$\epsilon_{ijk}$ tensor

Dru B. Renner
www.DruBryantRenner.org

30 April 2003

The $\epsilon_{ijk}$ tensor is defined by the following. The value is $0$ if any of the indices are equal, i.e. if $i=j$, $i=k$, or $j=k$. Given that the above conditions do not hold, then $\{i,j,k\}=\{1,2,3\}$ as sets. That is, $(i,j,k)$ is a permutation of $(1,2,3)$. In this case, the value is the sign of that permutation.

The $\epsilon_{ijk}$ tensor satisfies the following three contraction identities.

$$\begin{eqnarray*} &&\epsilon_{ija}\epsilon_{kla}=\delta_{ik}\delta_{jl}-\delta_{... ...}\epsilon_{jab}=2\delta_{ij}\\ &&\epsilon_{abc}\epsilon_{abc}=6 \end{eqnarray*}$$