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.

There are two basic algebraic structures jets can carry. The first is a product structure, although this ultimately turns out to be the least important. The second is the structure of the composition of jets.

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 functions between real or complex domains, to p-adic analysis, and to other areas of analysis.

The following definition uses ideas from algebraic geometry and commutative algebra to establish the notion of a jet and a jet space. Although this definition is not particularly suited for use in algebraic geometry per se, since it is cast in the smooth category, it can easily be tailored to such uses.

Intuitively, this means that we can express the jet of a curve through p in terms of its Taylor series in local coordinates on M.

is a smooth real-valued function of one variable whose 1-jet is given by
which proves that one can naturally identify tangent vectors at a point with the 1-jets of curves through that point.
Hence, the transformation law is given by evaluating these two expressions at t = 0.
Note that the transformation law for 2-jets is second-order in the coordinate transition functions.

We are now prepared to define the jet of a function from a manifold to a manifold.

John Mather introduced the notion of multijet. Loosely speaking, a multijet is a finite list of jets over different base-points. Mather proved the multijet transversality theorem, which he used in his study of stable mappings.

We work in local coordinates at a point and use the Einstein notation. Consider a vector field
Note that the transformation law is second-order in the coordinate transition functions.