One-form

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

d

f

{\displaystyle df}

for a smooth function

f

{\displaystyle f}

. The natural pairing means that if

v

{\displaystyle v}

is a tangent vector at a point

p

{\displaystyle p}

, and

α

{\displaystyle \alpha }

is a one-form, then

α

(

v

)

{\displaystyle \alpha (v)}

is a scalar. For the differential of a function,

d

f

(

v

)

{\displaystyle df(v)}

is the directional derivative, or the rate at which

f

{\displaystyle f}

is changing, at

p

{\displaystyle p}

and along

v

{\displaystyle 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 algebra of differential forms, and the differential of a function naturally extends to the exterior derivative: the exterior derivative of a one-form is a two-form, of a two-form is a three-form, and so on.

One-forms are widely used in differential geometry. Their duality with vector fields makes them suited to encode algebraic conditions on vector fields, and conditions such as integrability of systems of vector fields are often simpler to formulate in terms of one-forms, together with the exterior derivative.