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where <math>G_{abcd}</math> are a set of numbers. |
where <math>G_{abcd}</math> are a set of numbers. |
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===Bivector in three-dimensional Euclidean space=== |
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If '''a''' and '''b''' are any vectors in a real three-dimensional Euclidean vector space, then the complex vector '''A''' given by: |
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:<math>\boldsymbol A = \boldsymbol a + i \boldsymbol b \ , </math> |
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is a ''bivector'', where ''i'' = √−1. Two bivectors '''A''' = '''a''' + ''i'' '''b''' and '''A′''' = '''a′''' + ''i'' '''b′''' are equal if and only if '''a = a′''' and '''b = b′'''.<ref name= Hayes> |
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{{cite book |title=Bivectors and waves in mechanics and optics |url=http://books.google.com/books?id=QN0Ks3fTPpAC&pg=PA16 |author=Philippe Boulanger, Michael A. Hayes |chapter=Chapter 2: Bivectors |isbn=0412464608 |year=1993 |publisher=CRC Press}} |
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</ref> |
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Given an orthonormal triad of real unit vectors '''i, j, k''', |
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:<math>\boldsymbol A = A_1 \ \boldsymbol i + A_2\ \boldsymbol j + A_3\ \boldsymbol k \ </math> |
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::<math> = (a_1 + i\ b_1)\ \boldsymbol i + (a_2 + i\ b_2)\ \boldsymbol j + (a_3 + i\ b_3)\ \boldsymbol k \ . </math> |
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Introducing another bivector '''B''', dot and cross products are:<ref name=Hayes/> |
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:<math>\boldsymbol {A \cdot B} = A_1B_1 + A_2B_2 +A_3B_3 \ , </math> |
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:<math>\boldsymbol {A \times B} = (A_2B_3 - A_3B_2)\ \boldsymbol i +(A_3B_1 - A_1B_3)\ \boldsymbol j +(A_1B_2 - A_2B_1) \ \boldsymbol k \ . </math> |
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== Applications of ''p''-vectors == |
== Applications of ''p''-vectors == |
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(Alternatively, [[four-vector]] is used in [[theory of relativity|relativity]] to mean a quantity related to the four-dimensional [[spacetime]]. In analogy, the term ''three-vector'' is sometimes used as a synonym for a spatial [[vector (geometry)|vector]] in three dimensions. These meanings are different from ''p''-vectors for ''p'' equal to 3 or 4.) |
(Alternatively, [[four-vector]] is used in [[theory of relativity|relativity]] to mean a quantity related to the four-dimensional [[spacetime]]. In analogy, the term ''three-vector'' is sometimes used as a synonym for a spatial [[vector (geometry)|vector]] in three dimensions. These meanings are different from ''p''-vectors for ''p'' equal to 3 or 4.) |
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==Notes== |
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<references/> |
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{{DEFAULTSORT:P-Vector}} |
{{DEFAULTSORT:P-Vector}} |
Revision as of 18:13, 21 December 2009
In differential geometry, a p-vector is the tensor obtained by taking linear combinations of the wedge product of p tangent vectors, for some integer p ≥ 1. It is the dual concept to a p-form.
For p = 2 and 3, these are often called respectively bivectors and trivectors; they are dual to 2-forms and 3-forms.
Bivectors
A bivector is therefore an element of the 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 ∧ b has
- a norm which is its area, given by
- 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 pseudovectors, and are used to represent rotations in geometric algebra.
As bivectors are elements of a vector space Λ2V (where V is a finite-dimensional vector space with ), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ2V in terms of a basis (ei ∧ ej)1 ≤ i < j ≤ n of Λ2V as
where the Einstein summation convention is being used.
Now define a map G : Λ2V × Λ2V → R by insisting that
where are a set of numbers.
Bivector in three-dimensional Euclidean space
If a and b are any vectors in a real three-dimensional Euclidean vector space, then the complex vector A given by:
is a bivector, where i = √−1. Two bivectors A = a + i b and A′ = a′ + i b′ are equal if and only if a = a′ and b = b′.[1]
Given an orthonormal triad of real unit vectors i, j, k,
Introducing another bivector B, dot and cross products are:[1]
Applications of p-vectors
Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields.
(Alternatively, four-vector is used in relativity to mean a quantity related to the four-dimensional spacetime. In analogy, the term three-vector is sometimes used as a synonym for a spatial vector in three dimensions. These meanings are different from p-vectors for p equal to 3 or 4.)
Notes
- ^ a b Philippe Boulanger, Michael A. Hayes (1993). "Chapter 2: Bivectors". Bivectors and waves in mechanics and optics. CRC Press. ISBN 0412464608.