In differential geometry, the jet bundle is a certain construction which 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 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.
Read more about Jet Bundle: Jets, Jet Manifolds, Jet Bundles, Contact Forms, Vector Fields, Partial Differential Equations, Jet Prolongation, Infinite Jet Spaces, Remark
Famous quotes containing the words jet and/or bundle:
“But every jet of chaos which threatens to exterminate us is convertible by intellect into wholesome force. Fate is unpenetrated causes.”
—Ralph Waldo Emerson (1803–1882)
““There is Lowell, who’s striving Parnassus to climb
With a whole bale of isms tied together with rhyme,
He might get on alone, spite of brambles and boulders,
But he can’t with that bundle he has on his shoulders,
The top of the hill he will ne’er come nigh reaching
Till he learns the distinction ‘twixt singing and preaching;”
—James Russell Lowell (1819–1891)