Editing Jet (mathematics) (section)

==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 <math>f: {\mathbb R}\rightarrow{\mathbb R}</math> is a real-valued function having at least ''k''&nbsp;+&nbsp;1 [[derivative]]s in a [[neighbourhood (mathematics)|neighborhood]] ''U'' of the point <math>x_0</math>. Then by Taylor's theorem,

:<math>f(x)=f(x_0)+f'(x_0)(x-x_0)+\cdots+\frac{f^{(k)}(x_0)}{k!}(x-x_0)^{k}+\frac{R_{k+1}(x)}{(k+1)!}(x-x_0)^{k+1}</math>

where
:<math>|R_{k+1}(x)|\le\sup_{x\in U} |f^{(k+1)}(x)|.</math>
Then the '''''k''-jet''' of ''f'' at the point <math>x_0</math> is defined to be the polynomial
:<math>(J^k_{x_0}f)(z)
=\sum_{i=0}^k \frac{f^{(i)}(x_0)}{i!}z^i
=f(x_0)+f'(x_0)z+\cdots+\frac{f^{(k)}(x_0)}{k!}z^k.</math>

Jets are normally regarded as [[Polynomial#Abstract algebra|abstract polynomials]] in the variable ''z'', not as actual polynomial functions in that variable. In other words, ''z'' is an [[indeterminate (variable)|indeterminate variable]] allowing one to perform various [[abstract algebra|algebraic operations]] among the jets. It is in fact the base-point <math>x_0</math> 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 <math>f:{\mathbb R}^n\rightarrow{\mathbb R}^m</math> is a function from one Euclidean space to another having at least (''k''&nbsp;+&nbsp;1) derivatives. In this case, [[Taylor's theorem]] asserts that

:<math>
\begin{align}
f(x)=f(x_0)+ (Df(x_0))\cdot(x-x_0)+ {} & \frac{1}{2}(D^2f(x_0))\cdot (x-x_0)^{\otimes 2} + \cdots \\[4pt]
& \cdots +\frac{D^kf(x_0)}{k!}\cdot(x-x_0)^{\otimes k}+\frac{R_{k+1}(x)}{(k+1)!}\cdot(x-x_0)^{\otimes (k+1)}.
\end{align}
</math>

The ''k''-jet of ''f'' is then defined to be the polynomial

:<math>(J^k_{x_0}f)(z)=f(x_0)+(Df(x_0))\cdot z+\frac{1}{2}(D^2f(x_0))\cdot z^{\otimes 2} + \cdots + \frac{D^kf(x_0)}{k!}\cdot z^{\otimes k}</math>

in <math>{\mathbb R}[z]</math>, where <math>z=(z_1,\ldots,z_n)</math>.

===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 <math>f,g:{\mathbb R}^n\rightarrow {\mathbb R}</math> are a pair of real-valued functions, then we can define the product of their jets via

:<math>J^k_{x_0}f\cdot J^k_{x_0}g=J^k_{x_0}(f\cdot g).</math>

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 (jargon)|modulo]] <math>z^{k+1}</math>. In other words, it is multiplication in the ring <math>{\mathbb R}[z]/(z^{k+1})</math>, where <math>(z^{k+1})</math> is the [[Ideal (ring theory)|ideal]] generated by homogeneous polynomials of order&nbsp;&ge;&nbsp;''k''&nbsp;+&nbsp;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 <math>f:{\mathbb R}^m\rightarrow{\mathbb R}^\ell</math> and <math>g:{\mathbb R}^n\rightarrow{\mathbb R}^m</math> with ''f''(0)&nbsp;=&nbsp;0 and ''g''(0)&nbsp;=&nbsp;0, then <math>f\circ g:{\mathbb R}^n \rightarrow{\mathbb R}^\ell</math>. The ''composition of jets'' is defined by
<math>J^k_0 f\circ J^k_0 g=J^k_0 (f\circ g).</math>
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 homogeneous polynomials of order&nbsp;&ge;&nbsp;''k''&nbsp;+&nbsp;1.

''Examples:''
*In one dimension, let <math>f(x)=\log(1-x)</math> and <math>g(x)=\sin\,x</math>. Then

:<math>(J^3_0f)(x)=-x-\frac{x^2}{2}-\frac{x^3}{3}</math>

:<math>(J^3_0g)(x)=x-\frac{x^3}{6}</math>

and

:<math>
\begin{align}
& (J^3_0f)\circ (J^3_0g)=-\left(x-\frac{x^3}{6}\right)-\frac{1}{2}\left(x-\frac{x^3}{6}\right)^2-\frac{1}{3} \left(x-\frac{x^3}{6}\right)^3 \pmod{x^4} \\[4pt]
= {} & -x-\frac{x^2}{2}-\frac{x^3}{6}
\end{align}
</math>