In mathematics , the jet is an operation which 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 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-pointfrom 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.
Suppose thatis a function from one Euclidean space to another having at least (k+1) derivatives. In this case, Taylor's theorem asserts that
in, where.
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.
Ifare 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 . In other words, it is multiplication in the ring, whereis 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 which map the origin to the origin. Ifandwith f (0)=0 and g (0)=0, then. The composition of jets is defined byIt 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.
This subsection focuses on two different rigorous definitions of the jet of a function at a point, followed by a discussion of Taylor's theorem. These definitions shall prove to be useful later on during the intrinsic definition of the jet of a function between two manifolds.
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.
Letbe the vector space of smooth functions . Let k be a non-negative integer, and let p be a point of. We define an equivalence relation 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,iffto k -th order.
Theofat p is defined to be the set of equivalence classes of, and is denoted by.
Theat p of a smooth functionis defined to be the equivalence class of f in.
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.
Letbe the vector space of germs of smooth functions at a point p in. Letbe the ideal of functions which vanish at p . (This is the maximal ideal for the local ring .) Then the idealconsists of all function germs which vanish to order k at p . We may now define the jet space at p by
Ifis a smooth function, we may define the k -jet of f at p as the element ofby setting
If M and N are two smooth manifolds , how do we define the jet of a function? 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 equivariant fashion. Jets do not transform as tensors . Instead, jets of functions between two manifolds belong to a jet bundle .
This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a fibre bundle , analogous to the tangent bundle , which is an associated bundle of a jet group . It proceeds to address the problem of defining the jet of a function between two smooth manifolds. Throughout this section, we adopt an analytic approach to jets. Although an algebro-geometric approach is also suitable for many more applications, it is too subtle to be dealt with systematically here. See jet (algebraic geometry) for more details.
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 functionssuch that f (0)= p . Define an equivalence relationas 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,. Note that these jets are well-defined since the composite functionsandare 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 theof a curve f through p to be the equivalence class of f under, denotedor. Theis then the set of k -jets at p . This forms a real vector space.
As p varies over M ,forms a fibre bundle over M : the k -th order tangent bundle , often denoted in the literature by T 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 M is in fact a fibre bundle, it is instructive to examine the properties ofin local coordinates. Let ( x )= ( x ,..., x ) be a local coordinate system for M in a neighborhood U of p . Abusing notation slightly, we may regard ( x ) as a local diffeomorphism .
Claim. Two curves f and g through p are equivalent moduloif and only if.
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 spaceconsisting of smooth mapsdefined in some neighborhood of p . We define an equivalence relationonas 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 mappingsuch that), we haveon some neighborhood of.
The jet spaceis then defined to be the set of equivalence classes ofmodulo the equivalence relation. Note that because the target space N need not possess any algebraic structure,also need not have such a structure. This is, in fact, a sharp contrast with the case of Euclidean spaces.
Ifis a smooth function defined near p , then we define the k -jet of f at p ,, to be the equivalence class of f modulo.
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