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.
Jets of functions between Euclidean spaces.
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
One-dimensional case.
Suppose that formula_1 is a real-valued function having at least "k" + 1 derivatives in a neighborhood "U" of the point formula_2. Then by Taylor's theorem,
where
Then the "k"-jet of "f" at the point formula_2 is defined to be the polynomial
Jets are normally regarded as abstract polynomials in the variable "z", not as actual polynomial functions in that variable. In other words, "z" is an indeterminate variable allowing one to perform various algebraic operations among the jets. It is in fact the base-point formula_2 from which jets derive their functional dependency. 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. Jets, on the other hand, separate the algebraic properties of Taylor series from their functional properties. We shall deal with the reasons and applications of this separation later in the article.
Mappings from one Euclidean space to another.
Suppose that formula_8 is a function from one Euclidean space to another having at least ("k" + 1) derivatives. In this case, Taylor's theorem asserts that
The "k"-jet of "f" is then defined to be the polynomial
in formula_11, where formula_12.
Algebraic properties of jets.
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.
If formula_13 are a pair of real-valued functions, then we can define the product of their jets via
Here we have suppressed the indeterminate "z", since it is understood that jets are formal polynomials. This product is just the product of ordinary polynomials in "z", modulo formula_15. In other words, it is multiplication in the ring formula_16, where formula_17 is the ideal generated by polynomials homogeneous of order ≥ "k" + 1.
We now move to the composition of jets. To avoid unnecessary technicalities, we consider jets of functions that map the origin to the origin. If formula_18 and formula_19 with "f"(0) = 0 and "g"(0) = 0, then formula_20. The "composition of jets" is defined by
formula_21
It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin.
In fact, the composition of "k"-jets is nothing more than the composition of polynomials modulo the ideal of polynomials homogeneous of order formula_22.
"Examples:"
and
Jets at a point in Euclidean space: rigorous definitions.
Analytic definition.
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.
Let formula_28 be the vector space of smooth functions formula_29. Let "k" be a non-negative integer, and let "p" be a point of formula_30. We define an equivalence relation formula_31 on this space by declaring that two functions "f" and "g" are equivalent to order "k" if "f" and "g" have the same value at "p", and all of their partial derivatives agree at "p" up to (and including) their "k"-th-order derivatives. In short,formula_32 iff formula_33 to "k"-th order.
The "k"-th-order jet space of formula_28 at "p" is defined to be the set of equivalence classes of formula_35, and is denoted by formula_36.
The "k"-th-order jet at "p" of a smooth function formula_37 is defined to be the equivalence class of "f" in formula_36.
Algebro-geometric definition.
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.
Let formula_39 be the vector space of germs of smooth functions formula_29 at a point "p" in formula_30. Let formula_42 be the ideal consisting of germs of functions that vanish at "p". (This is the maximal ideal for the local ring formula_39.) Then the ideal formula_44 consists of all function germs that vanish to order "k" at "p". We may now define the jet space at "p" by
If formula_29 is a smooth function, we may define the "k"-jet of "f" at "p" as the element of formula_36 by setting
This is a more general construction. For an formula_49-space formula_50, let formula_51 be the stalk of the structure sheaf at formula_52 and let formula_42 be the maximal ideal of the local ring formula_51. The kth jet space at formula_52 is defined to be the ring formula_56(formula_44 is the product of ideals).
Taylor's theorem.
Regardless of the definition, Taylor's theorem establishes a canonical isomorphism of vector spaces between formula_36 and formula_59. So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism.
Jet spaces from a point to a point.
We have defined the space formula_36 of jets at a point formula_61. The subspace of this consisting of jets of functions "f" such that "f"("p") = "q" is denoted by
Jets of functions between two manifolds.
