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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. Jump to: navigation, search Image File history File links W21-1a. ...
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Jets of functions between Euclidean spaces
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
Example: One-dimensional case Suppose that f: is a real-valued function having at least k+1 derivatives in a neighborhood U of the point x0. Then by Taylor's theorem, In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
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where . Then the k-jet of f at the point x0 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 x0 from which jets derive their functional dependency. Thus, by varying the base-point, a jet yields a k-th order polynomial 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. Jump to: navigation, search In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
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Example: Mappings from one Euclidean space to another Suppose that f: is a function from one Euclidean space to another having at least (k+1) derivatives. In this case, the generalized Taylor theorem asserts that In this case, the k-jet of f is defined to be the polynomial
Example: 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 are a pair of real-valued functions, then we can define the product of their jets via
- .
Here we have supressed the indeterminate z, since it is understood that jets are formal polynomials. This product is just the product of ordinary polynomials in z, modulo zk + 1. In other words, it is multiplication in the ring where (zk + 1) is the ideal generated by polymials homogeneous of order ≥ k+1. The word modulo is the Latin ablative of modulus. ...
We now move to the composition of jets. To avoid unnecessary technicalities, we consider jets of functions which map the origin to the origin. If and with f(0)=0 and g(0)=0, then . The composition of jets is defined by It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin. In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...
In fact, the composition of k-jets is nothing more than the composition of polynomials modulo the ideal of polynomials homogeneous of order > k.
Examples:
- In one-dimension, let f(x) = log(1 − x) and . Then
and
Jets at a point in Euclidean space: Rigorous definitions 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.
An 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. Jump to: navigation, search Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...
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Let be 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. Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
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The k-th order jet space of at p is defined to be the set of equivalence classes of , and is denoted by .
The k-th order jet of a smooth function is defined to be the equivalence class of f in .
An 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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
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Let be the vector space of germs of smooth functions at a point p in . Let be the ideal of functions which vanish at p. (This is the maximal ideal for the local ring .) Then the ideal consists of all function germs which vanish to order k at p. We may now define the jet space at p by Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
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If is a smooth function, we may define the k-jet of f at p as the element of by setting
Taylor's theorem Regardless of the definition, Taylor's theorem (or its various generalizations: links please) establishes a canonical isomorphism of noncommutative monoids (under composition) between and . So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism.
Jets of functions between two manifolds 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 a equivariant fashion. Jets do not transform as tensors. In fact, jets of functions between two manifolds belong to a Jet bundle. In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
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This section begins by introducing the notion of jets of functions from a Euclidean space to a manifold, followed by a manifold to a Euclidean space. It proceeds to address the problem of definition the jet of a function between two smooth manifolds.
Jets of functions from the real line to a manifold Suppose that M is a smooth manifold containing a point p and is a curve in M such that f(0)=p. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
Jets of functions from a Euclidean space to a manifold
Jets of functions from a manifold to a the real line
Jets of functions from a manifold to a Euclidean space
Jets of sections This subsection deals with the notion of jets of sections of a vector bundle, followed by those of a fibred manifold.
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