## Proof using Levi-Civita symbol

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

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

3. 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.

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 Recognitions: Homework Help In your formula, replace the $$\delta_{jj}$$, $$\delta_{kk}$$ etc... where the variables are the same... with 1. Also, $$\delta_{ij}\delta_{jk} = \delta_{ik}$$

 Quote by learningphysics In your formula, replace the $$\delta_{jj}$$, $$\delta_{kk}$$ etc... where the variables are the same... with 1. Also, $$\delta_{ij}\delta_{jk} = \delta_{ik}$$
If $$\delta_{ij}\delta_{jk} = \delta_{ik}$$ does that mean that $$\delta_{lk}\delta_{kj} = \delta_{lj}$$ and so on?

Recognitions:
Homework Help

## Proof using Levi-Civita symbol

 Quote by cashmerelc If $$\delta_{ij}\delta_{jk} = \delta_{ik}$$ does that mean that $$\delta_{lk}\delta_{kj} = \delta_{lj}$$ and so on?
Yes, exactly.

 Okay, I think one more question will help me get it. $$\delta_{jk}$$$$\delta_{kj}$$ = ?

Recognitions:
Homework Help
 Quote by cashmerelc Okay, I think one more question will help me get it. $$\delta_{jk}$$$$\delta_{kj}$$ = ?
= $$\delta_{jj} = 1$$