One-form

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This article deals with differential one-forms, a global construction on differential manifolds. For one-forms on vector spaces, i.e. the corresponding local concept, see linear functional.

A one-form on a vector space is simply a linear functional. A (differential) one-form on a differential manifold is a section of the cotangent bundle--in other words, a map that sends each point of the manifold to an element of the cotangent space (i.e. a continuous linear functional on the tangent space) at the point in question. A one-form on a manifold may thus be viewed as a linear functional that varies from point to point, in the same way that a vector field (resp. tensor field) may be thought of as a vector (resp. tensor) that varies from point to point. (One then sees that the use of the term "one-form" in this context is actually an abuse of language of the same type as the use of "tensor" to mean "tensor field"--though presumably more acceptable than the latter, due to the awkwardness of a term such as "one-form-field".)

[edit] Special cases

Let $U \subseteq \mathbb{R}$ be open (e.g. an interval $(a, b)$ ), and consider a differentiable function $f: U \to \mathbb{R}$ , with derivative f'. The differential df of f, at a point $x_0\in U$ , is defined as a certain linear map of the variable dx. Specifically, $df(x_0, dx): dx \mapsto f'(x_0) dx$ . (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map $x \mapsto df(x,dx)$ sends each point x to a linear functional df(x,dx). This is the simplest example of a differential (one-)form.