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One-form

Encyclopedia : O : ON : ONE : One-form

A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space.

Contents

Introduction

A one-form is a tensor of type [ \begin 0 \\ 1 \end ]. It is the simplest non-scalar tensor.

Let [\tilde ] represent a one-form which acts on vectors of space V, including vectors [\vec u] and [\vec v]. Then the linearity properties of [\tilde ] are

[ \tilde (\vec u + \vec v) = \tilde (\vec u) + \tilde (\vec v)]
[ \tilde (\alpha \vec v) = \alpha \tilde (\vec v) ]
where α is a scalar.

The set of all one-forms definable on the vector space V can also itself be a vector space if one-forms can be added to each other or be multiplied by scalars in a pointwise linear manner. That is, if the vectors of the space V are position vectors of points, then for every point [\vec v] in the space V, the following should hold true:

[ (\tilde + \tilde) (\vec v) = \tilde(\vec v) + \tilde(\vec v) ]
[ (\alpha \tilde) (\vec v) = \alpha \tilde(\vec v). ]
If these last two conditions are true for every [\vec v \isin V] then the one-forms constitute a vector space.

If V is an inner-product space with inner product 〈 , 〉 then every vector [\vec v] can be mapped to a dual one-form [\tilde] defined by

[ \tilde := \langle \vec v, \ \rangle ]
(i.e. [ \tilde := \lambda x. \langle \vec v, x \rangle ] in lambda notation) so that the one-form [\tilde ] applied to a vector [\vec u] yields
[\tilde (\vec u) = \langle \vec v, \vec u\rangle. ]
Thus the inner product provides a bijection of each vector in V to a one-form of its dual vector space [\tilde].

Visualizing one-forms

A vector is usually visualized as an arrow extending from the origin to a point in space. A one-form can be visualized as a set of equally spaced parallel planes which partition the entire space. The magnitude of a one-form is directly proportional to the density of parallel planes and inversely proportional to the spacing between pairs of neighboring planes. To find the result of applying a one-form to a vector, basically count the number of planes which a vector cuts through. (Note: this visualization is discrete whereas one-forms and vectors have magnitudes which range continuously over the real numbers. The visualization can be interpolated linearly, as it were, to increase the precision.)

Unfortunately, the problem with visualizing a one-form as a set of planes is that there is no simple way to define the negative of the one-form, or addition of one-forms. Because of this, such a visualization must be seen as only a rudimentary concept.

Basis of the dual space

Let the vector space V have a basis [_1,\ _2], … , [ _n], not necessarily orthonormal nor even orthogonal. Then the dual space [\tilde] has a basis [\tilde^1, \ \tilde^2], … , [\ \tilde^n] which in the three-dimensional case (n = 3) can be defined by
[ \tilde^i = \, \left\langle \, (\vec e_j \times \vec e_k) \over \vec e_1 \cdot \vec e_2 \times \vec e_3} , \qquad \right\rangle ]
where [\epsilon\,\!] is the Levi-Civita symbol . This definition has the special property that
[ \tilde^i (\vec e_j) = \delta^i _j ]
where δ is the Kronecker delta. Thus, these two dual bases are mutually orthonormal even if each basis is not self-orthonormal.

N.B. The superscripts of the basis one-forms are not exponents but are instead contravariant indices.

A one-form [\tilde] belonging to the dual space [\tilde] can be expressed as a linear combination of basis one-forms, with coefficients ("components") ui ,

[\tilde = u_i \, \tilde^i ]
Then, applying one-form [\tilde] to a basis vector ej yields
[\tilde(\vec e_j) = (u_i \, \tilde^i) \vec e_j = u_i (\tilde^i (\vec e_j)) ]
due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then
[ \tilde(_j) = u_i (\tilde^i (_j)) = u_i \delta^i _j = u_j ]
that is
[\tilde (\vec e_j) = u_j. ]
This last equation shows that an individual component of a one-form can be extracted by applying the one-form to a corresponding basis vector.

Differential one-forms

A differential one-form is a one-form the components of which are all differential. It is the simplest non-scalar differential form.

See also

Reference

 

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
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