# Alternating tensor

The alternating tensor, also known as Levi-Civita symbol is defined by

$\epsilon_{ijk} = \begin{cases} 1, & \mbox{if i, j, k are all different and in cyclic order} \\ -1, & \mbox{if i, j, k are all different and in acyclic order} \\ 0, & \mbox{otherwise} \end{cases}$

Thus

$\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1$
$\epsilon_{321} = \epsilon_{132} = \epsilon_{213} = -1$

If any index is repeated then the value is zero, e.g.,

$\epsilon_{112} = \epsilon_{121} = 0$

If any two indices are interchanged then the sign changes, e.g.,

$\epsilon_{kji} = -\epsilon_{ijk}$

This tensor is useful in defining the cross product of two vectors. If $w := u \times v$, then

$w_i = \epsilon_{ijk} u_j v_k$