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- Jet_group abstract "In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ(0)=0. The following is a more precise definition of the jet group.Let k ≥ 2. The gradient of a function f: Rk → R can be interpreted as a section of the cotangent bundle of RK given by df: Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm(Rk) = Rk × W, whereand Si denotes the i-th symmetric power. A function f: Rk → R has a prolongation jmf: Rn → Jm(Rn) defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si((R*)k) component of W.Consider a point . There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is, . The differential data x′ may be transferred to lie over another point y ∈ Rn as jmfp(y) , the partials of fp over y. Provide Jm(Rn) with a group structure by takingWith this group structure, Jm(Rn) is a Carnot group of class m + 1.Because of the properties of jets under function composition, Gnk is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations.".
- Jet_group wikiPageID "3139577".
- Jet_group wikiPageRevisionID "579704669".
- Jet_group hasPhotoCollection Jet_group.
- Jet_group subject Category:Lie_groups.
- Jet_group type Abstraction100002137.
- Jet_group type Group100031264.
- Jet_group type LieGroups.
- Jet_group comment "In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ(0)=0. The following is a more precise definition of the jet group.Let k ≥ 2.".
- Jet_group label "Jet group".
- Jet_group sameAs m.08tyrw.
- Jet_group sameAs Q17098315.
- Jet_group sameAs Q17098315.
- Jet_group sameAs Jet_group.
- Jet_group wasDerivedFrom Jet_group?oldid=579704669.
- Jet_group isPrimaryTopicOf Jet_group.