One-form (differential geometry)

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 the atan2 function. Taking the derivative yields the following formula for the total derivative: $${\displaystyle {\begin{aligned}d\theta &=\partial _{x}\left(\operatorname {atan2} (y,x)\right)dx+\partial _{y}\left(\operatorname {atan2} (y,x)\right)dy\\&=-{\frac {y}{x^{2}+y^{2}}}dx+{\frac {x}{x^{2}+y^{2}}}dy\end{aligned}}}$$  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 the winding number times ${\displaystyle 2\pi .}$ 

In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact, 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 first de 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 a differentiable function ${\displaystyle f:U\to \mathbb {R} ,}$  with derivative ${\displaystyle f'.}$  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 ${\displaystyle f'(x_{0}).}$  This is the simplest example of a differential (one-)form.

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