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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.
where 
,
![eta in [-pi/2,pi/2]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOblateSpheroidalCoordinates%2FInline11.svg){kind=small}
,
and 
.
Arfken (1970) uses 
instead of 
. The scale factors
are
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)].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOblateSpheroidalCoordinates%2FNumberedEquation1.svg) |
(7) |
An alternate form useful for "two-center" problems is defined by
where ![xi_1 in [1,infty]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOblateSpheroidalCoordinates%2FInline36.svg){kind=small}
,
![xi_2 in [-1,1]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOblateSpheroidalCoordinates%2FInline37.svg){kind=small}
,
and 
.
In these coordinates,
(Abramowitz and Stegun 1972). 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)}.](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOblateSpheroidalCoordinates%2FNumberedEquation2.svg) |
(18) |
The Helmholtz differential equation is separable.
See also
Helmholtz Differential Equation--Oblate Spheroidal Coordinates, Latitude, Longitude, Prolate Spheroidal Coordinates, Spherical Coordinates
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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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Oblate Spheroidal Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OblateSpheroidalCoordinates.html