There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer (1988). The 
system defined in the Wolfram Language is
where 
.
Byerly (1959) uses a 
system which is essentially the same coordinate system
as above, but replacing 
with 
, 
with 
, and 
with 
. Moon and Spencer (1988) use 
instead of 
.
The above equations give
 |
(4) |
 |
(5) |
 |
(6) |
The scale factors are
The Laplacian is
 |
(10) |
The Helmholtz differential equation is separable in conical coordinates.
See also
Helmholtz Differential Equation--Conical Coordinates
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References
Arfken, G. "Conical Coordinates (
, 
, 
)." ยง2.16 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118-119,
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. 263, 1959.Moon, P. and Spencer, D. E. "Conical
Coordinates 
."
Table 1.09 in Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, pp. 37-40, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 659, 1953.Spence,
R. D. "Angular Momentum in Sphero-Conal Coordinates." Amer. J.
Phys. 27, 329-335, 1959.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Conical Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConicalCoordinates.html