oneforms

One-forms are linear functionals that map a tangent vector at a point on a differentiable manifold to a real number. At each point p, the set of all one-forms forms the cotangent space T_p^*M, the dual space to the tangent space T_pM. A one-form is thus a covector that varies smoothly from point to point, and a smooth choice of a covector across the manifold is a section of the cotangent bundle.

In a coordinate chart, the standard basis of one-forms is dx^1, dx^2, ..., dx^n. Any one-form ω can be written as ω = ∑ a_i dx^i, where the a_i are smooth functions. Given a tangent vector v = ∑ v^i ∂/∂x^i, the action of ω on v is ω(v) = ∑ a_i v^i. This evaluates the linear functional to a real number.

A central example is the differential of a function f, denoted df, which is a one-form. In coordinates, df = ∑ (∂f/∂x^i) dx^i, and df_p(v) equals the directional derivative of f along v at p. One-forms can be integrated along curves: the line integral ∫_C ω yields a scalar, independent of a particular parameterization but dependent on the path and the form.

More broadly, the space of all one-forms on a manifold forms the sections of the cotangent bundle, T^*M. One-forms are dual to vector fields and interact with exterior calculus, where they can be wedged to form higher-degree forms and differentiated via the exterior derivative.