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One-form (differential geometry)
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{{Short description|Differential form of degree one or section of a cotangent bundle}} {{redirect-distinguish|One-form|One-form (linear algebra)}}In [[differential geometry]], a '''one-form''' (or '''covector field''') on a [[differentiable manifold]] is a [[differential form]] of degree one, that is, a [[Smooth function|smooth]] [[Section (fiber bundle)|section]] of the [[cotangent bundle]].<ref>{{Cite web |title=2 Introducing Differential Geometry‣ General Relativity by David Tong |url=http://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S2.html |access-date=2022-10-04 |website=www.damtp.cam.ac.uk}}</ref> Equivalently, a one-form on a manifold <math>M</math> is a smooth mapping of the [[total space]] of the [[tangent bundle]] of <math>M</math> to <math>\R</math> whose restriction to each fibre is a linear functional on the tangent space.<ref>{{Cite book |last=McInerney |first=Andrew |url=https://books.google.com/books?id=nNK4BAAAQBAJ |title=First Steps in Differential Geometry: Riemannian, Contact, Symplectic |date=2013-07-09 |publisher=Springer Science & Business Media |isbn=978-1-4614-7732-7 |pages=136–155 |language=en}}</ref> Let <math>\omega</math> be a one-form. Then <math display=block>\begin{align} \omega: U & \rightarrow \bigcup_{p \in U} T^*_p(\R^n) \\ p & \mapsto \omega_p \in T_p^*(\R^n) \end{align}</math> Often one-forms are described [[Local property|locally]], particularly in [[local coordinates]]. In a local coordinate system, a one-form is a linear combination of the [[exterior derivative|differentials]] of the coordinates: <math display=block>\alpha_x = f_1(x) \, dx_1 + f_2(x) \, dx_2 + \cdots + f_n(x) \, dx_n ,</math> where the <math>f_i</math> are smooth functions. From this perspective, a one-form has a [[Covariance and contravariance of vectors|covariant]] transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant [[tensor field]]. ==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. ==Differential of a function== {{main|Differential of a function}} Let <math>U \subseteq \R</math> be [[Open set|open]] (for example, an interval <math>(a, b)</math>), and consider a [[differentiable function]] <math>f: U \to \R,</math> with [[derivative]] <math>f'.</math> The differential <math>df</math> assigns to each point <math>x_0\in U</math> a linear map from the tangent space <math>T_{x_0}U</math> to the real numbers. In this case, each tangent space is naturally identifiable with the real number line, and the linear map <math>\mathbb{R}\to\mathbb{R}</math> in question is given by scaling by <math>f'(x_0).</math> This is the simplest example of a differential (one-)form. ==See also== * {{annotated link|Differential form}} * {{annotated link|Inner product}} * {{annotated link|Reciprocal lattice}} * {{annotated link|Tensor}} ==References== {{reflist}} {{tensors}} {{Manifolds}} [[Category:Differential forms]] [[Category:1 (number)]]
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