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.
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)|
Parabolic coordinates. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Parabolic_coordinates&oldid=15697