How does onWhy is $\int\phi=0$ for $\phi\in\mathcal{D}$ for $\phi$ with co´nvex support equivalent to $\phi=\sum \partial_j F_j$?

Question

How does one prove $\int\phi=0$ for $\phi\in\mathcal{D}$ for $\phi$ with convex support is equivalent to $\phi=\sum \partial_j F_j$ ?

I actually, don't quite see why convex support is explicitly mentioned here and how it factors into a proof.