Antisymmetric tensor

An antisymmetric tensor changes sign under exchange of two indices in a specified subset of its indices. A totally antisymmetric tensor has this property for every pair of indices.

A tensor changes sign whenever two indices are interchanged. In three dimensions, a rank-two antisymmetric tensor is $A_{ij}=a_k{ϵ}_{kij}$, a rank-three one is $A_{ijk}=a{ϵ}_{ijk}$, and every totally antisymmetric tensor of rank greater than three vanishes.