Editing Jet bundle (section)

{{Short description|Construction in differential topology}}
{{Redirect-distinguish|Jet space|Space jet (disambiguation){{!}}space jet}}
{{technical|date=May 2025}}In [[differential topology]], the '''jet bundle''' is a certain construction that makes a new [[smooth manifold|smooth]] [[fiber bundle]] out of a given smooth fiber bundle. It makes it possible to write [[differential equation]]s on [[Fiber bundle#Sections|section]]s of a fiber bundle in an invariant form. [[Jet (mathematics)|Jets]] may also be seen as the coordinate free versions of [[Taylor expansions]].

Historically, jet bundles are attributed to [[Charles Ehresmann]], and were an advance on the method ([[Cartan's equivalence method|prolongation]]) of [[Élie Cartan]], of dealing ''geometrically'' with [[derivative|higher derivatives]], by imposing [[differential form]] conditions on newly introduced formal variables.  Jet bundles are sometimes called '''sprays''', although [[spray (mathematics)|sprays]] usually refer more specifically to the associated [[vector field]] induced on the corresponding bundle (e.g., the [[geodesic spray]] on [[Finsler manifold]]s.)

Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the [[calculus of variations]].<ref>{{cite book
 | last = Krupka
 | first = Demeter
 |author-link= Demeter Krupka
 | title = Introduction to Global Variational Geometry
 | year=  2015
 | publisher = Atlantis Press
 | isbn = 978-94-6239-073-7
 |url= https://www.springer.com/it/book/9789462390720
 }}</ref> Consequently, the jet bundle is now recognized as the correct domain for a [[covariant classical field theory|geometrical covariant field theory]] and much work is done in [[general relativity|general relativistic]] formulations of fields using this approach.