Editing Antisymmetric tensor (section)

==Notation==

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor '''M''',
<math display=block>M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}),</math>
and for an order 3 covariant tensor '''T''',
<math display=block>T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).</math>

In any 2 and 3 dimensions, these can be written as
<math display=block>\begin{align}
   M_{[ab]} &= \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} , \\[2pt]
  T_{[abc]} &= \frac{1}{3!} \, \delta_{abc}^{def} T_{def} .
\end{align}</math>
where <math>\delta_{ab\dots}^{cd\dots}</math> is the [[generalized Kronecker delta]], and the [[Einstein summation convention]] is in use.

More generally, irrespective of the number of dimensions, antisymmetrization over <math>p</math> indices may be expressed as
<math display=block>T_{[a_1 \dots a_p]} = \frac{1}{p!} \delta_{a_1 \dots a_p}^{b_1 \dots b_p} T_{b_1 \dots b_p}.</math>

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:
<math display=block>T_{ij} = \frac{1}{2}(T_{ij} + T_{ji}) + \frac{1}{2}(T_{ij} - T_{ji}).</math>

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.