Parabolic coordinates

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.

Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.

Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates $(\sigma, \tau)$ are defined by the equations

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

The curves of constant $\sigma$ form confocal parabolae

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

that open upwards (i.e., towards $+y$), whereas the curves of constant $\tau$ form confocal parabolae

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

that open downwards (i.e., towards $-y$). The foci of all these parabolae are located at the origin.

Two-dimensional scale factors

The scale factors for the parabolic coordinates $(\sigma, \tau)$ are equal

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

Hence, the infinitesimal element of area is

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

and the Laplacian equals

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

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

Three-dimensional parabolic coordinates

Coordinate surfaces of the three-dimensional parabolic coordinates. The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the $z$-direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, forming a coordinate system that is also known as "parabolic coordinates"

$x = \sigma \tau \cos \varphi$
$y = \sigma \tau \sin \varphi$
$z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right)$

where the parabolae are now aligned with the $z$-axis, about which the rotation was carried out. Hence, the azimuthal angle $\phi$ is defined

$\tan \varphi = \frac{y}{x}$

The surfaces of constant $\sigma$ form confocal paraboloids

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

that open upwards (i.e., towards $+z$) whereas the surfaces of constant $\tau$ form confocal paraboloids

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

that open downwards (i.e., towards $-z$). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

$g_{ij} = \begin{bmatrix} \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0 & \sigma^2\tau^2 \end{bmatrix}$

Three-dimensional scale factors

The three dimensional scale factors are:

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

It is seen that The scale factors $h_{\sigma}$ and $h_{\tau}$ are the same as in the two-dimensional case. The infinitesimal volume element is then

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

and the Laplacian is given by

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

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

Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 660. ISBN 0-07-043316-X. LCCN 52011515.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 185–186. LCCN 55010911.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 180. LCCN 59014456. ASIN B0000CKZX7.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 96. LCCN 67025285.
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk.
• Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 34–36 (Table 1.08). ISBN 978-0-387-18430-2.