Categories: Mathematics, Physics.

Parabolic cylindrical coordinates

Parabolic cylindrical coordinates are a coordinate system that describes a point in space using three coordinates \((\sigma, \tau, z)\). The \(z\)-axis is unchanged from the Cartesian system, hence it is called a cylindrical system. In the \(z\)-isoplane, however, confocal parabolas are used. These coordinates can be converted to the Cartesian \((x, y, z)\) as follows:

\[\begin{aligned} \boxed{ x = \frac{1}{2} (\tau^2 - \sigma^2 ) \qquad y = \sigma \tau \qquad z = z } \end{aligned}\]

Converting the other way is a bit trickier. It can be done by solving the following equations, and potentially involves some fiddling with signs:

\[\begin{aligned} 2 x = \frac{y^2}{\sigma^2} - \sigma^2 \qquad \quad 2 x = - \frac{y^2}{\tau^2} + \tau^2 \end{aligned}\]

Parabolic cylindrical coordinates form an orthogonal curvilinear system, so we would like to find its scale factors \(h_\sigma\), \(h_\tau\) and \(h_z\). The differentials of the Cartesian coordinates are as follows:

\[\begin{aligned} \dd{x} = - \sigma \dd{\sigma} + \tau \dd{\tau} \qquad \dd{y} = \tau \dd{\sigma} + \sigma \dd{\tau} \qquad \dd{z} = \dd{z} \end{aligned}\]

We calculate the line segment \(\dd{\ell}^2\), skipping many terms thanks to orthogonality:

\[\begin{aligned} \dd{\ell}^2 &= (\sigma^2 + \tau^2) \:\dd{\sigma}^2 + (\tau^2 + \sigma^2) \:\dd{\tau}^2 + \dd{z}^2 \end{aligned}\]

From this, we can directly read off the scale factors \(h_\sigma^2\), \(h_\tau^2\) and \(h_z^2\), which turn out to be:

\[\begin{aligned} \boxed{ h_\sigma = \sqrt{\sigma^2 + \tau^2} \qquad h_\tau = \sqrt{\sigma^2 + \tau^2} \qquad h_z = 1 } \end{aligned}\]

With these scale factors, we can use the general formulae for orthogonal curvilinear coordinates to easily to convert things from the Cartesian system. The basis vectors are:

\[\begin{aligned} \boxed{ \begin{aligned} \vu{e}_\sigma &= \frac{- \sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y \\ \vu{e}_\tau &= \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y \\ \vu{e}_z &= \vu{e}_z \end{aligned} } \end{aligned}\]

The basic vector operations (gradient, divergence, Laplacian and curl) are given by:

\[\begin{aligned} \boxed{ \nabla f = \frac{\mathbf{e}_\sigma}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\sigma} + \frac{\mathbf{e}_\tau}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\tau} + \mathbf{e}_z \pdv{f}{z} } \end{aligned}\]

\[\begin{aligned} \boxed{ \nabla \cdot \mathbf{V} = \frac{1}{\sigma^2 + \tau^2} \Big( \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{d\sigma} + \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{d\tau} \Big) + \pdv{V_z}{z} } \end{aligned}\]

\[\begin{aligned} \boxed{ \nabla^2 f = \frac{1}{\sigma^2 + \tau^2} \Big( \pdv[2]{f}{\sigma} + \pdv[2]{f}{\tau} \Big) + \pdv[2]{f}{z} } \end{aligned}\]

\[\begin{aligned} \boxed{ \begin{aligned} \nabla \times \mathbf{V} &= \mathbf{e}_\sigma \Big( \frac{\mathbf{e}_1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \Big) \\ &+ \mathbf{e}_\tau \Big( \pdv{V_\sigma}{z} - \frac{1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\sigma} \Big) \\ &+ \frac{\mathbf{e}_z}{\sigma^2 + \tau^2} \Big( \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\sigma} - \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) \end{aligned} } \end{aligned}\]

The differential element of volume \(\dd{V}\) in parabolic cylindrical coordinates is given by:

\[\begin{aligned} \boxed{ \dd{V} = (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} \dd{z} } \end{aligned}\]

The differential elements of the isosurfaces are as follows, where \(\dd{S_\sigma}\) is the \(\sigma\)-isosurface, etc.:

\[\begin{aligned} \boxed{ \begin{aligned} \dd{S_\sigma} &= \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} \\ \dd{S_\tau} &= \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} \\ \dd{S_z} &= (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} \end{aligned} } \end{aligned}\]

The normal element \(\dd{\vu{S}}\) of a surface and the tangent element \(\dd{\vu{\ell}}\) of a curve are respectively:

\[\begin{aligned} \boxed{ \dd{\vu{S}} = \mathbf{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} + \mathbf{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} + \mathbf{e}_z (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} } \end{aligned}\]

\[\begin{aligned} \boxed{ \dd{\vu{\ell}} = \mathbf{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\sigma} + \mathbf{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\tau} + \mathbf{e}_z \dd{z} } \end{aligned}\]

References

  1. M.L. Boas, Mathematical methods in the physical sciences, 2nd edition, Wiley.

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