One-form

Common card · No. 485960

In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one. One-forms are dual to vector fields on the same manifold, in the sense that a one-form pairs naturally with a vector field to produce a real valued function, or with a tangent vector at a point to produce a real number. The simplest example of a one-form is the differential of a function df for a smooth function f. The natural pairing means that if v is a tangent vector at a point p, and α is a one-form, then α(v) is a scalar. For the differential of a function, df(v) is the directional derivative, or the rate at which f is changing, at p and along v. A one-form defines a covector, also called a linear functional, at each point on the manifold. One-forms are generally assumed to vary smoothly from point to point on the manifold, or on a region of the manifold, meaning that they produce smooth functions when paired with smooth vector fields. One-forms generate the exterior...

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