Parabolic cylindrical coordinates

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Parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular <math>z</math>-direction. Hence, the coordinate surfaces are confocal parabolic cylinders.

Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of the edges.

1 Basic definition
2 Scale factors
3 Applications
4 References

[edit] Basic definition

The parabolic cylindrical coordinates <math>(\sigma, \tau, z)</math> are defined

<math>

x = \sigma \tau </math>

<math>

y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) </math>

<math>

z = z </math>

The surfaces of constant <math>\sigma</math> form confocal parabolic cylinders

<math>

2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2} </math>

that open towards <math>+y</math>, whereas the surfaces of constant <math>\tau</math> form confocal parabolic cylinders

<math>

2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2} </math>

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

<math>

r = \sqrt{x^{2} + y^{2}} = \frac{1}{2} \left( \sigma^{2} + \tau^{2} \right) </math>

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.

[edit] Scale factors

The scale factors for the parabolic cylindrical coordinates <math>\sigma</math> and <math>\tau</math> are equal

<math>

h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}} </math>

whereas the remaining scale factor is <math>h_{z}=1</math>. Hence, the infinitesimal element of volume is

<math>

dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz </math>

and the Laplacian equals

<math>

\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}} </math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] 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.