I'm struggling to prove the following: Given a volume $V$ bounded by a closed surface $S$, a vector $\bar{F}$ satisfies the following: $$\nabla \cdot \bar{F} = 0$$ $$\nabla^2 \bar{F} = \bar{0}$$ and the flux of $\bar{F} \times (\nabla \times \bar{F})$ through $S$ is zero.
Then there exists $\phi \in V$ such that $\bar{F} = \nabla \phi$.
I understand that this is equivalent to proving that $\bar{F}$ has zero curl, as the region is clearly simply connected.
Is there a clever application of Green's theorem that I am missing?