Index notation for vector id's

1. The problem statement, all variables and given/known data
Prove vector identity [tex]\nabla[/tex] x[tex]\nabla\phi[/tex] = 0 using index notation.

2. Relevant equations

[tex]\nabla[/tex] x A = Ejrt[tex]\partial[/tex]rAt

3. The attempt at a solution
I'm treating this as [tex]\nabla[/tex]x A, where A = del [tex]\Phi[/tex] = Etpq[tex]\partial[/tex]p[tex]\Phi[/tex]q

putting A back in the equation:

= EjrtEtpq[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]q

=EtrjEtpq[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]q

=([tex]\partial[/tex]jp[tex]\partial[/tex]rq - [tex]\partial[/tex]jq[tex]\partial[/tex]rp)[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]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

Science Advisor
Homework Helper
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Hi flame_m13! :smile:

(have a del: ∇ and a curly d: ∂ :wink:)
… where A = del [tex]\Phi[/tex] = Etpq[tex]\partial[/tex]p[tex]\Phi[/tex]q
Nooo … (∇φ)p = ∂pφ :wink:
 
thanks. i have skill at making things unnecessarily complicated. :)
 

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