Paraboloidal coordinates

Paraboloidal coordinates are a three-dimensional orthogonal coordinate system 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.

Coordinate surfaces of the three-dimensional paraboloidal coordinates.

Basic formulae

The Cartesian coordinates can be produced from the ellipsoidal coordinates by the equations

where the following limits apply to the coordinates

Consequently, surfaces of constant are elliptic paraboloids

and surfaces of constant are likewise

whereas surfaces of constant are hyperbolic paraboloids

Scale factors

The scale factors for the paraboloidal coordinates are

Hence, the infinitesimal volume element equals

Differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.

References

    Bibliography

    External links

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