Editing One-form (differential geometry) (section)

{{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]].