# Proof using Levi-Civita symbol

## Homework Statement

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

## Homework Equations

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

## 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.

Last edited:

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

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

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
Homework Helper
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.

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

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

$$\delta_{jk}$$$$\delta_{kj}$$ = ?
= $$\delta_{jj} = 1$$