“Antisymmetric tensor”的意思、由来-开放百科全书

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Antisymmetric tensor

释义

Antisymmetric and symmetric tensors
Notation
Examples
See also
Notes
References
External links

{{Short description|Tensor equal to the negative of any of its transpositions}}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 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

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 and a pair of indices i and j, 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

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 number of dimensions, these are equivalent to

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

In the above,

is the generalized Kronecker delta of the appropriate order.

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

Notes

1. ^{{cite book|author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}
2. ^{{cite book|author1=Juan Ramón Ruíz-Tolosa |author2=Enrique Castillo | title=From Vectors to Tensors | publisher=Springer| year=2005| isbn=978-3-540-22887-5 |url=https://books.google.com/books?id=vgGQUrQMzwYC&pg=PA225 |page=225}} section §7.

References

{{cite book |pages=85–86, §3.5|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=Gravitation| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
{{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}

External links

Antisymmetric Tensor – mathworld.wolfram.com

1 : Tensors

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