# Del operator crossed with a scalar times a vector proof

1. Jan 12, 2013

### galactic

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

1. The problem statement, all variables and given/known data
Prove the following identity (we use the summation convention notation)

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

2. Relevant equations

equation for del, the gradient, curl..

3. 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 heres what I did:

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

I dont know if this first step is right or if I decomposed the cross product right ?

Last edited by a moderator: Jan 13, 2013
2. Jan 13, 2013

### clamtrox

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

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

3. Jan 13, 2013

### dextercioby

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

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

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

4. Jan 13, 2013

### galactic

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

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 :/

5. Jan 13, 2013

### dextercioby

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

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.