Using suffix notation to find alternative expressions

[ppy]

Homework Statement

use the relationship ε_{ijk} ε_{klm}=δ_{il}δ_{jm}-δ_{im}δ_{jl} to find an alternative expression for ε_{ijk}ε_{ilm}. Hence simplify ε_{ijk}ε_{ijm}

2. Homework Equations

I know that the kronecker delta is 1 for i=j and 0 for i not equal to j. and that the alternating tensor is 0 for any I,j,k equal. +1 if (I,j,k)= (1,2,3) or (2,3,1) or (3,1,2) and -1 for (3,2,1) or (1,3,2) 0r (2,1,3). I also know that the alternating tensor is unchanged under cyclic permutations of its suffices. I know I am supposed to use all these facts but I am unsure how.


Help would be appreciated.

Thanks.

 

[tiny-tim]

hi ppy!

use the relationship ε_{ijk} ε_{klm}=δ_{il}δ_{jm}-δ_{im}δ_{jl} to find an alternative expression for ε_{ijk}ε_{ilm}.
…
I know that … the alternating tensor is unchanged under cyclic permutations of its suffices.

in ε_ijkε_klm, the common index is 3rd and 1st

in ε_ijkε_ilm, the common index is 1st and 1st

so use a cycllc permutation to make it 3rd and 1st