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, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, that is, a function), and in fact it 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 ${\displaystyle U\subseteq \mathbb {R} }$ be open (for example, an interval ${\displaystyle (a,b)}$), and consider a differentiable function ${\displaystyle f:U\to \mathbb {R} ,}$ with derivative ${\displaystyle f'.}$ The differential ${\displaystyle df}$ of ${\displaystyle f,}$ at a point ${\displaystyle x_{0}\in U,}$ is defined as a certain linear map of the variable ${\displaystyle dx.}$ Specifically, ${\displaystyle df(x_{0},\cdot ):dx\mapsto f'(x_{0})dx.}$ (The meaning of the symbol ${\displaystyle dx}$ is thus revealed: it is simply an argument, or independent variable, of the linear function ${\displaystyle df(x_{0},\cdot ).}$) Hence the map ${\displaystyle x\mapsto df(x)}$ sends each point ${\displaystyle x}$ to a linear functional ${\displaystyle df(x,\cdot ).}$ This is the simplest example of a differential (one-)form.

In terms of the de Rham cochain complex, one has an assignment from zero-forms (scalar functions) to one-forms; that is, ${\displaystyle f\mapsto df.}$

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