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 $f:{\mathbb {R} }\rightarrow {\mathbb {R} }$ $f:{{\mathbb R}}\rightarrow {{\mathbb R}}$ is a real-valued function having at least k+1 derivatives in a neighborhood U of the point $x_{0}$ $x_{0}$ . Then by Taylor's theorem,

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