In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the Taylor polynomial (truncated Taylor series) of f, at each point of its domain.
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
is a real-valued function having at least k + 1 derivatives in a neighborhood U of the point
In other words, z is an indeterminate variable allowing one to perform various algebraic operations among the jets.
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.
We shall deal with the reasons and applications of this separation later in the article.
In this case, Taylor's theorem asserts that The k-jet of f is then defined to be the polynomial in
The first is a product structure, although this ultimately turns out to be the least important.
It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin.
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.
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.
consists of all function germs that vanish to order k at p. We may now define the jet space at p by If
Regardless of the definition, Taylor's theorem establishes a canonical isomorphism of vector spaces between
So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism.
Let f and g be a pair of curves through p. We will then say that f and g are equivalent to order k at p if there is some neighborhood U of p, such that, for every smooth function
forms a fibre bundle over M: the k-th-order tangent bundle, often denoted in the literature by TkM (although this notation occasionally can lead to confusion).
To prove that TkM is in fact a fibre bundle, it is instructive to examine the properties of
Let (xi)= (x1,...,xn) be a local coordinate system for M in a neighborhood U of p. Abusing notation slightly, we may regard (xi) as a local diffeomorphism
Hence the ostensible fibre bundle TkM admits a local trivialization in each coordinate neighborhood.
At this point, in order to prove that this ostensible fibre bundle is in fact a fibre bundle, it suffices to establish that it has non-singular transition functions under a change of coordinates.
be the associated change of coordinates diffeomorphism of Euclidean space to itself.
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. Examples in local coordinates: We are now prepared to define the jet of a function from a manifold to a manifold.
Note that because the target space N need not possess any algebraic structure,
This is, in fact, a sharp contrast with the case of Euclidean spaces.
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.
Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection
Although this notation can lead to confusion with the more general jet spaces of functions between two manifolds, the context typically eliminates any such ambiguity.