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 ,
,
and .
Arfken (1970) uses instead of . The scale factors
are
The Laplacian is
(7)
An alternate form useful for "two-center" problems is defined by
where ,
,
and .
In these coordinates,
(Abramowitz and Stegun 1972). The scale factors are
and the Laplacian is
(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 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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