In mathematics and theoretical physics, a tensor is antisymmetric or alternating 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.

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 may be referred to as a 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 has the property that the contraction with a tensor B is identically 0.

For a general tensor U U has symmetric and antisymmetric parts defined as:

1. {|

|| ||||(antisymmetric part). |}

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

is the generalized Kronecker delta, and the Einstein summation convention is in use.

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

Examples

Totally antisymmetric tensors include:

• Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).

• The electromagnetic tensor in electromagnetism.

• The Riemannian volume form on a pseudo-Riemannian manifold.

See also

Notes

References

External links

• Antisymmetric Tensor – mathworld.wolfram.com