One-form
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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.
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Introduction
A one-form is a tensor of type <math> \begin{pmatrix} 0 \\ 1 \end{pmatrix} </math>. It is the simplest non-scalar tensor.
Let <math>\tilde{f} </math> represent a one-form which acts on vectors of space V, including vectors <math>\vec u</math> and <math>\vec v</math>. Then the linearity properties of <math>\tilde{f} </math> are
- <math> \tilde{f} (\vec u + \vec v) = \tilde{f} (\vec u) + \tilde{f} (\vec v)</math>
- <math> \tilde{f} (\alpha \vec v) = \alpha \tilde{f} (\vec v) </math>
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 <math>\vec v</math> in the space V, the following should hold true:
- <math> (\tilde{f} + \tilde{g}) (\vec v) = \tilde{f}(\vec v) + \tilde{g}(\vec v) </math>
- <math> (\alpha \tilde{f}) (\vec v) = \alpha \tilde{f}(\vec v). </math>
If these last two conditions are true for every <math>\vec v \isin V</math> then the one-forms constitute a vector space.
If V is an inner-product space with inner product 〈 , 〉 then every vector <math>\vec v</math> can be mapped to a dual one-form <math>\tilde{v}</math> defined by
- <math> \tilde{v} := \langle \vec v, \ \rangle </math>
(i.e. <math> \tilde{v} := \lambda x. \langle \vec v, x \rangle </math> in lambda notation) so that the one-form <math>\tilde{v} </math> applied to a vector <math>\vec u</math> yields
- <math>\tilde{v} (\vec u) = \langle \vec v, \vec u\rangle. </math>
Thus the inner product provides a bijection of each vector in V to a one-form of its dual vector space <math>\tilde{V}</math>.
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.)
Basis of the dual space
Let the vector space V have a basis <math>{\vec e}_1,\ {\vec e}_2</math>, … , <math> {\vec e}_n</math>, not necessarily orthonormal nor even orthogonal. Then the dual space <math>\tilde{V}</math> has a basis <math>\tilde{\omega}^1, \ \tilde{\omega}^2</math>, … , <math>\ \tilde{\omega}^n</math> which in the three-dimensional case (n = 3) can be defined by
- <math> \tilde{\omega}^i = {1 \over 2} \, \left\langle { \epsilon^{ijk} \, (\vec e_j \times \vec e_k) \over \vec e_1 \cdot \vec e_2 \times \vec e_3} , \qquad \right\rangle </math>
where <math>\epsilon\,\!</math> is the Levi-Civita symbol . This definition has the special property that
- <math> \tilde{\omega}^i (\vec e_j) = \delta^i {}_j </math>
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 <math>\tilde{u}</math> belonging to the dual space <math>\tilde{V}</math> can be expressed as a linear combination of basis one-forms, with coefficients ("components") ui ,
- <math>\tilde{u} = u_i \, \tilde{\omega}^i </math>
Then, applying one-form <math>\tilde{u}</math> to a basis vector ej yields
- <math>\tilde{u}(\vec e_j) = (u_i \, \tilde{\omega}^i) \vec e_j = u_i (\tilde{\omega}^i (\vec e_j)) </math>
due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then
- <math> \tilde{u}({\vec e}_j) = u_i (\tilde{\omega}^i ({\vec e}_j)) = u_i \delta^i {}_j = u_j </math>
that is
- <math>\tilde{u} (\vec e_j) = u_j. </math>
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
- Bernard F. Schutz (1985, 2002). A first course in general relativity. Cambridge University Press: Cambridge, UK. Chapter 3. ISBN 0-521-27703-5.