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:
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).