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

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.

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 Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates ${\displaystyle{\displaystyle(\sigma,\tau)}}$ are defined by the equations, in terms of Cartesian coordinates:

${\displaystyle{\displaystyle x=\sigma\tau}}$

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

The curves of constant ${\displaystyle{\displaystyle\sigma}}$ form confocal parabolae

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

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

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

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

The Cartesian coordinates ${\displaystyle{\displaystyle x}}$ and ${\displaystyle{\displaystyle y}}$ can be converted to parabolic coordinates by:

${\displaystyle{\displaystyle\sigma=\operatorname{sign}(x){\sqrt{{\sqrt{x^{2}+y^{2}}}-y}}}}$

${\displaystyle{\displaystyle\tau={\sqrt{{\sqrt{x^{2}+y^{2}}}+y}}}}$

 Two-dimensional scale factors

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

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

Hence, the infinitesimal element of area is

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

and the Laplacian equals

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

Other differential operators such as ${\displaystyle{\displaystyle\nabla\cdot\mathbf{F}}}$ and ${\displaystyle{\displaystyle\nabla\times\mathbf{F}}}$ can be expressed in the coordinates ${\displaystyle{\displaystyle(\sigma,\tau)}}$ 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 ${\displaystyle{\displaystyle z}}$-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:

${\displaystyle{\displaystyle x=\sigma\tau\cos\varphi}}$

${\displaystyle{\displaystyle y=\sigma\tau\sin\varphi}}$

${\displaystyle{\displaystyle z={\frac{1}{2}}\left(\tau^{2}-\sigma^{2}\right)}}$

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

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

The surfaces of constant ${\displaystyle{\displaystyle\sigma}}$ form confocal paraboloids

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

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

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

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

The Riemannian metric tensor associated with this coordinate system is

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

 Three-dimensional scale factors

The three dimensional scale factors are:

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

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

${\displaystyle{\displaystyle h_{\varphi}=\sigma\tau}}$

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

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

and the Laplacian is given by

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

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

 See also

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

 Bibliography

• 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 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", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• MathWorld description of parabolic coordinates