Antisymmetric tensor
From Free net encyclopedia
In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips the sign if the two indices are interchanged:
- <math>T_{ijk\dots} = -T_{jik\dots}</math>
An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric.
If the tensor changes the 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.
For each pair of indices a general tensor U, with components <math>U_{ijk\dots}</math>, has a symmetric and antisymmetric part, defined as:
<math>U_{(ij)k\dots}=(1/2)(U_{ijk\dots}+U_{jik\dots})</math> (symmetric part),
<math>U_{[ij]k\dots}=(1/2)(U_{ijk\dots}-U_{jik\dots})</math> (antisymmetric part),
and similarly for other indices.
As the term "part" suggests, <math>U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}</math>
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. Proof:
<math>A_{(ij)k\dots}B_{[ij]k\dots}=A_{(ji)k\dots}B_{[ji]k\dots} =-A_{(ij)k\dots}B_{[ij]k\dots}=0</math>
Important antisymmetric tensors in physics include the Faraday tensor F in electromagnetism.
