Proving the contracted epsilon identity

[demonelite123]

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: $ε_{ijk}ε_{lmn} = δ_{il}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{im}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{in}(δ_{jl}δ_{km} - δ_{jm}δ_{kl})$ and by contracting the first index in the product (so that i = l) it should be the case that i get $δ_{jm}δ_{kn} - δ_{jn}δ_{km}$.

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

but this turns out to be $- (δ_{jm}δ_{kn} - δ_{jn}δ_{km})$ 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.

 

[Muphrid]

Your problem is that $\delta_{ii} = 3$, not 1. The first term should have a leading factor of 3.

 

[demonelite123]

Your problem is that $\delta_{ii} = 3$, not 1. The first term should have a leading factor of 3.

doh! thank you for pointing that out.