- #1
Meggle
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Homework Statement
Evaluate the expression \epsilon_{ijk} \epsilon_{jmn} \epsilon_{nkp}
Homework Equations
\epsilon_{ijk} \epsilon_{ilj} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}
The Attempt at a Solution
Let \epsilon_{ijk} = \epsilon_{jki} by permutation of Levi-civita
\epsilon_{jki} \epsilon_{jmn} \epsilon_{nkp} = (\delta_{km}\delta_{in} - \delta_{kn}\delta_{im})\epsilon_{nkp}
\epsilon_{nkp}=0 if n=k, however \delta_{kn}=0 if n \neq k
(\delta_{km}\delta_{in} - \delta_{kn}\delta_{im})\epsilon_{nkp} = (\delta_{km}\delta_{in})\epsilon_{nkp}
At this point, can I just go through the possible values for each of the indicies and add it up?
(\delta_{km}\delta_{in})\epsilon_{nkp} = (\delta_{22}\delta_{11})\epsilon_{123} + (\delta_{33}\delta_{22})\epsilon_{231} + (\delta_{22}\delta_{33})\epsilon_{321} + (\delta_{11}\delta_{22})\epsilon_{213} + (\delta_{33}\delta_{11})\epsilon_{132}
As all other combinations result in zeros.
(\delta_{km}\delta_{in})\epsilon_{nkp} = 1+1+1-1-1-1 = 0 = \epsilon_{ijk} \epsilon_{jmn} \epsilon_{nkp}
Is that right?
I'm doing this paper extramurally and really struggling with it. My previous assignments are taking ages to come back to me, so I'm shaky about what I know and where I'm going wrong. Could someone have a look at this and tell me if it looks ok?
)
Sorry I'm being so dense. Do appreciate the help though.
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