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Oblate Spheroidal Coordinates

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OblateSpheroidalCoords3DOblateSpheroidalCoordinate

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.

x=acoshxicosetacosphi
(1)
y=acoshxicosetasinphi
(2)
z=asinhxisineta,
(3)

where xi in [0,infty), eta in [-pi/2,pi/2], and phi in [0,2pi). Arfken (1970) uses (u,v,phi) instead of (xi,eta,phi). The scale factors are

h_xi=asqrt(sinh^2xi+sin^2eta)
(4)
h_eta=asqrt(sinh^2xi+sin^2eta)
(5)
h_phi=acoshxicoseta.
(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

xi_1=sinhxi
(8)
xi_1^'=coshxi
(9)
xi_2=coseta
(10)
xi_3=phi,
(11)

where xi_1 in [1,infty], xi_2 in [-1,1], and xi_3 in [0,2pi). In these coordinates,

y=axi_1^'xi_2sinxi_3
(12)
z=asqrt((xi_1^'^2-1)(1-xi_2^2))
(13)
x=axi_1^'xi_2cosxi_3
(14)

(Abramowitz and Stegun 1972). The scale factors are

h_(xi_1)=asqrt((xi_1^2-xi_2^2)/(xi_1^2-1))
(15)
h_(xi_2)=asqrt((xi_1^2-xi_2^2)/(1-xi_2^2))
(16)
h_(xi_3)=axieta,
(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.

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 (u, v, phi)." §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 (eta,theta,psi)." 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.

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Oblate Spheroidal Coordinates

Cite this as:

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

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