Swain equation

2013-12-08T23:20:59Z

142.162.104.209:

{{Redirect|p-vector||K-vector (disambiguation)}}
In [[multilinear algebra]], a '''multivector''' or '''Clifford number'''<ref>John Snygg (2012), ''A New Approach to Differential Geometry Using Clifford’s Geometric Algebra'', Birkhäuser, p.5 §2.12</ref> is an element of the (graded) [[exterior algebra]] on a [[vector space]], {{math|Λ*''V''}}. This algebra consists of [[linear combination]]s of '''simple ''k''-vectors''' (also known as '''decomposable''' ''k''-vectors or [[Blade (geometry)|''k''-blades]])
:<math> v_1\wedge\cdots\wedge v_k.</math>
"Multivector" may mean either ''homogeneous'' elements (all terms of the sum have the same grade or degree ''k''), which are referred to as '''''k''-vectors''' or '''''p''-vectors''',<ref>Élie Cartan, ''The theory of spinors'', [http://books.google.com/books?id=AEZ1h7Cg3cwC&pg=PA16&dq=p-vector+multivectors p. 16], considers only homogeneous vectors, particularly simple ones, referring to them as "multivectors" (collectively) or ''p''-vectors (specifically).</ref> or may allow sums of terms in different degrees.

The ''k''-th exterior power,
:<math>\Lambda^k(V),</math>
is the vector space of formal sums of ''k''-multivectors. The product of a ''k''-multivector and an ''ℓ''-multivector is a {{math|(''k'' + ''ℓ'')}}-multivector. So, the direct sum <math>\bigoplus_k \Lambda^k(V)</math> forms an associative algebra, which is closed with respect to the wedge product. This algebra, commonly denoted by {{math|Λ(''V'')}}, is called the [[exterior algebra]] of ''V''.

In [[differential geometry]], a '''''p''-vector''' is the antisymmetric [[tensor]] obtained by taking [[linear combination]]s of the [[wedge product]] of ''p'' [[tangent vector]]s, for some integer ''p'' ≥ 0. It is the [[Dual space|dual]] concept to a [[p-form|''p''-form]].

For ''p'' = 0, 1, 2 and 3, these are often called respectively '''''[[Scalar (mathematics)|scalar]]s''''', '''''vectors''''', '''''[[bivector]]s''''' and '''''trivectors'''''; they are respectively dual to [[0-form]]s, [[1-form]]s, [[2-form]]s and 3-forms.<ref name="Ławrynowicz">

{{cite book |title=Deformations of mathematical structures II |url=http://books.google.com/books?id=KfNgBHNUW_cC&pg=PA131 |page=131 ''ff'' |isbn=0-7923-2576-1 |author=William M Pezzaglia Jr.|editor=Julian Ławrynowicz |year=1992 |publisher =Springer |chapter=Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations |quote=Hence in 3D we associate the alternate terms of ''pseudovector'' for [[bivector]], and ''pseudoscalar'' for the [[p-vector|trivector]]}}

</ref><ref name=Baylis>

{{cite book |author=Baylis |title=Theoretical methods in the physical sciences: an introduction to problem solving using Maple V |url=http://books.google.com/books?id=pEfMq1sxWVEC&pg=PA234 |page=234, see footnote |isbn=0-8176-3715-X |year=1994 |publisher=Birkhäuser}}

</ref>

==Examples==
[[File:N-vector.svg|thumb|125px|Geometric interpretation for the '''exterior product''' of ''n'' [[vector (geometry)|vector]]s ('''u''', '''v''', '''w''') to obtain an ''n''-vector ([[parallelotope]] elements), where ''n'' = [[graded algebra|grade]],<ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</ref> for ''n'' = 1, 2, 3. The "circulations" show [[Orientation (vector space)|orientation]].<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=83|isbn=0-7167-0344-0}}</ref>]]

* 0-vectors are scalars;
* 1-vectors are vectors;
* 2-vectors are [[bivector]]s;
* (''n'' − 1)-vectors are [[pseudovector]]s;
* ''n''-vectors are [[pseudoscalar]]s.
In the presence of a [[volume form]] (such as given an [[inner product]] and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in [[vector calculus]], but without a volume form this cannot be done without a choice.

In the [[Algebra of physical space]] (the geometric algebra of Euclidean 3-space, used as a model of 3+1 spacetime), a sum of a scalar and a vector is called a [[paravector]], and represents a point in spacetime (the vector the space, the scalar the time).

