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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[edit] 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> (Note that this mapping is not necessarily linear, but is conjugate linear for complex vector spaces).
[edit] 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 that 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 additional structure (a direction) needs to be included in order to define the negative of the one-form. Also, adding one-forms is not as straightforward as adding vectors. Because of this, such a visualization must be seen as only a rudimentary concept.
[edit] 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.
[edit] 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.
[edit] See also
[edit] References
- Bernard F. Schutz (1985, 2002). A first course in general relativity. Cambridge University Press: Cambridge, UK. Chapter 3. ISBN 0-521-27703-5.
- Richard Bishop and Samuel Goldberg(1968,1980). "Tensor Analysis on Manifolds" Dover Publications. Chapter 4. ISBN 0-486-64039-6bg:Ковектор
