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Vector Spherical Harmonics May 26, 2010

Posted by Sarah in Uncategorized.
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Spherical harmonics are the angular part of the solution to Laplace’s equation on the sphere: they have the property that
\Delta Y_{lm} = \frac{l (l+1)}{r^2} Y_{lm}

They form an orthonormal basis in Hilbert space and have many nice properties. In particular, the space of spherical harmonics is an irreducible representation of the group of rotations SO(3).

Vector spherical harmonics are an extension of the concept for use with vector fields. We define three vector spherical harmonics,
\mathbf{Y_{lm}} = Y_{lm} \hat{\mathbf{r}}
\mathbf{\Psi_{lm}} = r \nabla Y_{lm}
\mathbf{\Phi_{lm}} = \vec{r} \times \nabla Y_{lm}
These are orthogonal just like the usual spherical harmonics, and they allow every vector field to be expanded in vector spherical spherical harmonics.

They also turn out to be useful in the study of magnetostatics (that is, the study of static magnetic fields) as in this paper by Barrerra, Estevez, and Giraldo (Here.)

The gist: expanding functions on the sphere in spherical harmonics makes it possible to replace the Poisson equation with an ordinary differential equation. This means that if we define a charge distribution and require it to be bounded in extent, we can determine the potential outside the charge distribution. The key is canceling out the angular component to simplify the partial differential equation.

Extending this notion to vector fields allows us to do the same thing with differential equations involving the gradient (and Laplacian and curl etc), expanding in vector spherical harmonics and letting the angular components cancel. This will allow us to calculate magnetic multipoles analogous to electrical multipoles. When there is no charge, the magnetic induction field is the gradient of some function, the magnetic scalar potential.
\mathbf{B} = \nabla \mathbf{\Phi}.
The potential can be expanded in spherical harmonics; the coefficients are called the magnetostatic multipole moments.