Editing Antisymmetric tensor (section)

==Antisymmetric and symmetric tensors==

A tensor '''A''' that is antisymmetric on indices <math>i</math> and <math>j</math> has the property that the [[Tensor contraction|contraction]] with a tensor '''B''' that is symmetric on indices <math>i</math> and <math>j</math> is identically 0.

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

:{|
|-
| <math>U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})</math> ||&nbsp;|| (symmetric part)
|-
| <math>U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})</math> ||&nbsp;||(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
<math display=block>U_{ijk\dots} = U_{(ij)k\dots} + U_{[ij]k\dots}.</math>