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A system of curvilinear coordinates for which several different notations are commonly used. In this work 
is used, whereas Arfken (1970) uses 
and Moon and Spencer (1988) use 
. The toroidal coordinates are defined by
where 
is the hyperbolic sine and 
is the hyperbolic cosine.
The coordinates satisfy 
, 
, and 
.
Surfaces of constant 
are given by the toroids
 |
(4) |
surfaces of constant 
by the spherical bowls
 |
(5) |
spheres centered at 
with radii 
 |
(6) |
and surfaces of constant 
by
 |
(7) |
The scale factors are
The Laplacian is
![del ^2=(cschv(coshv-cosu)^3)/(a^2)[partial/(partialu)((sinhv)/(coshv-cosu)partial/(partialu))+partial/(partialv)((sinhv)/(coshv-cosu)partial/(partialv))+partial/(partialphi)((cschv)/(coshv-cosu)partial/(partialphi))].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FToroidalCoordinates%2FNumberedEquation5.svg) |
(11) |
The Helmholtz differential equation is not separable in toroidal coordinates, but Laplace's equation is.
See also
Bispherical Coordinates, Flat-Ring Cyclide Coordinates, Laplace's Equation--Toroidal Coordinates
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References
Arfken, G. "Toroidal Coordinates (
, 
, 
)." ยง2.13 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115,
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. 264, 1959.Moon, P. and Spencer, D. E. "Toroidal
Coordinates 
."
Fig. 4.04 in Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, pp. 112-115, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Toroidal Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ToroidalCoordinates.html