At the start of Evan;s textbook there is this statement:
How can we conclude 3 from the first equality? Why can't it be that the divergence is positive and negative over the volume in equal proportions so that the integral is 0 iver the full domain, while being 0 in only a small subset of it? i.e. why is it necessarily true that the divergence is 0 over the full domain?
Suppose $x_0 \in U$ is such that $div(F)(x_0) = h > 0$. Since $U$ is open and $div(F)$ is continuous, there is $r > 0$ such that $B(x_0, r) \subset U$ and $div(F)(x) > h/2$ for all $x \in V := B(x_0, r)$. Thus $$\int_{V}div(F)(x)\,dx \geq \int_{V}h/2\,dx = h/2 \cdot m(V) > 0,$$ since $m(V) > 0$.