Jet bundle | Wikipedia audio article
Channel: wikipedia tts
Date: 02/06/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Jet_bundle


00:01:09 1 Jets
00:03:26 2 Jet manifolds
00:05:58 3 Jet bundles
00:08:33 3.1 Algebraic-geometric perspective
00:11:09 3.2 Example
00:14:07 4 Contact structure
00:16:54 4.1 Example
00:19:14 5 Vector fields
00:25:17 6 Partial differential equations
00:25:29 6.1 Example
00:30:49 7 Jet prolongation
00:36:04 7.1 Example
00:45:09 8 Infinite jet spaces
00:45:56 8.1 Infinitely prolonged PDEs
00:52:49 9 Remark
01:08:05 10 See also
01:28:52 11 References
01:33:09 12 Further reading
01:35:54 Remark
01:36:19 (p, f(p))(grf is known as the graph of the function f) of the trivial bundle (M × N, π1, M). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π1.
01:36:37 See also



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SUMMARY
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In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. 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 (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables.  Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.)
More recently, 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. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.