Indifferential geometry, aone-form (orcovector field) on adifferentiable manifold is adifferential form of degree one, that is, asmoothsection of thecotangent bundle.^[1] Equivalently, a one-form on a manifold${\displaystyle M}$ is a smooth mapping of thetotal space of thetangent bundle of${\displaystyle M}$ to${\displaystyle \mathbb{R}}$ whose restriction to each fibre is a linear functional on the tangent space.^[2] Let${\displaystyle U}$ be an open subset of${\displaystyle M}$ and${\displaystyle p\in U}$. Thendefines a one-form${\displaystyle \omega }$.${\displaystyle {\omega }_p}$ is a covector.

Often one-forms are describedlocally, particularly inlocal coordinates. In a local coordinate system, a one-form is a linear combination of thedifferentials of the coordinates:$${\displaystyle {\alpha }_x=f_1(x)d{x}_1+f_2(x)d{x}_2+\cdots +f_n(x)d{x}_n,}$$where the${\displaystyle f_i}$ are smooth functions. From this perspective, a one-form has acovariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covarianttensor field.

The most basic non-trivial differential one-form is the "change in angle" form${\displaystyle d\theta .}$ This is defined as the derivative of the angle "function"${\displaystyle \theta (x,y)}$ (which is only defined up to an additive constant), which can be explicitly defined in terms of theatan2 function. Taking the derivative yields the following formula for thetotal derivative:While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative${\displaystyle y}$-axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local)changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives thewinding number times${\displaystyle 2\pi .}$

In the language ofdifferential geometry, this derivative is a one-form on thepunctured plane. It isclosed (itsexterior derivative is zero) but notexact, meaning that it is not the derivative of a 0-form (that is, a function): the angle${\displaystyle \theta }$ is not a globally defined smooth function on the entire punctured plane. In fact, this form generates the firstde Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

Let${\displaystyle U\subseteq \mathbb{R}}$ beopen (for example, an interval${\displaystyle (a,b)}$), and consider adifferentiable function${\displaystyle f:U\to \mathbb{R},}$ withderivative${\displaystyle f^{\prime }.}$ The differential${\displaystyle df}$ assigns to each point${\displaystyle x_0\in U}$ a linear map from the tangent space${\displaystyle T_{x_0}U}$ to the real numbers. In this case, each tangent space is naturally identifiable with the real number line, and the linear map${\displaystyle \mathbb{R}\to \mathbb{R}}$ in question is given by scaling by${\displaystyle f^{\prime }(x_0).}$ This is the simplest example of a differential (one-)form.

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