## Prolate Spheroidal Coordinates

A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the Elliptic Cylindrical Coordinates about the x-Axis, which is relabeled the z-Axis. The third set of coordinates consists of planes passing through this axis.

 (1) (2) (3)

where , , and . Arfken (1970) uses instead of . The Scale Factors are
 (4) (5) (6)

The Laplacian is

 (7) (8)

An alternate form useful for two-center'' problems is defined by

 (9) (10) (11)

where , , and (Abramowitz and Stegun 1972). In these coordinates,
 (12) (13) (14)

In terms of the distances from the two Foci,
 (15) (16) (17)

The Scale Factors are
 (18) (19) (20)

and the Laplacian is

 (21)
The Helmholtz Differential Equation is separable in prolate spheroidal coordinates.

See also Helmholtz Differential Equation--Prolate Spheroidal Coordinates, Latitude, Longitude, Oblate Spheroidal Coordinates, Spherical Coordinates

References

Abramowitz, M. and Stegun, C. A. (Eds.). Definition of Prolate Spheroidal Coordinates.'' §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.

Arfken, G. Prolate Spheroidal Coordinates (, , ).'' §2.10 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.

© 1996-9 Eric W. Weisstein
1999-05-26