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Relation between Levi-civita and Kronecker- delta symbol

  1. May 19, 2017 #1
    1. The problem statement, all variables and given/known data
    definition of εijk
    εijk=+1 if ijk = (123, 231, 312)
    εijk = −1if ijk = (213, 321, 132) , (1.1.1)
    εijk= 0,otherwise .
    That is,εijk is nonzero only when all three indices are different.

    From the definition in Eq. (1.1.1), show that
    εijkεinm= δjnδkm − δjmδkn , (1.1.4)
    where of course there is an implied sum over the i index in Eq. (1.1.4), but the indices j, k, n,and m are free.

    2. Relevant equations

    3. The attempt at a solution
    By using definition of εijk; I can show that under the values which i,j,k,n,m can take, L.H.S.=R.H.S.

    But is there any other better way to show it?
    Is there a way such that one will start from L.H.S. and reach to R.H.S.?
    Last edited: May 19, 2017
  2. jcsd
  3. May 19, 2017 #2


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    We do not have access to your book so we cannot tell how Eq. (1.1.1) defines the Levi-Civita symbol.
  4. May 19, 2017 #3
    I have written the definition.
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