Jet

$\def\a{\alpha}$ The jet (finite or infinite) of a smooth function at a point is the collection of partial derivatives of this function up until the specified order.

Formal definition

Let $K\Subset U\subseteq\R^n$ be a compact set considered with its open neighborhood. Two functions $f,g\in C^m(U)$ are said to be $k$-equivalent, if their difference is $k$-flat on $K$, that is, $$ |f(x)-g(x)|=o(\operatorname{dist}(x,K)^k)\qquad \text{as }x\to K. $$ In particular, $f\equiv g$ on $K$. This definition does not depend on the choice of the metric in $U$. The equivalence class is called the $k$-jet of a smooth function on the compact $K$.

Jets as Taylor polynomials

In the most important particular case where $K$ is a point, one can always choose the Taylor polynomial of order $k$ (centered at this point) as the representative in the equivalence class. Two different Taylor polynomials of degree $k$ cannot be $k$-equivalent, thus one can identify jets of functions with their Taylor polynomials.

Example.

Jets of order zero are identified with the values of functions. First jet is completely determined by the value $f(a)$ and the differential $\rd f(a)$.

Jets of maps between manifolds

For smooth maps between two smooth manifolds $f:M\to N$ one can define the $k$-equivalence relation near a compact $K\Subset M$ in the same way as before, using local coordinates. The equivalence classes are called jets of smooth maps.

Extension of jets

For each smooth function defined in a neighborhood of a compact $K$, one may restrict on $K$ all its partial derivatives $f^{(\a)}=\partial^\a f$ of order $|\a|\le k$; these will depend only on the $k$-jet of $f$ on $K$. The inverse problem is to recognize when a collection of functions $f^\a$ is a jet of a smooth function defined near $K$. The complete answer is given by the Whitney extension theorem.

Notation

There are several more or less "standard" notation systems for jets. One of the most popular is as follows. For a given point $a\in M$ the space of $k$-jets at $a$ is denored by $J^k_a(M,N)$. The ensemble of jets at all points may be denoted then $J^k(M,N)=\bigsqcup_{a\in M}J^k_a (M,N)$. The jet of a function $f$ at $a$ may be denoted by $j^k_a f$, $j^k f(a)$, $f^{(k)}(a)$ or by variety of other ways.

Remark

For a fixed point $a$ the jet space $J^k_a(M,\R)$ has a natural affine structure independent on the local coordinates on $M$ near $a$ (which, in particular, allows to introduce the class of linear differential operators). The set $J^k(M,\R)$ has only the structure of a smooth manifold.

References

[1]	P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[2]	W. Guillemin, "Stable mappings and their singularities" , Springer (1973) MR0341518 Zbl 0294.58004
[3]	T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) MR0501079 Zbl 0382.58006