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One-form

In differential geometry, a one-form on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space. Let be an open subset of and . Then defines a one-form . is a covector.

Examples
The most basic non-trivial differential one-form is the "change in angle" form d\theta. This is defined as the derivative of the angle "function" \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: \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} While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative 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) 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 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 \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. ==Differential of a function==
Differential of a function
Let U \subseteq \R be open (for example, an interval (a, b)), and consider a differentiable function f: U \to \R, with derivative f'. The differential df assigns to each point x_0\in U a linear map from the tangent space 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 \mathbb{R}\to\mathbb{R} in question is given by scaling by f'(x_0). This is the simplest example of a differential (one-)form. ==See also==
Source: Wikipedia ↗
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