Del operator crossed with a scalar times a vector proof

[galactic]

"Del" operator crossed with a scalar times a vector proof

Homework Statement

Prove the following identity (we use the summation convention notation)

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

Homework Equations

equation for del, the gradient, curl..

The Attempt at a Solution

im kind of confused on the first step...I broke it down into the following; however, levi civita symbols aren't my cup of tea and I get pretty confused about it...anyway here's what I did:

$$\bigtriangledown\times(\phi\vec{V})=(\epsilon_{ijk})\partial_i\vec{V}\phi\hat{x}_k$$

I don't know if this first step is right or if I decomposed the cross product right ?

 

[clamtrox]

You should have the j:th component of V in your expression,
$$\nabla \times (\phi \vec{V}) = \epsilon_{ijk} \partial_i (\phi V_j) \hat{x}_k$$

 

[dextercioby]

The form you've written is completely equal to the one seen in math/physics books

$$\nabla\times \left(\phi\vec{V}\right) = \nabla\phi\times\vec{V} + \phi\nabla\times\vec{V}$$

 

[galactic]

Thanks for the replies, I'm just not sure what to do after what clamtrox said to do, the whole proofing business if pretty new to me :/

 

[dextercioby]

You need two more steps. Apply the product rule for differentiation and then once you obtain a sum of two terms, reconstruct vectors from their components.