## Toroidal Coordinates

A system of Curvilinear Coordinates for which several different notations are commonly used. In this work (, , ) is used, whereas Arfken (1970) uses (, , ). The toroidal coordinates are defined by

 (1) (2) (3)

where is the Hyperbolic Sine and is the Hyperbolic Cosine. The Scale Factors are
 (4) (5) (6)

The Laplacian is

 (7) (8)
The Helmholtz Differential Equation is not separable in toroidal coordinates, but Laplace's Equation is.

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