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Antisymmetric tensor
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, a <a href="/facts/Tensor/pNjFo5tV">tensor</a> is antisymmetric or alternating on (or with respect to) an index subset if it alternates <a href="/facts/Sign_(mathematics)/z0xollg1">sign</a> (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all <i>covariant</i> or all <i>contravariant</i>. </p><p>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. </p><p>If a tensor changes sign under exchange of <i>each</i> pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a> of <a href="/facts/Tensor_order/pNjFo5tV">order</a> k {\displaystyle k} may be referred to as a <a href="/facts/Differential_form/T9gyHtMv">differential k {\displaystyle k} -form</a>, and a completely antisymmetric contravariant tensor field may be referred to as a <a href="/facts/Multivector/brM3rUny"> k {\displaystyle k} -vector</a> field. </p>