Permutation Symbol

A three-index object sometimes called the Levi-Civita Symbol defined by

$\begin{displaymath} \epsilon_{ijk} = \cases{ 0 & for $i = j, j = k$, or $k = i$... ...\cr -1 & for $(i,j,k) \in \{(1,3,2), (3,2,1), (2,1,3)\}$.\cr} \end{displaymath}$

(1)

The permutation symbol satisfies

$\displaystyle \delta_{ij}\epsilon_{ijk}$	$\textstyle =$	$\displaystyle 0$	(2)
$\displaystyle \epsilon_{ipq}\epsilon_{jpq}$	$\textstyle =$	$\displaystyle 2\delta_{ij}$	(3)
$\displaystyle \epsilon_{ijk}\epsilon_{ijk}$	$\textstyle =$	$\displaystyle 6$	(4)
$\displaystyle \epsilon_{ijk}\epsilon_{pqk}$	$\textstyle =$	$\displaystyle \delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp},$	(5)

where $\delta_{ij}$ is the Kronecker Delta. The symbol can be defined as the Scalar Triple Product of unit vectors in a right-handed coordinate system,

$\begin{displaymath} \epsilon_{ijk} \equiv \hat {\bf x}_i\cdot(\hat{\bf x}_j\times\hat{\bf x}_k). \end{displaymath}$

(6)

The symbol can also be interpreted as a Tensor, in which case it is called the Permutation Tensor.

See also Permutation Tensor

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 132-133, 1985.