Are there any nice known properties about the gradient of a spherical harmonic (i.e. $\vec\nabla Y_l^m(\theta,\phi)$) for arbitrary $l$ and $m$? I've tried searching for things online, but can't quite find anything about them. I understand that the divergences of these gradients have several nice properties, but can I say much about the gradients themselves?
By nice properties, I mean something that would hopefully allow me to write the components of the gradients in terms of the of only other spherical harmonics. Or any other property really that may be useful for setting up some other selection rules or anything of the sort.
Here by gradient I mean gradient on the sphere, or alternatively just the angular component of the gradient of $Y_l^m(\theta,\phi)$, which will trivially have no radial component.
Thanks.
Best Answer
Google "Vector Spherical Harmonics". There's a relationship for the gradient of a scalar SH function also expressed in SH that may do what you want.
Here's a link to a physics course-note webpage that has many of the relations developed. Alternatives are Mathematical Physics textbooks.