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antisymmetric
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| Of a relation R on a set S, having the property that for any two distinct elements of S, at least one is not related to the other via R. |
| Unaltered in magnitude but changed in sign by exchange of two variables or by a particular symmetry operation. |
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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 either be all covariant or all contravariant. * For example, * * * * * T * * i * j * k * … * * * = * − * * T * * j * i * k * … * * * = * * T * * j * k * i * … * * * = * − * * T * * k * j * i * … * * * = * * T * * k * i * j * … * * * = * − * * T * * i * k * j * … * * * * * {\displaystyle T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }} * holds when the tensor is antisymmetric with respect to its first three indices. * If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector. |