Antisymmetric tensor

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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,

T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}

holds when the tensor is antisymmetric on it first three indices.

If a tensor changes sign under exchange of any pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

Antisymmetric and symmetric tensors[edit]

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 U_{ijk\dots} and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:

U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})   (symmetric part)
U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})   (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

U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}.

Notation[edit]

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,

M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}) ,

and for an order 3 covariant tensor T,

T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}) .

In any number of dimensions, these are equivalent to

M_{[ab]} = \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} ,
T_{[abc]} = \frac{1}{3!} \, \delta_{abc}^{def} T_{def} .

More generally, irrespective of the number of dimensions, antisymmetrization over p indices may be expressed as

S_{[a_1 \dots a_p]} = \frac{1}{p!} \delta_{a_1 \dots a_p}^{b_1 \dots b_p} S_{b_1 \dots b_p} .

In the above,

\delta_{ab\dots}^{cd\dots}

is the generalized Kronecker delta of the appropriate order.

Examples[edit]

Antisymmetric tensors include:

See also[edit]

References[edit]

  1. ^ 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. 
  2. ^ Juan Ramón Ruíz-Tolosa, Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5.  section §7.

External links[edit]

  • [1] - mathworld, wolfram