Find $\nabla \cdot (F \times G)$ in terms of $\nabla \times F$ and $\nabla \times G$

Question

If $F,G$ are differentiable vector fields, Find $\nabla \cdot (F \times G)$ in terms of $\nabla \times F$ and $\nabla \times G$.

These are the types of questions I am stuck on and for past 3 hours have no idea on how to even start these, all I keep doing is trying to work it out only to get stuck, rub it all out and start all over again.

Any help or hints will be greatly appreciated, I am truly grateful.

Answer 1

With implicit summation over repeated indices,$$\nabla\cdot(F\times G)=\epsilon_{ijk}\partial_i(F_jG_k)\stackrel{\ast}{=}(\nabla\times F)\cdot G-F\cdot(\nabla\times G),$$where I leave you to work out why $\stackrel{\ast}{=}$ is correct using the product rule.