Multivector

In multilinear algebra, a multivector, sometimes called Clifford number or multor,^[1] is an element of the exterior algebra $Λ(V)$ of a vector space $V$ . This algebra is graded, associative and alternating, and consists of linear combinations of simple $k$ -vectors^[2] (also known as decomposable $k$ -vectors^[3] or $k$ -blades) of the form

v_{1}\wedge \cdots \wedge v_{k},

where $v_{1},\ldots ,v_{k}$ are in $V$ .

A $k$ -vector is such a linear combination that is homogeneous of degree $k$ (all terms are $k$ -blades for the same $k$ ). Depending on the authors, a "multivector" may be either a $k$ -vector or any element of the exterior algebra (any linear combination of $k$ -blades with potentially differing values of $k$ ).^[4]

In differential geometry, a $k$ -vector is a vector in the exterior algebra of the tangent vector space; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of $k$ tangent vectors, for some integer $k \geq 0$ . A differential $k$ -form is a $k$ -vector in the exterior algebra of the dual of the tangent space, which is also the dual of the exterior algebra of the tangent space.

For $k = 0, 1, 2$ and $3$ , $k$ -vectors are often called respectively scalars, vectors, bivectors and trivectors; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms.^[5]^[6]

^ John Snygg (2012), A New Approach to Differential Geometry Using Clifford’s Geometric Algebra, Birkhäuser, p. 5 §2.12
^ Cite error: The named reference Flanders was invoked but never defined (see the help page).
^ Wendell Fleming (1977) [1965] Functions of Several Variables, section 7.5 Multivectors, page 295, ISBN 978-1-4684-9461-7
^ Élie Cartan, The theory of spinors, p. 16, considers only homogeneous vectors, particularly simple ones, referring to them as "multivectors" (collectively) or p-vectors (specifically).
^ William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz (ed.). Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0-7923-2576-1. Hence in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector
^ Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X.

[1] John Snygg (2012), A New Approach to Differential Geometry Using Clifford’s Geometric Algebra, Birkhäuser, p. 5 §2.12

[Flanders-2] Cite error: The named reference Flanders was invoked but never defined (see the help page).

[3] Wendell Fleming (1977) [1965] Functions of Several Variables, section 7.5 Multivectors, page 295, ISBN 978-1-4684-9461-7

[4] Élie Cartan, The theory of spinors, p. 16, considers only homogeneous vectors, particularly simple ones, referring to them as "multivectors" (collectively) or p-vectors (specifically).

[Ławrynowicz-5] William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz (ed.). Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0-7923-2576-1. Hence in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector

[Baylis-6] Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X.

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