If $\nabla \cdot \vec{F} = 0$ show that $\vec{F}=\nabla \times \int_{0}^{1} \vec{F}(tx,ty,tz)\times(tx,ty,tz)dt$.

Suppose $ \vec{F}$ is a vector field on $ \mathbb{R}^3$ and $ \nabla \cdot \vec{F} = 0$ . Prove that:

$ \vec{F}=\nabla \times \int_{0}^{1} \vec{F}(tx,ty,tz)\times(tx,ty,tz)dt$ .

Tried doing it, but can’t seem to get it exactly right. Can anyone give an solution with working – naturally just doing 1st component should suffice.