Paraboloidal coordinates

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Numbers , and related to the rectangular Cartesian coordinates , and by the formulas

where , , . The coordinate surfaces are two systems of paraboloids of revolution with oppositely-directed axes ( and ) and half-planes (). The system of paraboloidal coordinates is orthogonal.

The Lamé coefficients (or scale factors) are

The element of surface area is

The volume element is

The fundamental operations of vector analysis are


These coordinates are also called rotation parabolic coordinates.


[a1] R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , 1 , Springer (1967) pp. 96
How to Cite This Entry:
Paraboloidal coordinates. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article