Antisymmetric tensor

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In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:

$T_{ijk\dots} = -T_{jik\dots}$

An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form.

A tensor A which is antisymmetric on indices i and j has the property that the contraction with a tensor B, which is symmetric on indices i and j, is identically 0.

For a general tensor U with components $U_{ijk\dots}$ and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

$U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})$ (symmetric part)

$U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})$ (antisymmetric part)

Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in $U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}$