If "M" and "N" are two smooth manifolds, how do we define the jet of a function formula_63? We could perhaps attempt to define such a jet by using local coordinates on "M" and "N". The disadvantage of this is that jets cannot thus be defined in an invariant fashion. Jets do not transform as tensors. Instead, jets of functions between two manifolds belong to a jet bundle.
Jets of functions from the real line to a manifold.
Suppose that "M" is a smooth manifold containing a point "p". We shall define the jets of curves through "p", by which we henceforth mean smooth functions formula_64 such that "f"(0) = "p". Define an equivalence relation formula_31 as follows. 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 formula_66, formula_67. Note that these jets are well-defined since the composite functions formula_68 and formula_69 are just mappings from the real line to itself. This equivalence relation is sometimes called that of "k"-th-order contact between curves at "p".
We now define the k"-jet of a curve "f" through "p" to be the equivalence class of "f" under formula_35, denoted formula_71 or formula_72. The k"-th-order jet space formula_73 is then the set of "k"-jets at "p".
As "p" varies over "M", formula_73 forms a fibre bundle over "M": the "k"-th-order tangent bundle, often denoted in the literature by "T""k""M" (although this notation occasionally can lead to confusion). In the case "k"=1, then the first-order tangent bundle is the usual tangent bundle: "T"1"M" = "TM".
To prove that "T""k""M" is in fact a fibre bundle, it is instructive to examine the properties of formula_73 in local coordinates. Let ("x""i")= ("x"1...,"x""n") be a local coordinate system for "M" in a neighborhood "U" of "p". Abusing notation slightly, we may regard ("x""i") as a local diffeomorphism formula_76.
"Claim." Two curves "f" and "g" through "p" are equivalent modulo formula_31 if and only if formula_78.
Hence the ostensible fibre bundle "T""k""M" 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. Let formula_88 be a different coordinate system and let formula_89 be the associated change of coordinates diffeomorphism of Euclidean space to itself. By means of an affine transformation of formula_30, we may assume without loss of generality that ρ(0)=0. With this assumption, it suffices to prove that formula_91 is an invertible transformation under jet composition. (See also jet groups.) But since ρ is a diffeomorphism, formula_92 is a smooth mapping as well. Hence,
which proves that formula_94 is non-singular. Furthermore, it is smooth, although we do not prove that fact here.
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:"
Jets of functions from a manifold to a manifold.
We are now prepared to define the jet of a function from a manifold to a manifold.
Suppose that "M" and "N" are two smooth manifolds. Let "p" be a point of "M". Consider the space formula_103 consisting of smooth maps formula_63 defined in some neighborhood of "p". We define an equivalence relation formula_35 on formula_103 as follows. Two maps "f" and "g" are said to be "equivalent" if, for every curve γ through "p" (recall that by our conventions this is a mapping formula_107 such that formula_108), we have formula_109 on some neighborhood of "0".
The jet space formula_110 is then defined to be the set of equivalence classes of formula_103 modulo the equivalence relation formula_35. Note that because the target space "N" need not possess any algebraic structure, formula_110 also need not have such a structure. This is, in fact, a sharp contrast with the case of Euclidean spaces.
If formula_63 is a smooth function defined near "p", then we define the "k"-jet of "f" at "p", formula_115, to be the equivalence class of "f" modulo formula_35.
Multijets.
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.
Jets of sections.
Suppose that "E" is a finite-dimensional smooth vector bundle over a manifold "M", with projection formula_117. Then sections of "E" are smooth functions formula_118 such that formula_119 is the identity automorphism of "M". The jet of a section "s" over a neighborhood of a point "p" is just the jet of this smooth function from "M" to "E" at "p".
The space of jets of sections at "p" is denoted by formula_120. 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.
Unlike jets of functions from a manifold to another manifold, the space of jets of sections at "p" carries the structure of a vector space inherited from the vector space structure on the sections themselves. As "p" varies over "M", the jet spaces formula_120 form a vector bundle over "M", the "k"-th-order jet bundle of "E", denoted by "J""k"("E").