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\partialrAt

The Attempt at a Solution

I'm treating this as \nablax A, where A = del \Phi = Etpq\partialp\Phiq

putting A back in the equation:

= EjrtEtpq\partialr\partialp\Phiq

=EtrjEtpq\partialr\partialp\Phiq

=(\partialjp\partialrq - \partialjq\partialrp)\partialr\partialp\Phiq

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\partialp\Phiq

Nooo … (∇φ)_p = ∂_pφ

 

[flame_m13]

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