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Levi-Civita and Kronecker delta identity, proof with determinants

  1. Jul 8, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel.
    What is the author saying in the last line of text? How can we sum the deltas in the upper right corner, shouldn't we sum the three determinants as a whole, since we're doing that on the other side of the equation? And how does the sum 3 come into the picture anyway?

    2. Relevant equations

    See attachment.

    3. The attempt at a solution

    It seems I should write out the whole determinant for all i,j,k,r,s,t, but that would not make the proof any easier than doing it by "brute force" in the first place!

    Thanks for any help.
     

    Attached Files:

    Last edited: Jul 8, 2012
  2. jcsd
  3. Jul 8, 2012 #2

    tiny-tim

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    Welcome to PF!

    Hi Pifagor! Welcome to PF! :smile:
    erm :redface:

    no attachment! :biggrin:
     
  4. Jul 8, 2012 #3
    .../four letter word /... now it should be OK
     
  5. Jul 8, 2012 #4

    tiny-tim

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    that's better! :biggrin:

    if you use the δit on the top right to replace all the is by ts in its cofactor, you get minus the cofactor for δii

    the same for δis, so you have (3 - 1 - 1) times that cofactor :wink:
     
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