Does this notation S^2 denotes the surface area of the sphere?
I believe so yes.
Some insights into the notation used here: in Maths (oftentimes as told in math classes in Berkeley) ϕ\phiϕ and θ\thetaθ are swapped. This notation is used more in Physics.
This slide shows a proof of sphere's area value is 4pir^2. By calculate the integration of a related to omega (further related with theta and phi), we end up with a value of entire sphere surface area.
@RCD-Y is correct. In general, S^n represents the surface of an (n+1)-dimensional sphere (so for example, S^1 is a circle).
Why is dphi from 0 to pi? I have forgotten multivar calc
Or oops, dtheta instead of dphi
Ah, it's because you're supposed to visualize it kinda like a half circle on the right doing a sweep around the center (2pi for phi).