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Surface harmonics and solid harmonics

A surface harmonic of degree $ n$ (where $ n$ is a non-negative integer), denoted $ {\cal S}_n(\theta,\phi)$ , is defined as a well-behaved solution to

$\displaystyle r^{\,2}\,\nabla^{\,2} {\cal S}_n + n\,(n+1)\,{\cal S}_n = 0$ (C.1)

on the surface of a sphere (i.e., $ r=$ constant). Here, $ r$ , $ \theta $ , $ \phi $ are standard spherical coordinates. It follows that

$\displaystyle {\cal S}_n(\theta,\phi) = \sum_{m=-n,+n}\,\alpha_{n,m}\,Y_n^{\,m}(\theta,\phi),$ (C.2)

where the $ \alpha_{n,m}$ are arbitrary coefficients, and the $ Y_n^{\,m}(\theta,\phi)$ are spherical harmonics. (See Section A.12.)

A solid harmonic of degree $ n$ (where $ n$ is a non-negative integer), denoted $ {\cal R}_n(r,\theta,\phi)$ , is defined as a well-behaved solution to

$\displaystyle \nabla^{\,2} {\cal R}_n = 0$ (C.3)

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

$\displaystyle {\cal R}_n(r,\theta,\phi) \propto r^{\,n}\,{\cal S}_n(\theta,\phi).$ (C.4)

Note that the Cartesian coordinates $ x_i$ (where $ i$ runs from $ 1$ to $ 3$ ) are solid harmonics of degree $ 1$ . Moreover, $ \partial {\cal R}_n/\partial x_i$ is a solid harmonic of degree $ n-1$ . Here, we have employed standard tensor notation (Riley 1974e).

The following results regarding solid harmonics are helpful:

$\displaystyle x_i\,\frac{\partial {\cal R}_n}{\partial x_i}$ $\displaystyle = r\,\frac{\partial {\cal R}_n}{\partial r} = n\,{\cal R}_n,$ (C.5)
$\displaystyle \nabla^{\,2}(x_i\,{\cal R}_n)$ $\displaystyle = \nabla\cdot({\cal R}_n\,\nabla x_i + x_i\,\nabla{\cal R}_n) = 2\,\nabla x_i\cdot\nabla{\cal R}_n = 2\,\frac{\partial {\cal R}_n}{\partial x_i},$ (C.6)
$\displaystyle \nabla^{\,2}(r^{\,m}\,{\cal R}_n)$ $\displaystyle = \nabla^{\,2}(r^{\,m+n}\,{\cal S}_n) = \frac{1}{r^{\,2}}\,\frac{...
...,\frac{d}{dr}(r^{\,m+n}\,{\cal S}_n)\right] - n\,(n+1)\,r^{\,m+n-2}\,{\cal S}_n$    
  $\displaystyle = m\,(m+2\,n+1)\,r^{\,m-2}\,{\cal R}_n.$ (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).


next up previous
Next: Gravitational potential theory Up: Yielding of an elastic Previous: Introduction
Richard Fitzpatrick 2016-03-31