One-form (differential geometry)

"One-form" redirects here. Not to be confused withOne-form (linear algebra).

Indifferential geometry,aone-form(orcovector field) on adifferentiable manifoldis adifferential formof degree one, that is, asmoothsectionof thecotangent bundle.Equivalently, a one-form on a manifold$M$is a smooth mapping of thetotal spaceof thetangent bundleof$M$to$\mathbb {R}$whose restriction to each fibre is a linear functional on the tangent space.Symbolically,

$$\ Alpha:TM\rightarrow {\mathbb {R} },\quad \ Alpha _{x}=\ Alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },$$ where$\ Alpha _{x}$is linear.

Often one-forms are describedlocally,particularly inlocal coordinates.In a local coordinate system, a one-form is a linear combination of thedifferentialsof the coordinates: $$\ Alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},$$ where the$f_{i}$are smooth functions. From this perspective, a one-form has acovarianttransformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covarianttensor field.

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 theatan2function. Taking the derivative yields the following formula for thetotal derivative: $${\begin{aligned}d\theta &=\partial _{x}\left(\operatorname {atan2} (y,x)\right)dx+\partial _{y}\left(\operatorname {atan2} (y,x)\right)dy\\&=-{\frac {y}{x^{2}+y^{2}}}dx+{\frac {x}{x^{2}+y^{2}}}dy\end{aligned}}$$ While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative$y$-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)changesin 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 numbertimes$2\pi.$

In the language ofdifferential geometry,this derivative is a one-form on thepunctured plane.It isclosed(itsexterior derivativeis zero) but notexact,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 firstde Rham cohomologyof the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.

Differential of a function

Main article:Differential of a function

Let$U\subseteq \mathbb {R}$beopen(for example, an interval$(a,b)$), and consider adifferentiable function$f:U\to \mathbb {R},$withderivative$f'.$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

• Differential form– Expression that may be integrated over a region
• Inner product– Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
• Reciprocal lattice– Fourier transform of a real-space lattice, important in solid-state physics
• Tensor– Algebraic object with geometric applications