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Index notation identity

  1. Nov 10, 2015 #1
    1. The problem statement, all variables and given/known data
    The antisymmetric tensor is constructed from a vector ##\vec a## according to ##A_{ij} = k\varepsilon_{ijk}a_k##.
    For which values of ##k## is ##A_{ij}A_{ij} = |\vec a|^2##?

    2. Relevant equations
    Identity
    ##\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}##

    3. The attempt at a solution
    ##A_{ij}A_{ij} = k^2\varepsilon_{ijk}\varepsilon_{ijm}a_ka_m = k^2\varepsilon_{jki}\varepsilon_{ijm}a_ka_m = k^2(\delta_{jj}\delta_{km}-\delta_{jm}\delta_{kj})a_ka_m = k^2(\delta_{km}-\delta_{km})a_ka_m =0##
    Which I obviously shouldn't get but I can't see where I'm making an error.
     
  2. jcsd
  3. Nov 10, 2015 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You are using the summation convention, so
    $$\delta_{jj} \equiv \sum_{j=1}^3 \delta_{jj}.$$
    This gives a different numerical factor in front of the first ##\delta_{km}## term.
     
  4. Nov 10, 2015 #3
    Right thanks!
     
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