P-vector

From Wikipedia, the free encyclopedia

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.

[edit] 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 \mathbf a \wedge \mathbf b has

  • a norm which is its area, given by
\Vert \mathbf a \wedge \mathbf b \Vert = \Vert \mathbf{a} \Vert \, \Vert \mathbf{b} \Vert \, \sin(\phi_{a,b})
  • 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 \dim V =n), it makes sense to define an inner product on this vector space as follows. First, write any element F \in \Lambda ^2 V in terms of a basis (e_a \wedge e_b)_{1 \le a < b \le n} of Λ2V as

F = F^{ab} e_a \wedge e_b \quad (1 \le a < b \le n)

where the Einstein summation convention is being used.

Now define a map G : \Lambda ^2 V \times \Lambda ^2 V \rightarrow \R by insisting that

G(F, H) := \, G_{abcd}F^{ab}H^{cd}

where Gabcd are a set of numbers.

[edit] 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.)

Languages