Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all "covariant" or all "contravariant".
For example,
holds when the tensor is antisymmetric with respect to its first three indices.
If a tensor changes sign under exchange of "each" pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order "k" may be referred to as a differential "k"-form, and a completely antisymmetric contravariant tensor field may be referred to as a "k"-vector field.
Antisymmetric and symmetric tensors.
A tensor A that is antisymmetric on indices "i" and "j" has the property that the contraction with a tensor B that is symmetric on indices "i" and "j" is identically 0.
For a general tensor U with components formula_2 and a pair of indices "i" and "j", U has symmetric and antisymmetric parts defined as:
 
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
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,
and for an order 3 covariant tensor T,
In any 2 and 3 dimensions, these can be written as
where formula_8 is the generalized Kronecker delta,
and we use the Einstein notation to summation over like indices.
More generally, irrespective of the number of dimensions, antisymmetrization over "p" indices may be expressed as
In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:
This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.
Examples.
Totally antisymmetric tensors include: