Orthonormality of spherical harmonics
PROFESSOR: How do we state the issue of normalization? See, the spherical harmonics are functions of theta and phi. So it makes sense that you would integrate over theta and phi-- solid angle. The solid angle is the natural integration. And it's a helpful integration, because if you have solid angle integrals and then radial integrals, you will have integrated over all volume. So for this spherical harmonic, solid angle is the right variable. And you may remember, if you have solid angle, you have to integrate over theta and phi. Solid angle, you think of it as a radius of one. Here is sine theta. So what is solid angle? It's really the area on a sphere of radius one. The definition of solid angle is area over radius squared, the part of the solid angle that you have. If you're working with a sphere of radius one, it's the area element is the solid angle. So the area element in here would be, or the integral over solid angle-- this is solid angle, d omega-...