Jets of Functions Between Two Manifolds
If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an equivariant fashion. Jets do not transform as tensors. Instead, jets of functions between two manifolds belong to a jet bundle.
This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a fibre bundle, analogous to the tangent bundle, which is an associated bundle of a jet group. It proceeds to address the problem of defining the jet of a function between two smooth manifolds. Throughout this section, we adopt an analytic approach to jets. Although an algebro-geometric approach is also suitable for many more applications, it is too subtle to be dealt with systematically here. See jet (algebraic geometry) for more details.
Read more about this topic: Jet (mathematics)
Famous quotes containing the word functions:
“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)