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.

Basic definition

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The parabolic cylindrical coordinates $(\backslash sigma,\; \backslash tau,\; z)$ are defined

$$

x = \sigma \tau

$$

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

$$

z = z

The surfaces of constant $\backslash sigma$ form confocal parabolic cylinders

$$

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

that open towards $+y$, whereas the surfaces of constant $\backslash tau$ 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 $x=y=0$.

Scale factors

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The scale factors for the parabolic cylindrical coordinates $\backslash sigma$ and $\backslash tau$ 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

$$

dA = \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 $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$ and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash sigma,\; \backslash tau)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Applications

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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

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

Coordinate systems