Main Question or Discussion Point
In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the [itex]\sin \theta[/itex] 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 [itex]\phi, \theta[/itex]. So, this is a surface integral and the surface element (as I would've done it naively) is [itex]r^2\sin\theta\, d\phi\, d\theta[/itex]. I am aware that this isn't a function of [itex]r[/itex] 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 [itex]\sin\theta\, d\phi\, d\theta[/itex], 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 [itex]r[/itex].