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Evaluating expression involving permutation symbol and Kronecker delta

  1. Sep 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following expression:
    [itex]\sum_{j}\sum_{k}\epsilon_{ijk}\delta_{jk}[/itex]

    2. Relevant equations
    [itex]\delta_{ij}[/itex] = [itex][i = j][/itex]


    3. 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.
    _____________________

    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]
    ______________________

    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
     
  2. jcsd
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