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
(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 ( , , )." §2.13 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 Toroidal Coordinates
Cite this as:
Weisstein, Eric W. "Toroidal Coordinates."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ToroidalCoordinates.html
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