Proof using Levi-Civita symbol

cashmerelc

1. The problem statement, all variables and given/known data
Prove [tex]\sum_{j,k}[/tex] [tex]\epsilon_{ijk}[/tex] [tex]\epsilon_{ljk}[/tex] = 2[tex]\delta_{il}[/tex]



2. Relevant equations
[tex]\epsilon_{ijk}[/tex] [tex]\epsilon_{ljk} = [tex]\delta_{il}[/tex]([tex]\delta_{jj}[/tex][tex]\delta_{kk}[/tex] - [tex]\delta_{jk}[/tex][tex]\delta_{kj}[/tex]) + [tex]\delta_{ij}[/tex]([tex]\delta_{jk}[/tex][tex]\delta_{kl}[/tex] - [tex]\delta_{jl}[/tex][tex]\delta_{kk}[/tex]) + [tex]\delta_{ik}[/tex]([tex]\delta_{jl}[/tex][tex]\delta_{kk}[/tex] - [tex]\delta_{jj}[/tex][tex]\delta_{kl}[/tex])


3. The attempt at a solution

Okay, in cases where subscripts of the Kronecker delta are equal, then [tex]\delta_{jj}[/tex] = 1.

If the subscripts are not equal, then [tex]\delta_{il}[/tex] = 0.

So plugging those into the parenthesis of the above equation gives me:

[tex]\delta_{il}[/tex]([tex]\delta_{jj}[/tex][tex]\delta_{kk}[/tex]) ?

If that is the case, then how could the two inside the parenthesis equal 2? I know I must be missing something.

learningphysics

In your formula, replace the [tex]\delta_{jj}[/tex], [tex]\delta_{kk}[/tex] etc... where the variables are the same... with 1.

Also, [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex]

cashmerelc

If [tex]\delta_{ij}\delta_{jk} = \delta_{ik}[/tex] does that mean that [tex]\delta_{lk}\delta_{kj} = \delta_{lj}[/tex] and so on?

learningphysics

Yes, exactly.

cashmerelc

Okay, I think one more question will help me get it.

[tex]\delta_{jk}[/tex][tex]\delta_{kj}[/tex] = ?

learningphysics

= [tex]\delta_{jj} = 1[/tex]