oneform

A one-form, or covector field, is an object in differential geometry that assigns to each point p of a manifold M a linear functional on the tangent space T_pM. Equivalently, it is an element of the cotangent space T_p^*M, and a smoothly varying selection of such linear functionals across points forms a one-form field on M. One-forms are dual to vector fields and are key objects in the study of manifolds and their differential structure.

In a local coordinate chart (x^1, ..., x^n), a one-form can be written as ω = ∑_i ω_i dx^i, where the dx^i are the dual basis to the coordinate vector fields ∂/∂x^i. Given a tangent vector v = ∑_i v^i ∂/∂x^i, the action of ω on v is the scalar ω(v) = ∑_i ω_i v^i. Thus a one-form represents a linear measurement of vectors at each point.

A basic example is the differential df of a smooth function f: M → R, which is a canonical one-form. More generally, any linear combination of the coordinate one-forms with smooth coefficients is a one-form.

Operations on one-forms include the exterior derivative d, which maps k-forms to (k+1)-forms and satisfies d^2 = 0. One-forms can be wedged to form higher-degree forms, pulled back along smooth maps, and contracted with vector fields to yield functions. The integration of a one-form along a curve γ gives a line integral ∫_γ ω, which depends on the path and orientation; if ω = df, the integral equals f(γ(b)) − f(γ(a)).

One-forms play a central role in de Rham cohomology, Stokes' theorem, and the geometric formulation of physics, where they interpret quantities like work and flux. The concept and formalism were developed in tensor calculus late in the 19th and early 20th centuries by Ricci-Curbastro and Levi-Civita.