Parabolic cylindrical coordinates

Parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in theperpendicular <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.

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.

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.

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.

References

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.

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