# Homework Help: Proof of a vectoral differentation identity by levi civita symbol

1. Mar 2, 2013

1. The problem statement, all variables and given/known data

prove,
∇x(ψv)=ψ(∇xv)-vx(∇ψ)
using levi civita symbol and tensor notations

2. Relevant equations

εijkεimnjnδkmknδjm

3. The attempt at a solution

i tried for nth component

εnjk (d/dxjklm ψl vm

εknjεklm (d/dxj) ψl vm

using εijkεimnjnδkmknδjm

i got,

(d/dxjn vj - (d/dxjj vn

But, i can't go further. I think only one simple step is left to show it is equal to the right hand side of the given identity. But how?

2. Mar 2, 2013

### dx

That identity is not applicable here. Why do you have two levi-civita symbols? And ψ is a scalar, not a vector.

(∇ x ψv)i = εijk(∂/∂xj)ψvk

3. Mar 2, 2013

Because i may not know any other identity. :D

I thought for (ψv)k term i may have one more levi civita symbol.

4. Mar 2, 2013

### dx

Just evaluate the derivative using the product rule, and remember that (∇ψ)i = ∂ψ/∂xi

5. Mar 2, 2013

### dx

ψv is a vector with components ψvi, so the cross product ∇ x ψv is simply the vector with components εijk(∂/∂xj)(ψvk)

6. Mar 2, 2013