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Using suffix notation to find alternative expressions

  • Thread starter ppy
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  • #1
ppy
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Homework Statement


use the relationship ε[itex]_{ijk}[/itex] ε[itex]_{klm}[/itex]=δ[itex]_{il}[/itex]δ[itex]_{jm}[/itex]-δ[itex]_{im}[/itex]δ[itex]_{jl}[/itex] to find an alternative expression for ε[itex]_{ijk}[/itex]ε[itex]_{ilm}[/itex]. Hence simplify ε[itex]_{ijk}[/itex]ε[itex]_{ijm}[/itex]

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.
 

Answers and Replies

  • #2
tiny-tim
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hi ppy! :smile:
use the relationship ε[itex]_{ijk}[/itex] ε[itex]_{klm}[/itex]=δ[itex]_{il}[/itex]δ[itex]_{jm}[/itex]-δ[itex]_{im}[/itex]δ[itex]_{jl}[/itex] to find an alternative expression for ε[itex]_{ijk}[/itex]ε[itex]_{ilm}[/itex].

I know that … the alternating tensor is unchanged under cyclic permutations of its suffices.


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

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

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

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