Can I use divergence theorem on a plane?

Question

I have a surface integral over a plane $z=1-x-y$ above $x^2 + y^2 \leq 1$. Can I apply divergence theorem or do I need to do it directly? I'm getting the same value either way, but I'm not sure

Answer 1

There is a two-dimensional divergence theorem referring to domains $S\subset{\mathbb R}^2$, their boundaries $\partial S$, and a vector field ${\bf v}$ defined in a domain $\Omega\supset S$. This theorem is equivalent to "Greens theorem in the plane".

From your question one gets the impression that in fact you have a three-dimensinonal problem with a $${\bf v}(x,y,z):=\bigl(u(x,y,z),v(x,y,z),w(x,y,z)\bigr)\ ,\tag{1}$$ and you are told to compute the flux of ${\bf v}$ through a certain elliptical disc $E$. You have to compute this flux as a standard flux integral, using a parametrization of $E$.

The three-dimensional divergence theorem applies to three-diemnsional bodies $B$, their boundaries $\partial B$, and a vector field $(1)$. There is no body $B$ present in your problem.