Editing One-form (differential geometry) (section)

==Examples==

The most basic non-trivial differential one-form is the "change in angle" form <math>d\theta.</math> This is defined as the derivative of the angle "function" <math>\theta(x,y)</math> (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]]:
<math display=block>\begin{align}
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{align}</math>
While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative <math>y</math>-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) {{em|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 <math>2 \pi.</math>

In the language of [[differential geometry]], this derivative is a one-form on the [[punctured plane]]. It is [[Closed differential form|closed]] (its [[exterior derivative]] is zero) but not [[Exact differential form|exact]], meaning that it is not the derivative of a 0-form (that is, a function): the angle <math>\theta</math> 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.