The problem is:
Let $S$ be a simple 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 $\Omega(S)$ subtended by $S$ at $P$ is the set of lines starting at $P$ and passing through $S$. Let $S(a)$ be the intersection of $\Omega(S)$ with the surface with the surface of the sphere with center $P$ and radius $a$. Then the measure of the solid angle (measured in steradians) is defined to be $$ |\Omega(S)| = \frac{\text{area of } S(a)}{a^2} $$ Apply the Divergence Theorem to the part of $\Omega(S)$ between $S(a)$ and $S$ to show that $$ |\Omega(S)| = \iint\limits_S \frac{\mathbf{r} \cdot \mathbf{n}}{r^3} \,dS $$ where $\mathbf{r}$ is the radius vector from $P$ to any point on $S$, $r=|\mathbf{r}|$, and the unit normal vector $\mathbf{n}$ is directed away from $P$.
A picture of the stuff described is below:
And here is my interpretation of the things described:
Is my interpretation correct? And before I get my hands dirty I just want to double-check this thing that's confusing me:
The Divergence Theorem requires a vector field $\mathbf{F}$ with component functions defined on an open region containing $E$ with surface $S'$. Let $E$ be the shaded region above and let $S'$ be the surface that encloses $E$. There clearly is no given vector field to use. The closest thing to a vector field is $(\mathbf{r} \cdot \mathbf{n} / r^3)$, but it is not defined throughout $E$, only on $S$. Is it my job to determine which vector field to use? If so, please don't tell me the answer, I only want a yes/no answer because I would like to do this problem with minimal help.
And as always, this other notice: I can't believe I always have to put this in every post, but please do not explain the answer in terms of topology or differential geometry or whatever. I have zero experience in those subjects and answers explained in terms of them aren't helpful. How would you even do that?, you ask. There are those who always find a way.