Parabolic cylindrical coordinates



In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular $$z$$-direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.

Basic definition


The parabolic cylindrical coordinates $(σ, τ, z)$ are defined in terms of the Cartesian coordinates $(x, y, z)$ by:


 * $$\begin{align}

x &= \sigma \tau \\ y &= \frac{1}{2} \left( \tau^2 - \sigma^2 \right) \\ z &= z \end{align}$$

The surfaces of constant $σ$ form confocal parabolic cylinders



2 y = \frac{x^2}{\sigma^2} - \sigma^2 $$

that open towards $+y$, whereas the surfaces of constant $τ$ form confocal parabolic cylinders



2 y = -\frac{x^2}{\tau^2} + \tau^2 $$

that open in the opposite direction, i.e., towards $−y$. The foci of all these parabolic cylinders are located along the line defined by $x = y = 0$. The radius $r$ has a simple formula as well



r = \sqrt{x^2 + y^2} = \frac{1}{2} \left( \sigma^2 + \tau^2 \right) $$

that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article.

Scale factors
The scale factors for the parabolic cylindrical coordinates $σ$ and $τ$ are:


 * $$\begin{align}

h_\sigma &= h_\tau = \sqrt{\sigma^2 + \tau^2} \\ h_z &= 1 \end{align}$$

Differential elements
The infinitesimal element of volume is


 * $$dV = h_\sigma h_\tau h_z d\sigma d\tau dz = ( \sigma^2 + \tau^2 ) d\sigma \, d\tau \, dz

$$

The differential displacement is given by:


 * $$d\mathbf{l} = \sqrt{\sigma^2 + \tau^2} \, d\sigma \, \boldsymbol{\hat{\sigma}} + \sqrt{\sigma^2 + \tau^2} \, d\tau \, \boldsymbol{\hat{\tau}} + dz \, \mathbf{\hat{z}}$$

The differential normal area is given by:


 * $$d\mathbf{S} = \sqrt{\sigma^2 + \tau^2}      \, d\tau   \, dz  \boldsymbol{\hat{\sigma}} + \sqrt{\sigma^2 + \tau^2}       \, d\sigma \, dz    \boldsymbol{\hat{\tau}} + \left(\sigma^2 + \tau^2\right) \, d\sigma \, d\tau \mathbf{\hat{z}}

$$

Del
Let $f$ be a scalar field. The gradient is given by


 * $$\nabla f = \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat{\sigma}} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat{\tau}} + {\partial f \over \partial z}\mathbf{\hat{z}}$$

The Laplacian is given by


 * $$\nabla^2 f = \frac{1}{\sigma^{2} + \tau^{2}}

\left(\frac{\partial^{2} f}{\partial \sigma^{2}} + \frac{\partial^{2} f}{\partial \tau^{2}} \right) + \frac{\partial^{2} f}{\partial z^{2}} $$

Let $A$ be a vector field of the form:


 * $$\mathbf A = A_\sigma \boldsymbol{\hat{\sigma}} + A_\tau \boldsymbol{\hat{\tau}} + A_z \mathbf{\hat{z}}$$

The divergence is given by


 * $$\nabla \cdot \mathbf A = \frac{1}{\sigma^{2} + \tau^{2}}\left({\partial (\sqrt{\sigma^2+\tau^2} A_\sigma) \over \partial \sigma} + {\partial (\sqrt{\sigma^2+\tau^2} A_\tau) \over \partial \tau}\right) + {\partial A_z \over \partial z}$$

The curl is given by


 * $$\nabla \times \mathbf A =

\left(   \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau}  - \frac{\partial A_\tau}{\partial z}  \right) \boldsymbol{\hat{\sigma}} - \left(   \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma}  - \frac{\partial A_\sigma}{\partial z}  \right) \boldsymbol{\hat{\tau}} + \frac{1}{\sigma^2 + \tau^2} \left(   \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\tau \right)}{\partial \sigma}  - \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\sigma\right)}{\partial \tau}  \right) \mathbf{\hat{z}} $$

Other differential operators can be expressed in the coordinates $(σ, τ)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Relationship to other coordinate systems
Relationship to cylindrical coordinates $(ρ, φ, z)$:


 * $$\begin{align}

\rho\cos\varphi &= \sigma \tau\\ \rho\sin\varphi &= \frac{1}{2} \left( \tau^2 - \sigma^2 \right) \\ z &= z \end{align}$$

Parabolic unit vectors expressed in terms of Cartesian unit vectors:


 * $$\begin{align}

\boldsymbol{\hat{\sigma}} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \boldsymbol{\hat{\tau}} &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \mathbf{\hat{z}} &= \mathbf{\hat{z}} \end{align}$$

Parabolic cylinder harmonics
Since all of the surfaces of constant $σ$, $τ$ and $z$ are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:


 * $$V = S(\sigma) T(\tau) Z(z)$$

and Laplace's equation, divided by $V$, is written:


 * $$\frac{1}{\sigma^2 + \tau^2} \left[\frac{\ddot{S}}{S} + \frac{\ddot{T}}{T}\right] + \frac{\ddot{Z}}{Z} = 0$$

Since the $Z$ equation is separate from the rest, we may write


 * $$\frac{\ddot{Z}}{Z}=-m^2$$

where $m$ is constant. $Z(z)$ has the solution:


 * $$Z_m(z)=A_1\,e^{imz}+A_2\,e^{-imz}$$

Substituting $−m^{2}$ for $$\ddot{Z} / Z$$, Laplace's equation may now be written:


 * $$\left[\frac{\ddot{S}}{S} + \frac{\ddot{T}}{T}\right] = m^2 (\sigma^2 + \tau^2)$$

We may now separate the $S$ and $T$ functions and introduce another constant $n^{2}$ to obtain:


 * $$\ddot{S} - (m^2\sigma^2 + n^2) S = 0$$
 * $$\ddot{T} - (m^2\tau^2 - n^2) T = 0$$

The solutions to these equations are the parabolic cylinder functions


 * $$S_{mn}(\sigma) = A_3 y_1(n^2 / 2m, \sigma \sqrt{2m}) + A_4 y_2(n^2 / 2m, \sigma \sqrt{2m})$$
 * $$T_{mn}(\tau)  = A_5 y_1(n^2 / 2m, i \tau \sqrt{2m}) + A_6 y_2(n^2 / 2m, i \tau \sqrt{2m})$$

The parabolic cylinder harmonics for $(m, n)$ are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:


 * $$V(\sigma, \tau, z) = \sum_{m, n} A_{mn} S_{mn} T_{mn} Z_m$$

Applications
The classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate.