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Toroidal Coordinates

\begin{figure}\begin{center}\BoxedEPSF{ToroidalCoordinates.epsf scaled 1000}\end{center}\end{figure}

A system of Curvilinear Coordinates for which several different notations are commonly used. In this work ($u$, $v$, $\phi$) is used, whereas Arfken (1970) uses ($\xi$, $\eta$, $\varphi$). The toroidal coordinates are defined by

$\displaystyle x$ $\textstyle =$ $\displaystyle {a\sinh u\cos\phi\over\cosh u-\cos v}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {a\sinh u\sin\phi\over\cosh u-\cos v}$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle {a\sin v\over \cosh u-\cos v},$ (3)

where $\sinh z$ is the Hyperbolic Sine and $\cosh z$ is the Hyperbolic Cosine. The Scale Factors are
$\displaystyle h_u$ $\textstyle =$ $\displaystyle {a\over\cosh u-\cos v}$ (4)
$\displaystyle h_v$ $\textstyle =$ $\displaystyle {a\over\cosh u-\cos v}$ (5)
$\displaystyle h_\phi$ $\textstyle =$ $\displaystyle {a\sinh u\over\cosh u-\cos v}.$ (6)

The Laplacian is

$\nabla^2f={\sinh u\over(\cosh u-\cos v)^3}\left[{{\partial\over\partial u}\left({{\sinh u\over\cosh u-\cos v}{\partial f\over\partial u}}\right)}\right.$
$ +\left.{{\partial\over\partial v}\left({{\sinh u\over\cosh u-\cos v}{\partial ...
...olimits u\over\cosh u-\cos v}{\partial f\over\partial\phi}}\right)}\right]\quad$ (7)
$= (\cos v-\cosh u)\left[{\sin v{\partial f\over\partial v}+(\cos v-\cosh u)\lef...
...partial^2 f\over\partial\phi^2}+{\partial^2 f\over\partial v^2}}\right)}\right.$
$ +\left.{(\cos v\cosh u-1)\mathop{\rm csch}\nolimits u{\partial f\over\partial u}+(\cos v-\cosh u){\partial^2f\over\partial u^2}}\right].\quad$ (8)
The Helmholtz Differential Equation is not separable in toroidal coordinates, but Laplace's Equation is.

See also Bispherical Coordinates, Laplace's Equation--Toroidal Coordinates


Arfken, G. ``Toroidal Coordinates ($\xi$, $\eta$, $\phi$).'' §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.

© 1996-9 Eric W. Weisstein