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:
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 
and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:
(symmetric part)
![U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})](../../../math/f/6/c/f6cb1993bc1c2ff5c610f30fb18398f8.png)
(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}](../../../math/9/b/2/9b2135de1fb09a26aed76c2714382fb7.png)
An important antisymmetric tensor in physics is the Faraday tensor F in electromagnetism.
[edit] See also
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