How Do You Prove \sum_{j,k} \epsilon_{ijk} \epsilon_{ljk} = 2\delta_{il}?

[cashmerelc]

Homework Statement

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

Homework Equations

\epsilon_{ijk} \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})<h2>The Attempt at a Solution</h2><br /> <br /> Okay, in cases where subscripts of the Kronecker delta are equal, then \delta_{jj} = 1. <br /> <br /> If the subscripts are not equal, then \delta_{il} = 0. <br /> <br /> So plugging those into the parenthesis of the above equation gives me:<br /> <br /> \delta_{il}(\delta_{jj}\delta_{kk}) ?<br /> <br /> 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