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Permutation Symbol

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

\epsilon_{ijk} = \cases{
0 & for $i = j, j = k$, or $k = i$...
-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,
\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


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

© 1996-9 Eric W. Weisstein