Kronecker delta and Levi-Civita epsilon

Question

Evaluate the sum $\sum_{k}\epsilon_{ijk}\varepsilon _{lmk}$ by considering the result for all possible combinations of i , j, l, m ; that is: a)i=j; b)i=l; c)i=m; d)j=l; e)j=m; f)l=m; g)i=!l or m; h)j=! or m;

Show that: $$\sum_{k}\epsilon_{ijk}\varepsilon _{lmk}=\delta _{il}\delta_{jm}-\delta_{im}\delta_{jl}$$

Well, i already know how to prove this last identity. My only problem is with evaluating the sum. I would be thankful if someone do the first and second case ( a) and b) ) for i get the point of it.

Answer 1

Firstly, your index notation is extremely important.

$\epsilon_{ijk}\epsilon _{lmk}$ gives you a rank 6 tensor.

$\sum_{k}\epsilon_{ijk}\epsilon^{lmk}$ is the product you're looking for.

As for the sums, express $\sum_{k}\epsilon_{ijk}\epsilon^{lmk}$ as a sum of as many products of Krönecker deltas as is needed to express the correct values of each combination, i.e., for a) and f) your deltas should cancel to give you 0, because the Levi-Civita tensor is completely antisymmetric.