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In [[mathematics]], a '''multivector field''', '''polyvector field''' of '''degree ''k'' ''', or ''' ''k''-vector field''', on a [[manifold]] <math>M</math>, is a generalization of the notion of a [[vector field]] on a manifold. Whereas a vector field <math>X \in \Gamma(TM)</math>is a [[Section (mathematics)|section]]{{dn|date=May 2019}} of tangent bundle, which assigns to each point on the manifold a [[tangent vector]] <math>X_p \in T_pM</math>, a multivector field is a section of the ''k''th [[exterior power]] of the [[tangent bundle]], <math>\Lambda^k TM</math>, and to each point <math>p \in M</math>it assigns a [[Multivector|''k-''vector]] in <math>\Lambda^k T_p M</math>. Just as the smooth sections of the tangent bundle (vector fields) make up a vector space, the space of smooth ''k''-vector fields over ''M'' make up a vector space <math>\Gamma(\Lambda^k TM)</math>. Furthermore, since the tangent bundle is dual to the [[cotangent bundle]], multivector fields of degree ''k'' are dual to ''k''-[[differential forms|forms]], and both are subsumed in the general concept of a [[tensor field]], which is a section of some [[Tensor field#Tensor bundles|tensor bundle]], often consisting of exterior powers of the tangent and cotangent bundles. A (k,0)-tensor field is a differential k-form, a (0,1)-tensor field is a vector field, and a (0,k)-tensor field is ''k''-vector field. While differential forms are widely studied as such in [[differential geometry]] and [[differential topology]], multivector fields are often encountered as tensor fields of type (0,k), except in the context of the [[geometric algebra]] (see also [[Clifford algebra]]).<ref>{{Cite book|url=https://www.worldcat.org/oclc/213362465|title=Geometric algebra for physicists|last=Doran, Chris (Chris J. L.)|date=2007|publisher=Cambridge University Press|others=Lasenby, A. N. (Anthony N.), 1954-|isbn=9780521715959|edition=1st pbk. ed. with corr|location=Cambridge|oclc=213362465}}</ref><ref>{{Cite book|url=https://www.worldcat.org/oclc/757486966|title=Geometric algebra|last=Artin, Emil, 1898-1962.|year=1988|origyear=1957|publisher=Interscience Publishers|isbn=9781118164518|location=New York|oclc=757486966}}</ref><ref>{{Cite book|url=https://www.worldcat.org/oclc/769755408|title=A new approach to differential geometry using Clifford's geometric algebra|last=Snygg, John.|date=2012|publisher=Springer Science+Business Media, LLC|isbn=9780817682835|location=New York|oclc=769755408}}</ref> |
In [[mathematics]], a '''multivector field''', '''polyvector field''' of '''degree ''k'' ''', or ''' ''k''-vector field''', on a [[manifold]] <math>M</math>, is a generalization of the notion of a [[vector field]] on a manifold. Whereas a vector field <math>X \in \Gamma(TM)</math>is a [[Section (mathematics)|section]]{{dn|date=May 2019}} of tangent bundle, which assigns to each point on the manifold a [[tangent vector]] <math>X_p \in T_pM</math>, a multivector field is a section of the ''k''th [[exterior power]] of the [[tangent bundle]], <math>\Lambda^k TM</math>, and to each point <math>p \in M</math>it assigns a [[Multivector|''k-''vector]] in <math>\Lambda^k T_p M</math>. Just as the smooth sections of the tangent bundle (vector fields) make up a vector space, the space of smooth ''k''-vector fields over ''M'' make up a vector space <math>\Gamma(\Lambda^k TM)</math>. Furthermore, since the tangent bundle is dual to the [[cotangent bundle]], multivector fields of degree ''k'' are dual to ''k''-[[differential forms|forms]], and both are subsumed in the general concept of a [[tensor field]], which is a section of some [[Tensor field#Tensor bundles|tensor bundle]], often consisting of exterior powers of the tangent and cotangent bundles. A (k,0)-tensor field is a differential k-form, a (0,1)-tensor field is a vector field, and a (0,k)-tensor field is ''k''-vector field. While differential forms are widely studied as such in [[differential geometry]] and [[differential topology]], multivector fields are often encountered as tensor fields of type (0,k), except in the context of the [[geometric algebra]] (see also [[Clifford algebra]]).<ref>{{Cite book|url=https://www.worldcat.org/oclc/213362465|title=Geometric algebra for physicists|last=Doran, Chris (Chris J. L.)|date=2007|publisher=Cambridge University Press|others=Lasenby, A. N. (Anthony N.), 1954-|isbn=9780521715959|edition=1st pbk. ed. with corr|location=Cambridge|oclc=213362465}}</ref><ref>{{Cite book|url=https://www.worldcat.org/oclc/757486966|title=Geometric algebra|last=Artin, Emil, 1898-1962.|year=1988|origyear=1957|publisher=Interscience Publishers|isbn=9781118164518|location=New York|oclc=757486966}}</ref><ref>{{Cite book|url=https://www.worldcat.org/oclc/769755408|title=A new approach to differential geometry using Clifford's geometric algebra|last=Snygg, John.|date=2012|publisher=Springer Science+Business Media, LLC|isbn=9780817682835|location=New York|oclc=769755408}}</ref> |
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In mathematics, a multivector field, polyvector field of degree k , or k-vector field, on a manifold , is a generalization of the notion of a vector field on a manifold. Whereas a vector field is a section[disambiguation needed] of tangent bundle, which assigns to each point on the manifold a tangent vector , a multivector field is a section of the kth exterior power of the tangent bundle, , and to each point it assigns a k-vector in . Just as the smooth sections of the tangent bundle (vector fields) make up a vector space, the space of smooth k-vector fields over M make up a vector space . Furthermore, since the tangent bundle is dual to the cotangent bundle, multivector fields of degree k are dual to k-forms, and both are subsumed in the general concept of a tensor field, which is a section of some tensor bundle, often consisting of exterior powers of the tangent and cotangent bundles. A (k,0)-tensor field is a differential k-form, a (0,1)-tensor field is a vector field, and a (0,k)-tensor field is k-vector field. While differential forms are widely studied as such in differential geometry and differential topology, multivector fields are often encountered as tensor fields of type (0,k), except in the context of the geometric algebra (see also Clifford algebra).[1][2][3]
See also
References
- ^ Doran, Chris (Chris J. L.) (2007). Geometric algebra for physicists. Lasenby, A. N. (Anthony N.), 1954- (1st pbk. ed. with corr ed.). Cambridge: Cambridge University Press. ISBN 9780521715959. OCLC 213362465.
- ^ Artin, Emil, 1898-1962. (1988) [1957]. Geometric algebra. New York: Interscience Publishers. ISBN 9781118164518. OCLC 757486966.
{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ Snygg, John. (2012). A new approach to differential geometry using Clifford's geometric algebra. New York: Springer Science+Business Media, LLC. ISBN 9780817682835. OCLC 769755408.
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