Let $D$ be an open subset in the $(x, y)$ plane with smooth boundary and let $X = (X1, X2)$ be a continuously differentiable vector field on $\bar{D}$. Make a solid cylinder $C$ by $C = \left \{ (x, y, z) : (x, y) ∈ D, 0 < z < 1 \right \}$. Use $X$ to define a vector field $A$ on $\bar{C}$ by $A = (X1, X2, 0)$. Apply the Divergence theorem to C and A, with the the answer in terms of an integral over D and an integral over the boundary of D.
I think this question is interesting, but I am completely lost as to where one would start with it?