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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 global section 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 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.
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In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
References
- ↑ John Vince: Geometric Algebra: An Algebraic System for Computer Games and Animation, Springer, 2009, ISBN 978-1-84882-379-2, p. 3
- ↑ Val L. Fitch, Daniel R. Marlow, Margit Ann Elisabeth Dementi: Critical problems in physics: proceedings of a conference celebrating the 250th anniversary of Princeton university, Princeton University Press, 1997, ISBN 0-691-05785-0, p. 165
- ↑ J. Lasenby, A. N. Lasenby, C. J. L. Doran: A unified mathematical language for physics and engineering in the 21st century, Phil. Trans. R. Soc. Lond. A 358, 21-39 (2000) (sbtract, full text)
- ↑ Lasenby, Anthony; Chris Doran; Stephen Gull (1998), "Gravity, gauge theories and geometric algebra", Philosophical Transactions of the Royal Society A , 356: 487–582, arXiv: gr-qc/0405033 , Bibcode:1998RSPTA.356..487L, doi:10.1098/rsta.1998.0178
