What is "one-form"

n. (context mathematics English) The dual of a vector; a linear functional which when applied to a vector yields a scalar, in a manner similar to the dot product

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

α : TM → R,  α = α∣ : TM → R

where α is linear.

Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:

α = f(x) dx + f(x) dx + ⋯ + f(x) dx

where the f are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.