# Surface harmonics and solid harmonics

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

 (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:

 (C.5) (C.6) (C.7)

In deriving these results, use has been made of standard vector field theory (Fitzpatrick 2008). In addition, we have adopted the well-known Einstein summation convention that repeated indices are implicitly summed from 1 to 3 (Riley 1974e).