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

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A one-form is a tensor of type $\backslash begin\{pmatrix\}\; 0\; \backslash \backslash \; 1\; \backslash end\{pmatrix\}$. It is the simplest non-scalar tensor.

Let $\backslash tilde\{f\}$ represent a one-form which acts on vectors of space V, including vectors $\backslash vec\; u$ and $\backslash vec\; v$. Then the linearity properties of $\backslash tilde\{f\}$ are

$\backslash tilde\{f\}\; (\backslash vec\; u\; +\; \backslash vec\; v)\; =\; \backslash tilde\{f\}\; (\backslash vec\; u)\; +\; \backslash tilde\{f\}\; (\backslash vec\; v)$

$\backslash tilde\{f\}\; (\backslash alpha\; \backslash vec\; v)\; =\; \backslash alpha\; \backslash tilde\{f\}\; (\backslash vec\; v)$

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 $\backslash vec\; v$ in the space V, the following should hold true:

$(\backslash tilde\{f\}\; +\; \backslash tilde\{g\})\; (\backslash vec\; v)\; =\; \backslash tilde\{f\}(\backslash vec\; v)\; +\; \backslash tilde\{g\}(\backslash vec\; v)$

$(\backslash alpha\; \backslash tilde\{f\})\; (\backslash vec\; v)\; =\; \backslash alpha\; \backslash tilde\{f\}(\backslash vec\; v).$

If these last two conditions are true for every $\backslash vec\; v\; \backslash isin\; V$ then the one-forms constitute a vector space.

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

$\backslash tilde\{v\}\; :=\; \backslash langle\; \backslash vec\; v,\; \backslash \; \backslash rangle$

(i.e. $\backslash tilde\{v\}\; :=\; \backslash lambda\; x.\; \backslash langle\; \backslash vec\; v,\; x\; \backslash rangle$ in lambda notation) so that the one-form $\backslash tilde\{v\}$ applied to a vector $\backslash vec\; u$ yields

$\backslash tilde\{v\}\; (\backslash vec\; u)\; =\; \backslash langle\; \backslash vec\; v,\; \backslash vec\; u\backslash rangle.$

Thus the inner product provides a bijection of each vector in V to a one-form of its dual vector space $\backslash tilde\{V\}$.

Visualizing one-forms

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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

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Let the vector space V have a basis $\{\backslash vec\; e\}\_1,\backslash \; \{\backslash vec\; e\}\_2$, … , $\{\backslash vec\; e\}\_n$, not necessarily orthonormal nor even orthogonal. Then the dual space $\backslash tilde\{V\}$ has a basis $\backslash tilde\{\backslash omega\}^1,\; \backslash \; \backslash tilde\{\backslash omega\}^2$, … , $\backslash \; \backslash tilde\{\backslash omega\}^n$ which in the three-dimensional case (n = 3) can be defined by

$\backslash tilde\{\backslash omega\}^i\; =\; \{1\; \backslash over\; 2\}\; \backslash ,\; \backslash left\backslash langle\; \{\; \backslash epsilon^\{ijk\}\; \backslash ,\; (\backslash vec\; e\_j\; \backslash times\; \backslash vec\; e\_k)\; \backslash over\; \backslash vec\; e\_1\; \backslash cdot\; \backslash vec\; e\_2\; \backslash times\; \backslash vec\; e\_3\}\; ,\; \backslash qquad\; \backslash right\backslash rangle$

where $\backslash epsilon\backslash ,\backslash !$ is the Levi-Civita symbol . This definition has the special property that

$\backslash tilde\{\backslash omega\}^i\; (\backslash vec\; e\_j)\; =\; \backslash delta^i\; \{\}\_j$

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 $\backslash tilde\{u\}$ belonging to the dual space $\backslash tilde\{V\}$ can be expressed as a linear combination of basis one-forms, with coefficients ("components") u_i ,

$\backslash tilde\{u\}\; =\; u\_i\; \backslash ,\; \backslash tilde\{\backslash omega\}^i$

Then, applying one-form $\backslash tilde\{u\}$ to a basis vector e_j yields

$\backslash tilde\{u\}(\backslash vec\; e\_j)\; =\; (u\_i\; \backslash ,\; \backslash tilde\{\backslash omega\}^i)\; \backslash vec\; e\_j\; =\; u\_i\; (\backslash tilde\{\backslash omega\}^i\; (\backslash vec\; e\_j))$

due to linearity of scalar multiples of one-forms and pointwise linearity of sums of one-forms. Then

$\backslash tilde\{u\}(\{\backslash vec\; e\}\_j)\; =\; u\_i\; (\backslash tilde\{\backslash omega\}^i\; (\{\backslash vec\; e\}\_j))\; =\; u\_i\; \backslash delta^i\; \{\}\_j\; =\; u\_j$

that is

$\backslash tilde\{u\}\; (\backslash vec\; e\_j)\; =\; u\_j.$

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

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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

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• Two-form
• Intermediate treatment of tensors

Reference

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• Bernard F. Schutz (1985, 2002). A first course in general relativity. Cambridge University Press: Cambridge, UK. Chapter 3. ISBN 0-521-27703-5.

Differential forms

Ковектор