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Thread: Permutation Tensor (Explanation Needed)

  1. #1
    Oct 2010

    Permutation Tensor (Explanation Needed)

    Can somebody help me understand the Permutation Tensor?

    I know what it means when it says "if i=j or j=k or i=k, then the permutation tensor is 0".

    However, I don't understand the concepts of even and odd permutation (ie, those cases when the permutation tensor equals -1 or 1). Can somebody explain?

    Attached Thumbnails Attached Thumbnails Permutation Tensor (Explanation Needed)-permutationtensor.jpg  
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  2. #2
    A Plied Mathematician
    Jun 2010
    CT, USA

    Re: Permutation Tensor (Explanation Needed)

    A permutation is when you've swapped two indices. So, if I start with the ordering (1,2,3), and I perform one swap (a swap has to be on adjacent numbers), I could get, for example, (2,1,3). Since I've performed one swap, the permutation (2,1,3) is an odd permutation. Now, let's suppose I do another swap from (2,1,3) to get to (2,3,1). I've now done two swaps, so (2,3,1) is an even permutation. The even-ness or odd-ness of a permutation is identified with the even-ness or odd-ness of the number of swaps I must do to get to that permutation. It's a theorem of abstract algebra that, although there are non-unique ways to get to a particular permutation (for example, I might do two swaps and get right back to where I started), the even-ness or odd-ness of a permutation is constant.

    Does that help? Incidentally, in practice, the most important thing to know about the Levi-Civita symbol $\displaystyle \varepsilon_{ijk}$ is that

    $\displaystyle \sum_{i}\varepsilon_{ijk}\varepsilon_{imn}=\delta_ {jm}\delta_{kn}-\delta_{jn}\delta_{km}.$

    And you should know the relationship between the vector cross product and the Levi-Civita symbol as well.
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