Surface harmonics and solid harmonics

(C.1) |

on the surface of a sphere (i.e., constant). Here, , , are standard spherical coordinates. It follows that

(C.2) |

where the are arbitrary coefficients, and the are spherical harmonics. (See Section A.12.)

A *solid harmonic* of degree
(where
is a non-negative integer), denoted
, is defined as a well-behaved solution
to

(C.3) |

in the interior of a sphere (i.e., the region constant). It follows that (Riley 1974c)

(C.4) |

Note that the Cartesian coordinates (where runs from to ) are solid harmonics of degree . Moreover, is a solid harmonic of degree . Here, we have employed standard tensor notation (Riley 1974e).

The following results regarding solid harmonics are helpful:

In deriving these results, use has been made of standard vector field theory (Fitzpatrick 2008). In addition, we have adopted the well-known