Antisymmetric Tensor

Antisymmetric Tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign when any two indices of the subset are interchanged. The index subset must generally be either be all covariant or all contravariant.

For example,

holds when the tensor is antisymmetric on it first three indices.

If a tensor changes sign under exchange of any pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.