Vector Calculus Identity help

Question

I am having some issues with the following question:

Prove the following vector calculus identity in $\mathbb{R}^3$, where $f$ is a twice continuously differentiable scalar field and $F$ is a twice continuously differentiable vector field: $$\nabla*(fF) = (\nabla f)*F+f(\nabla*F)$$

Appreciate all and any help!

Answer 1

Hint: Write $F=(F_1, F_2, F_3)$. Now compute $\nabla(fF_1, fF_2, fF_3)$ by product rule.

Answer 2

One approach to this proof is to remember that $\nabla$ is an operator that exhibits both vector and differential properties. So we could write:

$\nabla.(fF)\equiv\nabla_f.(fF)+\nabla_F.(fF)$ with $\nabla_f$ only acting on f and $\nabla_F$ on F.

Then we can expand the expression as a dot product of vectors:

$\nabla_f.(fF)+\nabla_F.(fF)= (\nabla f).F+(\nabla.F).f$

which is the same as the expression you are tying to proof.