===Bivectors===
{{main|Bivector}}
A '''bivector''' is therefore an element of the [[antisymmetric tensor|antisymmetric]] [[tensor product]] of a [[tangent space]] with itself.

In [[geometric algebra]], also, a '''bivector''' is a grade 2 element (a 2-vector) resulting from the [[wedge product]] of two vectors, and so it is geometrically an ''oriented area'', in the same way a ''vector'' is an oriented line segment.
If '''a''' and '''b''' are two vectors, the bivector '''a''' &and; '''b''' has
* a [[norm (mathematics)|norm]] which is its area, given by
::<math>\Vert \mathbf a \wedge \mathbf b \Vert = \Vert \mathbf{a} \Vert \,
\Vert \mathbf{b} \Vert \, \sin(\phi_{a,b})</math>
* a direction: the plane where that area lies on, i.e., the plane determined by '''a''' and '''b''', as long as they are linearly independent;
* an orientation (out of two), determined by the order in which the originating vectors are multiplied.
Bivectors are connected to [[pseudovector]]s, and are used to represent rotations in geometric algebra.

As bivectors are elements of a vector space Λ2''V'' (where ''V'' is a finite-dimensional vector space with <math>\dim V =n</math>), it makes sense to define an [[inner product]] on this vector space as follows. First, write any element ''F'' ∈ Λ2''V'' in terms of a basis {{nowrap|1=('''e'''''i'' &and; '''e'''''j'')1 ≤ ''i'' < ''j'' ≤ ''n'' of Λ2''V''}} as

: <math>F = F^{ab} e_a \wedge e_b \quad (1 \le a 

where the [[Einstein summation convention]] is being used.

Now define a map {{nowrap|G : Λ2''V'' × Λ2''V'' → '''R'''}} by insisting that

: <math>G(F, H) := \, G_{abcd}F^{ab}H^{cd}
</math>

where <math>G_{abcd}</math> are a set of numbers.

==Geometric algebra==
{{See also|Blade (geometry)}}
In [[geometric algebra]], a multivector is defined to be the sum of different-grade [[blade (geometry)|''k''-blades]], such as the summation of a [[scalar (mathematics)|scalar]], a [[Vector (geometric)|vector]], and a ''2''-vector.<ref name= Rodrigues>

{{cite book |title=Invariants for pattern recognition and classification |author=Marcos A. Rodrigues |chapter=§1.2 Geometric algebra: an outline |url=http://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3 |page=3 ''ff'' |isbn=981-02-4278-6 |year=2000 |publisher=World Scientific}}
</ref> A sum of only ''k''-grade components is called a ''k''-vector,<ref name=Sommer>

{{cite book |title=Computer algebra and geometric algebra with applications |editor=Hongbo Li, Peter J. Olver, Gerald Sommer |url=http://books.google.com/books?id=uxofVAQE3LoC&pg=PA330 |chapter=Applications of conformal geometric algebra in computer vision and graphics |page=330 |author=R Wareham, J Cameron & J Lasenby |isbn=3-540-26296-2 |year=2005 |publisher=Springer}}
</ref> or a ''homogeneous'' multivector.<ref name=Sanfeliu>

{{cite book |title=Progress in pattern recognition, image analysis and applications |url=http://books.google.com/books?id=gsnXS1xdeekC&pg=PA25 |page=25 |editor=Alberto Sanfeliu, José Francisco Martínez Trinidad, Jesús Ariel Carrasco Ochoa |author=Eduardo Bayro-Corrochano |chapter = Clifford geometric algebra: A promising framework for computer vision, robotics and learning |isbn=3-540-23527-2 |publisher=Springer |year=2004}}
</ref>

The highest grade element in a space is called a ''[[pseudoscalar]]''.

If a given element is homogeneous of a grade ''k'', then it is a ''k''-vector, but not necessarily a ''k''-blade. Such an element is a ''k''-blade when it can be expressed as the wedge product of ''k'' vectors. A geometric algebra generated by a 4-dimensional Euclidean vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

== Applications ==

Bivectors play many important roles in physics, for example, in the [[classification of electromagnetic fields]].

==See also==
* [[Blade (geometry)]]
* [[Paravector]]

==Notes==
{{reflist}}

{{tensors}}

[[Category:Multilinear algebra]]
[[Category:Tensors]]
[[Category:Differential geometry]]
[[Category:Geometric algebra]]

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Swain equation