Jet mathematics - Example: Algebraic properties of jets

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. In fact, 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 vector bundle, analog ...

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Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between two manifolds

This subsection deals with the notion of jets of local sections a vector bundle. Almost everything in this section generalizes mutatis mutandis to the case of local sections of a fibre bundle, a Banach bundle over a Banach manifold, a fibered manifold, or quasi-coherent sheaves over schemes. Furthermore, these examples of possible generalizations are certainly not exhaustive. Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection . Then sections of E are smooth functions su ...

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Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of sections

This subsection focuses on two different rigorous definitions of the jet of a function at a point, followed by a discussion of Taylor's theorem. These definitions shall prove to be useful later on during the intrinsic definition of the jet of a function between two manifolds. Jet mathematics - An analytic definition. The following definition uses ideas from mathematical analysis to define jets and jet spaces. It can be generalized to smooth functions between Banach spaces, analytic fu ...

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Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions

Before giving a rigorous definition of a jet, it is useful to examine some special cases. Jet mathematics - Example: One-dimensional case. Suppose that is a real-valued function having at least k+1 derivatives in a neighborhood U of the point x0. Then by Taylor's theorem, where Then the k-jet of f at the point See also:

Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between Euclidean spaces