Vector spherical harmonics provide a convenient representation to solve linear problems involving the Laplacian operator acting on vector fields in spherical symmetry, and may be considered as the natural extension of spherical harmonics. They have been used extensively for example in electrodynamics, and the textbook by Morse and Feshbach [^1]provides several examples of application.
When studying vector fields in the sphere, one may be tempted to introduce the standard spherical coordinate system and expand in spherical harmonics the components of this field into spherical harmonics. It turns out, however, that a much more convenient choice is to adopt the following functions:
where is the position vector with respect to the origin and and are the spherical harmonics . An alternative definition of the third spherical harmonic oftentimes found in the literature is Both the choices of and as the third vector spherical harmonic are equivalent. It is seen that indeed that is to say,
A virtue of ) over the third of 125 ) is that it makes it apparent that and are perpendicular. We can then expand a vector field as follows:
An alternative representation of a real-valued field is through complex spherical harmonics:
In this case, the field is represented as
[1]: Morse, Philip M., and Herman Feshbach. 1981. Methods of Theoretical Physics. Pt. 2: Chapters 9 to 13