- #1

xWaffle

- 30

- 0

## Homework Statement

Evaluate the following expression:

[itex]\sum_{j}\sum_{k}\epsilon_{ijk}\delta_{jk}[/itex]

## Homework Equations

[itex]\delta_{ij}[/itex] = [itex][i = j][/itex]

## The Attempt at a Solution

I don't have a solution attempt to this one yet, because somehow I completely missed out on what the permutation thing has to do with anything.

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This is the second expression given on this homework assigment. The first one was a little easier, which I did work out, and came up with the solution. I'm going to show you guys this first problem so you know I at least know a little of what I'm doing..

Evaluate expression:

[itex]\sum_{i}\sum_{k}\delta_{ij}\delta_{ji}[/itex]

I used my knowlege of the Kronecker delta to say that:

[itex]\delta_{ij}\delta_{ji} = \delta_{ii} = \delta_{jj}[/itex]

Then using my knowledge of the trace of an n x n matrix (since I'm only dealing with square matrices), the trace of an n x n matrix is just n. So the final solution to the expression I found to be:

[itex]\sum_{i}\sum_{k}\delta_{ij}\delta_{ji} = \sum_{i}\delta_{ii} = tr(I_{i}) = i[/itex]

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So I do have some of the knowledge I'm expected to have, but I really have no idea how to progress further, with the [itex]\epsilon_{ijk}[/itex] thrown in there. Any help is greatly appreciated. Thanks

edit: I should have mentioned that I do at least know what the permutation symbol is.. it is valued at 0 if any of the i,j,k are the same, it is valued at +1 if the indicies are in cyclic order (123,231,312), -1 if they are are not in cyclic order