial \eta} \left(\eta \frac{\partial}{\partial{\eta}} \right) \right] + \frac{1}{\xi\eta} \frac{\partial^{2}{u}}{\partial{\phi}^{2}} ~.
\end{equation}
1. ^ 参考 [1] Chap 3.5 和 Wikipedia 相关页面。
[1] ^ Bransden, Physics of Atoms and Molecules, 2ed