How Do You Prove the Vector Identity ∇ x ∇φ = 0 Using Index Notation?

[flame_m13]

Homework Statement

Prove vector identity $$\nabla$$ x$$\nabla\phi$$ = 0 using index notation.

Homework Equations

$$\nabla$$ x A = Ejrt$$\partial$$rAt

The Attempt at a Solution

I'm treating this as $$\nabla$$x A, where A = del $$\Phi$$ = Etpq$$\partial$$p$$\Phi$$q

putting A back in the equation:

= EjrtEtpq$$\partial$$r$$\partial$$p$$\Phi$$q

=EtrjEtpq$$\partial$$r$$\partial$$p$$\Phi$$q

=($$\partial$$jp$$\partial$$rq - $$\partial$$jq$$\partial$$rp)$$\partial$$r$$\partial$$p$$\Phi$$q

It's here that I'm stuck. I know it should be simple after this, and somehow I need to get zero, but I'm completely stuck. Maybe I messed up somewhere?

 

[tiny-tim]

Hi flame_m13!

(have a del: ∇ and a curly d: ∂ )

… where A = del $$\Phi$$ = Etpq$$\partial$$p$$\Phi$$q

Nooo … (∇φ)_p = ∂_pφ

 

[flame_m13]

thanks. i have skill at making things unnecessarily complicated. :)