Difference between revisions of "Skew-symmetric tensor"
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| − | The most important skew-symmetric tensors are those that remain invariant under alternation with respect to the entire group of covariant or contravariant indices. A contravariant (covariant) skew-symmetric tensor of valency | + | {{TEX|auto}} |
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| + | A tensor over an $ n $- | ||
| + | dimensional vector space $ E $ | ||
| + | that is invariant under the operation of [[Alternation|alternation]] with respect to some group of its indices. The components of a skew-symmetric tensor are skew-symmetric with respect to the corresponding group of indices, i.e. if two indices are exchanged the components change sign (in the sense of the additive law of the field $ K $ | ||
| + | over which $ E $ | ||
| + | is defined), and if two indices are equal the components vanish. | ||
| + | |||
| + | The most important skew-symmetric tensors are those that remain invariant under alternation with respect to the entire group of covariant or contravariant indices. A contravariant (covariant) skew-symmetric tensor of valency $ r $ | ||
| + | is an $ r $- | ||
| + | vector or multi-vector over $ E $( | ||
| + | respectively, over $ E ^ {*} $, | ||
| + | the space dual to $ E $); | ||
| + | they are elements of the exterior algebra of the vector space $ E $. | ||
| + | The exterior algebra over $ E ^ {*} $ | ||
| + | is usually called the algebra of exterior forms, identifying covariant skew-symmetric tensors of valency $ r $ | ||
| + | with $ r $- | ||
| + | forms. | ||
For references see [[Exterior algebra|Exterior algebra]]. | For references see [[Exterior algebra|Exterior algebra]]. | ||
Latest revision as of 08:14, 6 June 2020
A tensor over an $ n $-
dimensional vector space $ E $
that is invariant under the operation of alternation with respect to some group of its indices. The components of a skew-symmetric tensor are skew-symmetric with respect to the corresponding group of indices, i.e. if two indices are exchanged the components change sign (in the sense of the additive law of the field $ K $
over which $ E $
is defined), and if two indices are equal the components vanish.
The most important skew-symmetric tensors are those that remain invariant under alternation with respect to the entire group of covariant or contravariant indices. A contravariant (covariant) skew-symmetric tensor of valency $ r $ is an $ r $- vector or multi-vector over $ E $( respectively, over $ E ^ {*} $, the space dual to $ E $); they are elements of the exterior algebra of the vector space $ E $. The exterior algebra over $ E ^ {*} $ is usually called the algebra of exterior forms, identifying covariant skew-symmetric tensors of valency $ r $ with $ r $- forms.
For references see Exterior algebra.
How to Cite This Entry:
Skew-symmetric tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew-symmetric_tensor&oldid=48726
Skew-symmetric tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew-symmetric_tensor&oldid=48726
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article