Parabolic coordinates
From Encyclopedia of Mathematics
The numbers and
related to rectangular Cartesian coordinates
and
by the formulas
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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:
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The area element is given by:
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The fundamental operators of vector analysis are given by:
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In parabolic coordinates the Laplace equation allows separation of variables.
Comments
Using complex functions the coordinate transformation can be described by , where
and
.
For parabolic coordinates in space see [a1].
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_coordinates&oldid=48105
Parabolic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_coordinates&oldid=48105
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article