In Toroidal Coordinates, attempt Separation of Variables to Laplace's Equation by plugging in the
trial solution

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

Laplace's Equation is partially separable, although the Helmholtz Differential Equation is not.

**References**

Arfken, G. ``Toroidal Coordinates
.'' §2.13 in
*Mathematical Methods for Physicists, 2nd ed.* Orlando, FL: Academic Press, pp. 112-114, 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.

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

© 1996-9

1999-05-26