Proof of a vectoral differentation identity by levi civita symbol

[advphys]

Homework Statement

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

Homework Equations

ε_ijkε_imn=δ_jnδ_km-δ_knδ_jm

The Attempt at a Solution

i tried for nth component

ε_njk (d/dx_j)ε_klm ψ_l v_m

ε_knjε_klm (d/dx_j) ψ_l v_m

using ε_ijkε_imn=δ_jnδ_km-δ_knδ_jm

i got,

(d/dx_j)ψ_n v_j - (d/dx_j)ψ_j v_n

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?

 

[dx]

Hi advphys,

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 = ε_ijk(∂/∂x_j)ψv_k

 

[advphys]

Because i may not know any other identity. :D

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

 

[dx]

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

 

[dx]

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

ψv is a vector with components ψv_i, so the cross product ∇ x ψv is simply the vector with components ε_ijk(∂/∂x_j)(ψv_k)

 

[advphys]

Oh, yes. Definitely.
Thanks a lot, i got that.