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.[1][2] 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 may be referred to as a differential -form, and a completely antisymmetric contravariant tensor field may be referred to as a -vector field.
Antisymmetric and symmetric tensorsedit
A tensor A that is antisymmetric on indices and has the property that the contraction with a tensor B that is symmetric on indices and is identically 0.
For a general tensor U with components and a pair of indices and U has symmetric and antisymmetric parts defined as:
(symmetric part)
(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
Notationedit
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
Ricci calculus – Tensor index notation for tensor-based calculations
Symmetric tensor – Tensor invariant under permutations of vectors it acts on
Symmetrization – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback
Notesedit
^K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
^Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.