Say we have a gradient field $\vec{F}(x, y) = \nabla \phi$ and a closed curve $C$.
We know that the curl $\nabla \times \vec{F}(x,y)$ will be $0$ and the net flow of this field along the closed curve $C$ is $0$.
Can we say anything about the net flow across the curve? Can we say anything about the divergence $\nabla \cdot \vec{F}(x,y)$?