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Alternating tensor

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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
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