Tensor form of $\nabla \times (\phi \vec{V})$

Question

So i'm given to find the tensor/index notation of this Vector identity:

(1) $\nabla\times(\phi{V})=\phi(\nabla\times\vec{V})-\vec{V}\times(\nabla\phi)$

would this just be

$$=\partial_i\epsilon_{ijk}\phi_i\vec{V}_je_k$$

or since $\phi$ is a scalar

$$=\phi\big(\partial_i\epsilon_{ijk}\vec{V}_je_k\big)$$

I'm not pretty good with tensors or index in general so any help would be appreciated.

Answer 1

If $e_k$ is the ordinary constant ON-basis in $\mathbb{R}^3$ then we have $$ [\nabla \times (\phi V)]_k = \epsilon_{ijk} \partial_i (\phi V)_j = \epsilon_{ijk} \partial_i (\phi V_j) = \epsilon_{ijk} ((\partial_i \phi) V_j + \phi (\partial_i V_j)) \\ = \epsilon_{ijk} (\partial_i \phi) V_j + \epsilon_{ijk} \phi (\partial_i V_j) = \epsilon_{ijk} (\partial_i \phi) V_j + \phi \epsilon_{ijk} (\partial_i V_j) \\ = [\nabla \phi \times V]_k + \phi [\nabla \times V]_k $$ so $$ \nabla \times (\phi V) = \nabla \phi \times V + \phi \nabla \times V. $$