Paraboloidal coordinates

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Paraboloidal coordinates are a three-dimensional orthogonal coordinate system $(λ,μ,ν)$ that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

1 Basic formulae
2 Scale factors
3 See also
4 References
5 Bibliography
6 External links

[edit] Basic formulae

The Cartesian coordinates $(x, y, z)$ can be produced from the ellipsoidal coordinates $(λ,μ,ν)$ by the equations

$x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}$

$y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}$

$z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)$

where the following limits apply to the coordinates

λ < B < μ < A < ν

Consequently, surfaces of constant $λ$ are elliptic paraboloids

$\frac{x^{2}}{\lambda - A} + \frac{y^{2}}{\lambda - B} = 2z + \lambda$

and surfaces of constant $ν$ are likewise

$\frac{x^{2}}{\nu - A} + \frac{y^{2}}{\nu - B} = 2z + \nu$

whereas surfaces of constant $μ$ are hyperbolic paraboloids

$\frac{x^{2}}{\mu - A} + \frac{y^{2}}{\mu - B} = 2z + \mu$

[edit] Scale factors

The scale factors for the paraboloidal coordinates $(λ,μ,ν)$ are

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}$

$h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}$

$h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}$

Hence, the infinitesimal volume element equals

$dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu - B \right) }} \ d\lambda d\mu d\nu$

Differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(λ,μ,ν)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] See also

Orthogonal coordinates
Two dimensional orthogonal coordinate systems

Three dimensional orthogonal coordinate systems

Elliptic cylindrical coordinates
Toroidal coordinates
Bispherical coordinates
Bipolar cylindrical coordinates
Conical coordinates
Flat-Ring cyclide coordinates
Flat-Disk cyclide coordinates
Bi-cyclide coordinates
Cap-cyclide coordinates

[edit] References

[edit] Bibliography

Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 664. ISBN 0-07-043316-X, LCCN 52-11515.

Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand, pp. 184–185. LCCN 55-10911.

Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 180. LCCN 59-14456, ASIN B0000CKZX7.

Arfken G (1970). Mathematical Methods for Physicists, 2nd ed., Orlando, FL: Academic Press, pp. 119–120.

Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, p. 98. LCCN 67-25285.

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). "Paraboloidal Coordinates (μ, ν, λ)", Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, corrected 2nd ed., 3rd print ed., New York: Springer-Verlag, pp. 44–48 (Table 1.11). ISBN 978-0387184302.