 | Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between Euclidean spaces
Jet mathematics - Jets of functions between Euclidean spaces
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
Jet mathematics - Example: One-dimensional case
Suppose that is a real-valued function having at least k+1 derivatives in a neighborhood U of the point x0. Then by Taylor's theorem,
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.
Jet mathematics - Example: Mappings from one Euclidean space to another
Suppose that 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
Jet mathematics - 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 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 zk + 1. In other words, it is multiplication in the ring where (zk + 1) 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 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 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
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 Adapted from the Wikipedia article "Jets of functions between Euclidean spaces", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
