An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that
(1)

is antisymmetric.
The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank2 tensor, which satisfies
(2)

Furthermore, any rank2 tensor can be written as a sum of symmetric and antisymmetric parts as
(3)

The antisymmetric part of a tensor is sometimes denoted using the special notation
(4)

For a general rank tensor,
(5)

where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example
(6)

(Wald 1984, p. 26).