An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor A^(x_1, ..., x_n) such that A^(x_1, ..., x_i, ..., x_j, ..., x_n) = - A^(x_n, ..., x_i, ..., x_j, ..., x_1) is antisymmetric. The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies A^(m n) = - A^(n m). Furthermore, any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as A^(m n) = 1/2(A^(m n) + A^(n m)) + 1/2(A^(m n) - A^(n m)).
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