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The Levi-Civita Tensor

The Levi-Civita tensor is also know as the third rank fully antisymmetric unit tensor and is defined by:

    $\displaystyle \ \ 1 \quad\textrm{if $ijk$\ are any cyclic permutation of 123 }$  
$\displaystyle \epsilon_{ijk}$ $\displaystyle =$ $\displaystyle -1 \quad\textrm{if $ijk$\ are any cyclic permutation
of 321 }$  
    $\displaystyle \ \ 0 \quad\textrm{otherwise (if any pair of indices are repeated).}$  

Using this we can reduce the cross product to the following tensor contraction, using the Einstein summation convention:

$\displaystyle (\vA \times \vB)_k = \sum_{i = 1}^3 \sum_{j = 1}^3 A_i B_j
\epsilon_{ijk} = A_i B_j \epsilon_{ijk} $

where (as before) we sum repeated indices over all of the orthogonal cartesian coordinate indices. Note well that it is understood that any leftover index in a contraction of this sort represents a component in a vector answer.

Robert G. Brown 2017-07-11