Parabolic coordinates

In green, confocal parabolae opening upwards, [math]\displaystyle{ 2y = \frac {x^2}{\sigma^2}-\sigma^2 }[/math] In red, confocal parabolae opening downwards, [math]\displaystyle{ 2y =-\frac{x^2}{\tau^2}+\tau^2 }[/math]

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 [math]\displaystyle{ (\sigma, \tau) }[/math] are defined by the equations, in terms of Cartesian coordinates:

[math]\displaystyle{ x = \sigma \tau }[/math]

[math]\displaystyle{ y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) }[/math]

The curves of constant [math]\displaystyle{ \sigma }[/math] form confocal parabolae

[math]\displaystyle{ 2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2} }[/math]

that open upwards (i.e., towards [math]\displaystyle{ +y }[/math]), whereas the curves of constant [math]\displaystyle{ \tau }[/math] form confocal parabolae

[math]\displaystyle{ 2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2} }[/math]

that open downwards (i.e., towards [math]\displaystyle{ -y }[/math]). The foci of all these parabolae are located at the origin.

The Cartesian coordinates [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] can be converted to parabolic coordinates by:

[math]\displaystyle{ \sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y} }[/math]

[math]\displaystyle{ \tau = \sqrt{\sqrt{x^{2} +y^{2}}+y} }[/math]

Two-dimensional scale factors

The scale factors for the parabolic coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] are equal

[math]\displaystyle{ h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}} }[/math]

Hence, the infinitesimal element of area is

[math]\displaystyle{ dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau }[/math]

and the Laplacian equals

[math]\displaystyle{ \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]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] 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 [math]\displaystyle{ 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]\displaystyle{ x = \sigma \tau \cos \varphi }[/math]

[math]\displaystyle{ y = \sigma \tau \sin \varphi }[/math]

[math]\displaystyle{ z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) }[/math]

where the parabolae are now aligned with the [math]\displaystyle{ z }[/math]-axis, about which the rotation was carried out. Hence, the azimuthal angle [math]\displaystyle{ \phi }[/math] is defined

[math]\displaystyle{ \tan \varphi = \frac{y}{x} }[/math]

The surfaces of constant [math]\displaystyle{ \sigma }[/math] form confocal paraboloids

[math]\displaystyle{ 2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2} }[/math]

that open upwards (i.e., towards [math]\displaystyle{ +z }[/math]) whereas the surfaces of constant [math]\displaystyle{ \tau }[/math] form confocal paraboloids

[math]\displaystyle{ 2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2} }[/math]

that open downwards (i.e., towards [math]\displaystyle{ -z }[/math]). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

[math]\displaystyle{ 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 factors

The three dimensional scale factors are:

[math]\displaystyle{ h_{\sigma} = \sqrt{\sigma^2+\tau^2} }[/math]

[math]\displaystyle{ h_{\tau} = \sqrt{\sigma^2+\tau^2} }[/math]

[math]\displaystyle{ h_{\varphi} = \sigma\tau }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\sigma, \tau, \phi) }[/math] by substituting the scale factors into the general formulae found in orthogonal coordinates.

See also

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 660. ISBN 0-07-043316-X. 
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 185–186. https://archive.org/details/mathematicsofphy0002marg. 
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 180. ASIN B0000CKZX7. 
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 96. 
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. pp. 114. ISBN 0-86720-293-9.  Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• 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. 

External links

• Hazewinkel, Michiel, ed. (2001), "Parabolic coordinates", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p071170 
• MathWorld description of parabolic coordinates

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