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 coordinatesEdit
Two-dimensional parabolic coordinates <math>(\sigma, \tau)</math> are defined by the equations, in terms of Cartesian coordinates:
- <math>
x = \sigma \tau </math>
- <math>
y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) </math>
The curves of constant <math>\sigma</math> form confocal parabolae
- <math>
2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2} </math>
that open upwards (i.e., towards <math>+y</math>), whereas the curves of constant <math>\tau</math> form confocal parabolae
- <math>
2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2} </math>
that open downwards (i.e., towards <math>-y</math>). The foci of all these parabolae are located at the origin.
The Cartesian coordinates <math>x</math> and <math>y</math> can be converted to parabolic coordinates by:
- <math>
\sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y} </math>
- <math>
\tau = \sqrt{\sqrt{x^{2} +y^{2}}+y} </math>
Two-dimensional scale factorsEdit
The scale factors for the parabolic coordinates <math>(\sigma, \tau)</math> are equal
- <math>
h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}} </math>
Hence, the infinitesimal element of area is
- <math>
dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau </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) </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.
Three-dimensional parabolic coordinatesEdit
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 <math>z</math>-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:
- <math>
x = \sigma \tau \cos \varphi </math>
- <math>
y = \sigma \tau \sin \varphi </math>
- <math>
z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) </math>
where the parabolae are now aligned with the <math>z</math>-axis, about which the rotation was carried out. Hence, the azimuthal angle <math>\varphi</math> is defined
- <math>
\tan \varphi = \frac{y}{x} </math>
The surfaces of constant <math>\sigma</math> form confocal paraboloids
- <math>
2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2} </math>
that open upwards (i.e., towards <math>+z</math>) whereas the surfaces of constant <math>\tau</math> form confocal paraboloids
- <math>
2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2} </math>
that open downwards (i.e., towards <math>-z</math>). The foci of all these paraboloids are located at the origin.
The Riemannian metric tensor associated with this coordinate system is
- <math> g_{ij} = \begin{bmatrix} \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0 & \sigma^2\tau^2 \end{bmatrix} </math>
Three-dimensional scale factorsEdit
The three dimensional scale factors are:
- <math>h_{\sigma} = \sqrt{\sigma^2+\tau^2}</math>
- <math>h_{\tau} = \sqrt{\sigma^2+\tau^2}</math>
- <math>h_{\varphi} = \sigma\tau</math>
It is seen that the scale factors <math>h_{\sigma}</math> and <math>h_{\tau}</math> are the same as in the two-dimensional case. The infinitesimal volume element is then
- <math>
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 </math>
and the Laplacian is given by
- <math>
\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} </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, \phi)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.
See alsoEdit
BibliographyEdit
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- Template:Cite book Same as Morse & Feshbach (1953), substituting uk for ξk.
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