Definition
Several conventions have been used to define the VSH. We follow that of Barrera et al.. Given a scalar spherical harmonic we define three VSH:
being the unitary vector along the radial direction and the position vector of the point with spherical coordinates, and. The radial factors are included to guarantee that the dimensions of the VSH are the same as the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.
The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion
The labels on the components reflect that is the radial component of the vector field, while and are transverse components.
Read more about this topic: Vector Spherical Harmonics
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