# Computing inner products of spherical harmonics

• I

In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the $\sin \theta$ in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a sphere, hence varying only $\phi, \theta$. So, this is a surface integral and the surface element (as I would've done it naively) is $r^2\sin\theta\, d\phi\, d\theta$. I am aware that this isn't a function of $r$ and it isn't even really defined here so that brings some problems. However his explanation is that this is a volume integral and the volume element is $\sin\theta\, d\phi\, d\theta$, which doesn't really make sense to me either. Can someone help me out with what exactly is happening here? Mostly why the surface/volume element, and furthermore his entire explanation here, is missing any reference to $r$.

## Answers and Replies

kuruman
Science Advisor
Homework Helper
Gold Member
He misspoke. It is not a volume integral, it is a surface integral.

• beefbrisket