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.
where xi in [0,infty), eta in [0,pi], and phi in [0,2pi). Note that several conventions are in common use; Arfken (1970) uses (u,v,phi) instead of (xi,eta,phi), and Moon and Spencer (1988, p. 28) use (eta,theta,psi).
In this coordinate system, the scale factors are
The Laplacian is
![画像:del ^2f=1/(sinetasinhxi(sin^2eta+sinh^2xi)){partial/(partialxi)(sinetasinhxi(partialf)/(partialxi))+partial/(partialeta)(sinetasinhxi(partialf)/(partialeta))+partial/(partialphi)[(cschxisineta+cscetasinhxi)(partialf)/(partialphi)]}.](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/ProlateSpheroidalCoordinates/Inline25.svg){kind=link}
![画像:=1/(sin^2eta+sinh^2xi)[(csc^2eta+csch^2xi)(partial^2f)/(partialphi^2)+coteta(partialf)/(partialeta)+(partial^2f)/(partialeta^2)+cothxi(partialf)/(partialxi)+(partial^2f)/(partialxi^2)]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/ProlateSpheroidalCoordinates/Inline26.svg){kind=link}
An alternate form useful for "two-center" problems is defined by
where xi_1 in [1,infty], xi_2 in [-1,1], and xi_3 in [0,2pi) (Abramowitz and Stegun 1972). In these coordinates,
In terms of the distances from the two foci,
The scale factors are
and the Laplacian is
| del ^2f=1/(a^2){1/(xi_1^2-xi_2^2)partial/(partialxi_1)[(xi_1^2-1)(partialf)/(partialxi_1)]+1/(xi_1^2-xi_2^2)partial/(partialxi_2)[(1-xi_2^2)(partialf)/(partialxi_2)]+1/((xi_1^2-1)(1-xi_2^2))(partial^2f)/(partialxi_3^2)}. |
(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 CoordinatesExplore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. 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 (u, v, phi)." §2.10 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107, 1970.Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 243-244, 1959.Moon, P. and Spencer, D. E. "Prolate Spheroidal Coordinates (eta,theta,psi)." Table 1.06 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 28-30, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.Wrinch, D. M. "Inverted Prolate Spheroids." Philos. Mag. 280, 1061-1070, 1932.Referenced on Wolfram|Alpha
Prolate Spheroidal CoordinatesCite this as:
Weisstein, Eric W. "Prolate Spheroidal Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html