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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.
Introduction
A one-form is a tensor of type . It is the simplest non-scalar tensor.
Let represent a one-form which acts on vectors of space V, including vectors and . Then the linearity properties of are
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 in the space V, the following should hold true:
If these last two conditions are true for every then the one-forms constitute a vector space.
If V is an inner-product space with inner product 〈 , 〉 then every vector can be mapped to a dual one-form defined by
(i.e. in lambda notation) so that the one-form applied to a vector yields
Thus the inner product provides a bijection of each vector in V to a one-form of its dual vector space (Note that this mapping is not necessarily linear, but is conjugate linear for complex vector spaces).
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.
Basis of the dual space
Let the vector space V have a basis , … , , not necessarily orthonormal nor even orthogonal. Then the dual space has a basis , … , which in the three-dimensional case (n = 3) can be defined by
where is the Levi-Civita symbol . This definition has the special property that
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 belonging to the dual space can be expressed as a linear combination of basis one-forms, with coefficients ("components") ui ,
Then, applying one-form to a basis vector ej yields
due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then
that is
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
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 0-486-64039-6