Dual Tensor

Given an antisymmetric second Rank Tensor $C_{ij}$ , a dual pseudotensor is defined by

$\begin{displaymath} C_i \equiv {\textstyle{1\over 2}}\epsilon_{ijk}C_{jk}, \end{displaymath}$

(1)

where

$\displaystyle C_i$	$\textstyle \equiv$	$\displaystyle \left[\begin{array}{c}C_{23}\\ C_{31}\\ C_{12}\end{array}\right]$	(2)
$\displaystyle C_{jk}$	$\textstyle \equiv$	$\displaystyle \left[\begin{array}{ccccccc} 0 & C_{12} & -C_{31}\\ -C_{12} & 0 & C_{23}\nonumber\\ C_{31} & -C_{23} & 0\end{array}\right].$

See also Dual Scalar

References

Arfken, G. ``Pseudotensors, Dual Tensors.'' §3.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 128-137, 1985.