Prove: ∇⋅ϕF = ϕ∇⋅F + F⋅∇ϕ

Question

I am asked to prove this identity using tensor notation. However, I am not sure where to even begin the problem.

Answer 1

Using indices summation convention, we have simply

$$\nabla\cdot(\phi F)=\partial_i(\phi F^i)=(\partial_i\phi)F^i+\phi(\partial_iF^i)=(\nabla\phi)\cdot F+\phi(\nabla\cdot F)$$ In case you're dealing with a covariant derivative, just use $\nabla_i$ instead of $\partial_i$, as the product rule also holds.