I have a homework problem to prove about the solid angle. The book says:
Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. The solid angle Ω(S) subteded by S at P is the set of lines starting at P and passing through S. Let S(a) be the intersection of Ω(S) with the surface of the sphere with center P and radius a. Then the measure of the solid angle (in steardians) is defined to be
$$ |Ω(S)| = \frac{\text{area of }S(a)}{a^2}$$
Apply the Divergence Theorem to the part of Ω(S) between S(a) and S to show that
$$ |Ω(S)| = \iint_S \frac{\mathbf r \cdot \mathbf n}{r^3} dS$$
where r is the radius vector from P to any point on S, r = r, and the unit normal vector n is directed away from P.
First let me say that it is O.K. for me to ask for some help here according to my university rules.
I try to do this problem but have some questions. First, problem says it is the set of lines starting at P but should this be rays? It seems to me like in the case of a sphere there would be no lines that start at P and intersect the sphere at most once if the point is inside the circle, because each line can go in to directions. So I assume it is rays.
After that I have some questions like, how am I supposed to calculate the area of S(a)? I know that the area of S(a) is equal to $\iint_{S(a)} dS$ but I have to get that as an integral of S, not S(a). I had the idea that because I know it intersects at most once I could map each point on S onto S(a), but I don't know how to do that. I thought maybe could just divide by the radius of the sphere a but that clearly doesn't work.
I think this is an interesting problem and could be fun to solve but I don't know where to start. Could you please give a hint -- NOT the whole answer?
By the way this is Stewart's Calculus 7e on page 1163, the "Problems Plus" problem #1.
Sorry for my bad English and thanks in advance! :)
EDIT I have seen on Wikipedia (http://en.wikipedia.org/wiki/Solid_angle) that it says $$Ω = \iint_S \frac{\mathbf r \cdot \hat n}{r^3} dS$$ which means that this is correct, but it does not say how the proof works. It says "can be calculated as the surface integral" but does not explain how "can be calculated" is true. Could someone elaborate here?