Surface harmonics and solid harmonics
A surface harmonic of degree (where is a nonnegative integer), denoted
, is defined as a wellbehaved 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 nonnegative integer), denoted
, is defined as a wellbehaved 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 wellknown Einstein summation convention that repeated indices are implicitly summed from 1 to 3 (Riley 1974e).