CFD Online Logo CFD Online URL
Home > Wiki > Alternating tensor

Alternating tensor

From CFD-Wiki

Jump to: navigation, search

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}


\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
My wiki