Jet (mathematics)

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces. It concludes with a description of jets between manifolds, and how these jets can be constructed intrinsically. In this more general context, it summarizes some of the applications of jets to differential geometry and the theory of differential equations. Before giving a rigorous definition of a jet, it is useful to examine some special cases. Suppose that is a real-valued function having at least k + 1 derivatives in a neighborhood U of the point . Then by Taylor's theorem, where Then the k-jet of f at the point is defined to be the polynomial Jets are normally regarded as abstract polynomials in the variable z, not as actual polynomial functions in that variable. In other words, z is an indeterminate variable allowing one to perform various algebraic operations among the jets. It is in fact the base-point from which jets derive their functional dependency. Thus, by varying the base-point, a jet yields a polynomial of order at most k at every point. This marks an important conceptual distinction between jets and truncated Taylor series: ordinarily a Taylor series is regarded as depending functionally on its variable, rather than its base-point. Jets, on the other hand, separate the algebraic properties of Taylor series from their functional properties. We shall deal with the reasons and applications of this separation later in the article. Suppose that is a function from one Euclidean space to another having at least (k + 1) derivatives.