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 $z$ -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 $(σ,τ, z)$ are defined

x = στ

$y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)$

z = z

The surfaces of constant $σ$ form confocal parabolic cylinders

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

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

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

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.

[edit] Scale factors

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

$h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}$

whereas the remaining scale factor is $h z = 1$ . Hence, the infinitesimal element of volume is

$dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz$

and the Laplacian equals

$\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}$

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(σ,τ)$ 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.