Parabolic coordinates

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The numbers and related to rectangular Cartesian coordinates and by the formulas

where and . The coordinate lines are two systems of mutually orthogonal parabolas with oppositely-directed axes.

Figure: p071170a

The Lamé coefficients (or scale factors) are given by:

The area element is given by:

The fundamental operators of vector analysis are given by:

In parabolic coordinates the Laplace equation allows separation of variables.


Using complex functions the coordinate transformation can be described by , where and .

For parabolic coordinates in space see [a1].


[a1] R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , 1 , Springer (1967) pp. 98
[a2] L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1960) pp. 151ff (Translated from Russian)
How to Cite This Entry:
Parabolic coordinates. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098