Proof using Levi-Civita symbol

[cashmerelc]

Homework Statement

Prove $$\sum_{j,k}$$ $$\epsilon_{ijk}$$ $$\epsilon_{ljk}$$ = 2$$\delta_{il}$$

Homework Equations

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

The Attempt at a Solution

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

If the subscripts are not equal, then $$\delta_{il}$$ = 0.

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

$$\delta_{il}$$($$\delta_{jj}$$$$\delta_{kk}$$) ?

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 $$\delta_{jj}$$, $$\delta_{kk}$$ etc... where the variables are the same... with 1.

Also, $$\delta_{ij}\delta_{jk} = \delta_{ik}$$

 

[cashmerelc]

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

Also, $$\delta_{ij}\delta_{jk} = \delta_{ik}$$

If $$\delta_{ij}\delta_{jk} = \delta_{ik}$$ does that mean that $$\delta_{lk}\delta_{kj} = \delta_{lj}$$ and so on?

 

[learningphysics]

If $$\delta_{ij}\delta_{jk} = \delta_{ik}$$ does that mean that $$\delta_{lk}\delta_{kj} = \delta_{lj}$$ and so on?

Yes, exactly.

 

[cashmerelc]

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

$$\delta_{jk}$$$$\delta_{kj}$$ = ?

 

[learningphysics]

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

$$\delta_{jk}$$$$\delta_{kj}$$ = ?

= $$\delta_{jj} = 1$$