An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that
(1)
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is antisymmetric.
The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies
(2)
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Furthermore, any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as
(3)
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The antisymmetric part of a tensor is sometimes denoted using the special notation
(4)
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For a general rank- tensor,
(5)
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where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example
(6)
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(Wald 1984, p. 26).