Parabolic coordinates explained

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 = (1)/(2) \left( \tau ^{2} - \sigma ^{2} \right)

The curves of constant

\sigma form confocal parabolae2 y = (x ^{2})/(\sigma ^{2}) - \sigma ^{2}

that open upwards (i.e., towards

+ y ), whereas the curves of constant \tau form confocal parabolae2 y =- (x ^{2})/(\tau ^{2}) + \tau ^{2}

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 = \operatorname{sign}(x)\sqrt{\sqrt{x ^{2} + y ^{2} }-y}\tau = \sqrt{\sqrt{x ^{2} + y ^{2} }+y}

Two-dimensional scale factors

The scale factors for the parabolic coordinates

(\sigma, \tau) are equalh _{\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 = (1)/(\sigma ^{2} + \tau ^{2}) \left( (\partial ^{2} \Phi)/(\partial \sigma ^{2}) + (\partial ^{2} \Phi)/(\partial \tau ^{2}) \right)

Other differential operators such as

\nabla ⋅ F and \nabla x 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

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 = (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 \varphi is defined\tan \varphi = (y)/(x)

The surfaces of constant

\sigma form confocal paraboloids2 z = (x ^{2} + y ^{2})/(\sigma ^{2}) - \sigma ^{2}

that open upwards (i.e., towards

+ z ) whereas the surfaces of constant \tau form confocal paraboloids 2 z =- (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 thendV = h _{\sigma} h _{\tau} h _{\varphi} d\sigmad\taud\varphi = \sigma\tau \left( \sigma ^{2} + \tau ^{2} \right)d\sigmad\taud\varphi

and the Laplacian is given by

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

Other differential operators such as

\nabla ⋅ F and \nabla x F 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

• Book: . 1953 . Methods of Theoretical Physics, Part I . McGraw-Hill . New York . 0-07-043316-X. 52011515 . 660.
• Book: Margenau H, Murphy GM . 1956 . The Mathematics of Physics and Chemistry . registration. D. van Nostrand . New York . 185–186 . 55010911 .
• Book: . 1961 . Mathematical Handbook for Scientists and Engineers . McGraw-Hill . New York . ASIN B0000CKZX7 . 180 . 59014456.
• Book: Sauer R, Szabó I . 1967 . Mathematische Hilfsmittel des Ingenieurs . Springer Verlag . New York . 96 . 67025285.
• Book: Zwillinger D . 1992 . Handbook of Integration . Jones and Bartlett . Boston, MA . 0-86720-293-9 . 114. Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• Book: 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 . Springer-Verlag . New York . 34–36 (Table 1.08) . 978-0-387-18430-2.

External links

• • MathWorld description of parabolic coordinates