Oblate Spheroidal Coordinates -- from Wolfram MathWorld

Oblate 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 y-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

del ^2f=1/(a^3(sinh^2xi+sin^2eta)coshxicoseta)[(partialf)/(partialxi)(acoshxicoseta(partialf)/(partialxi))+(partialf)/(partialeta)(acoshxicoseta(partialf)/(partialeta))+(a^2(sinh^2xi+sin^2eta))/(acoshxicoseta)(partial^2f)/(partialphi^2)]
=1/(a^3(sinh^2xi+sin^2eta)coshxicoseta)[asinhxicoseta(partialf)/(partialxi)+acoshxicoseta(partial^2f)/(partialxi^2)+asinhxicoseta(partialf)/(partialeta)+acoshxicoseta(partial^2f)/(partialeta^2)]+1/(a^2(sinh^2xi+sin^2eta))(partial^2f)/(partialphi^2)
=1/(a^2(sinh^2xi+sin^2eta))[1/(coshxi)partial/(partialxi)(coshxi(partialf)/(partialxi))+1/(coseta)partial/(partialeta)(coseta(partialf)/(partialeta))]+1/(a^2(cosh^2xi+cos^2eta))(partial^2f)/(partialphi^2)
=1/(sin^2eta+sinh^2xi)[(sech^2xitan^2eta+sec^2tanh^2xi)(partial^2)/(partialphi^2)+tanhxipartial/(partialxi)+(partial^2)/(partialxi^2)-tanetapartial/eta+(partial^2)/(eta^2)].

(7)

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

			(8)
			(9)
			(10)
			(11)

where , , and . In these coordinates,

			(12)
			(13)
			(14)

(Abramowitz and Stegun 1972). The scale factors are

			(15)
			(16)
			(17)

and the Laplacian is

$del ^2f=1/(a^2){1/(xi_1^2+xi_2^2)partial/(partialxi_1)[(xi_1^2+1)(partialf)/(partialxi_1)]+1/(xi_1^2+xi_2^2)partial/(partialxi_2)[(1-xi_2^2)(partialf)/(partialxi_2)]+1/((xi_1^2+1)(1-xi_2^2))(partial^2f)/(partialxi_3^2)}.$

(18)

The Helmholtz differential equation is separable.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Definition of Oblate 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 (, , )." §2.11 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 107-109, 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, p. 242, 1959.

Moon, P. and Spencer, D. E. "Oblate Spheroidal Coordinates ." Table 1.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 31-34, 1988.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.

CITE THIS AS:

Weisstein, Eric W. "Oblate Spheroidal Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OblateSpheroidalCoordinates.html