Английская Википедия:Antisymmetric tensor
Шаблон:Short descriptionIn 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, <math display=block>T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}</math> 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 <math>k</math> may be referred to as a differential <math>k</math>-form, and a completely antisymmetric contravariant tensor field may be referred to as a <math>k</math>-vector field.
Antisymmetric and symmetric tensors
A tensor A that is antisymmetric on indices <math>i</math> and <math>j</math> has the property that the 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> (symmetric part) <math>U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})</math> (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>
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 we use the Einstein notation to summation over like indices.
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.
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, <math>F_{\mu\nu}</math> in electromagnetism.
- The Riemannian volume form on a pseudo-Riemannian manifold.
See also
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
Notes
References
External links
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book section §7.
