# 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.

## Contents

### 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 |

**How to Cite This Entry:**

Jet.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jet&oldid=30978