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, in terms of Cartesian coordinates:

:$x = \sigma \tau$

:$y = \frac \left( \tau^ - \sigma^ \right)$

The curves of constant $\sigma$ form confocal parabolae

:$2y = \frac - \sigma^$

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

:$2y = -\frac + \tau^$

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

The Cartesian coordinates $x$ and $y$ can be converted to parabolic coordinates by:
:$\sigma = \sqrt$

:$\tau = \sqrt.$

Two-dimensional scale factors

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

:$h_ = h_ = \sqrt$

Hence, the infinitesimal element of area is

:$dA = \left( \sigma^ + \tau^ \right) d\sigma d\tau$

and the Laplacian equals

:$\nabla^ \Phi = \frac \left( \frac + \frac \right)$

Other differential operators such as $\nabla \cdot \mathbf$
and $\nabla \times \mathbf$ 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

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, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

:$x = \sigma \tau \cos \varphi$

:$y = \sigma \tau \sin \varphi$

:$z = \frac \left(\tau^ - \sigma^ \right)$

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

:$\tan \varphi = \frac$

The surfaces of constant $\sigma$ form confocal paraboloids

:$2z = \frac - \sigma^$

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

:$2z = -\frac + \tau^$

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

:$g_ = \begin \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0 & \sigma^2\tau^2 \end$

Three-dimensional scale factors

The three dimensional scale factors are:

:$h_ = \sqrt$
:$h_ = \sqrt$
:$h_ = \sigma\tau$

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

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

and the Laplacian is given by

:
\nabla^2 \Phi = \frac
\left \frac \frac
\left( \sigma \frac \right) +
\frac \frac
\left( \tau \frac \right)\right+
\frac\frac

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

See also

* Parabolic cylindrical coordinates
* Orthogonal coordinate system
* Curvilinear coordinates

Bibliography

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* Same as Morse & Feshbach (1953), substituting ''u''_''k'' for ξ_''k''.
*

External links

*
MathWorld description of parabolic coordinates

{{Orthogonal coordinate systems
Orthogonal coordinate systems