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Proving the contracted epsilon identity

  1. Oct 1, 2012 #1
    proving the "contracted epsilon" identity

    in the wikipedia page for the Levi Civita symbol, they have a definition of the product of 2 permutation symbols as: [itex] ε_{ijk}ε_{lmn} = δ_{il}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{im}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{in}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) [/itex] and by contracting the first index in the product (so that i = l) it should be the case that i get [itex] δ_{jm}δ_{kn} - δ_{jn}δ_{km} [/itex].

    however, when i actually replace all the i's with l's i get: [itex] ε_{ijk}ε_{imn} = δ_{ii}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{lm}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{ln}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) [/itex] and then using the fact that δ will be 0 unless both of its indices match, i get [itex] ε_{ijk}ε_{imn} = (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) + (δ_{jn}δ_{km} - δ_{jm}δ_{kn}) [/itex],

    but this turns out to be [itex] - (δ_{jm}δ_{kn} - δ_{jn}δ_{km}) [/itex] which is the negative of the answer that I expected. did i do something wrong? I don't know why i picked up an extra minus sign.
     
  2. jcsd
  3. Oct 3, 2012 #2
    Re: proving the "contracted epsilon" identity

    Your problem is that [itex]\delta_{ii} = 3[/itex], not 1. The first term should have a leading factor of 3.
     
  4. Oct 3, 2012 #3
    Re: proving the "contracted epsilon" identity

    doh! thank you for pointing that out.
     
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