- #1
LoopQG
- 22
- 0
This isn't a HW question just something I am curious about. I was looking on wikipedia and found a way to prive the Levi-Citiva Kronecker Delta relation that I hadn't seen before.
The site claims
[tex]
\epsilon_{ijk}\epsilon_{lmn} = \det \begin{bmatrix}
\delta_{il} \delta_{im} \delta_{in}\\
\delta_{jl} \delta_{jm} \delta_{jn}\\
\delta_{kl} \delta_{km} \delta_{kn}\\
\end{bmatrix}
[/tex]
= [tex]
\delta_{il}\left( \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\right) - \delta_{im}\left( \delta_{jl}\delta_{kn} - \delta_{jn}\delta_{kl} \right) + \delta_{in} \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \right) \
[/tex]
the website is:
http://en.wikipedia.org/wiki/Levi-Civita_symbol
The way I have proved the relation before is showing that all 81 components of the tensor are zero accept the [tex] \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}
[/tex] terms.
Taking the determinate of the matrix I do get the correct answer,just not sure why you can write that matrix down.
Can anybody offer a proof of the matrix ? If so it is much easier than the way I have previously done it.
The site claims
[tex]
\epsilon_{ijk}\epsilon_{lmn} = \det \begin{bmatrix}
\delta_{il} \delta_{im} \delta_{in}\\
\delta_{jl} \delta_{jm} \delta_{jn}\\
\delta_{kl} \delta_{km} \delta_{kn}\\
\end{bmatrix}
[/tex]
= [tex]
\delta_{il}\left( \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\right) - \delta_{im}\left( \delta_{jl}\delta_{kn} - \delta_{jn}\delta_{kl} \right) + \delta_{in} \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \right) \
[/tex]
the website is:
http://en.wikipedia.org/wiki/Levi-Civita_symbol
The way I have proved the relation before is showing that all 81 components of the tensor are zero accept the [tex] \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}
[/tex] terms.
Taking the determinate of the matrix I do get the correct answer,just not sure why you can write that matrix down.
Can anybody offer a proof of the matrix ? If so it is much easier than the way I have previously done it.