Namespaces
Variants
Actions

Difference between revisions of "Jet"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m
Line 1: Line 1:
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542301.png" /> obtained by the truncation of the (formal) [[Taylor series|Taylor series]] of a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542302.png" />. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542304.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542305.png" />-manifolds. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542306.png" /> of equivalent triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542308.png" /> is open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j0542309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423010.png" /> is a mapping of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423011.png" />, is then called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423013.png" />-jet from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423014.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423015.png" />. Equivalence is defined thus:
+
The jet (finite or infinite) of a smooth function at a point is the collection of partial derivatives of this function up until the specified order.
 +
===Formal definition===
 +
Let $K\Subset U\subseteq\R^n$ be a compact set considered with its open neighborhood. Two functions $f,g\in C^m(U)$ are said to be $k$-equivalent, if their difference is $k$-flat on $K$, that is,
 +
$$
 +
|f(x)-g(x)|=o(\operatorname{dist}(x,K)^k)\qquad \text{as }x\to K.
 +
$$
 +
In particular, $f\equiv g$ on $K$. This definition does not depend on the choice of the metric in $U$. The equivalence class is called the $k$-jet of a smooth function on the compact $K$.
 +
===Jets as Taylor polynomials===
 +
In the most important particular case where $K$ is a point, one can always choose the Taylor polynomial of order $k$ (centered at this point) as the representative in the equivalence class. Two different Taylor polynomials of degree $k$ cannot be $k$-equivalent, thus one can identify jets of functions with their Taylor polynomials.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423016.png" /></td> </tr></table>
+
;Example.
 +
Jets of order zero are identified with the values of functions. First jet is completely determined by the value $f(a)$ and the differential $\rd f(a)$.
 +
=== Jets of maps between manifolds===
 +
For smooth maps between two smooth manifolds $f:M\to N$ one can define the $k$-equivalence relation near a compact $K\Subset M$  in the same way as before, using local coordinates. The equivalence classes are called jets of smooth maps.
 +
=== Notation ===
 +
There are several "standard" notation systems for jets. One of the most popular is as follows.
 +
For a given point $a\in M$ the space of $k$-jets at $a$ is denored by $J^k_a(M,N)$. The ensemble of jets at all points may be denoted then $J^k(M,N)=\bigsqcup_{a\in M}J^k_a (M,N)$. The jet of a function $f$ at $a$ may be denoted by $j^k_a f$, $j^k f(a)$, $f^{(k)}(a)$ or by variety of other ways.
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423017.png" /> and if the local images of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423019.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423020.png" /> in relation to a pair of charts have identical derivatives up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423021.png" />, inclusive. The space of jets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423022.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054230/j05423023.png" />-manifold.
+
===Remark===
 +
For a fixed point $a$ the jet space $J^k_a(M,\R)$ has a natural affine structure independent on the local coordinates on $M$ near $a$ (which, in particular, allows to introduce the class of linear differential operators). The set $J^k(M,\R)$ has only the structure of a smooth manifold.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) {{MR|0494220}} {{ZBL|0302.58006}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Guillemin, "Stable mappings and their singularities" , Springer (1973) {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) {{MR|0501079}} {{ZBL|0382.58006}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) {{MR|0494220}} {{ZBL|0302.58006}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Guillemin, "Stable mappings and their singularities" , Springer (1973) {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) {{MR|0501079}} {{ZBL|0382.58006}} </TD></TR></table>

Revision as of 14:35, 30 April 2012

The jet (finite or infinite) of a smooth function at a point is the collection of partial derivatives of this function up until the specified order.

Formal definition

Let $K\Subset U\subseteq\R^n$ be a compact set considered with its open neighborhood. Two functions $f,g\in C^m(U)$ are said to be $k$-equivalent, if their difference is $k$-flat on $K$, that is, $$ |f(x)-g(x)|=o(\operatorname{dist}(x,K)^k)\qquad \text{as }x\to K. $$ In particular, $f\equiv g$ on $K$. This definition does not depend on the choice of the metric in $U$. The equivalence class is called the $k$-jet of a smooth function on the compact $K$.

Jets as Taylor polynomials

In the most important particular case where $K$ is a point, one can always choose the Taylor polynomial of order $k$ (centered at this point) as the representative in the equivalence class. Two different Taylor polynomials of degree $k$ cannot be $k$-equivalent, thus one can identify jets of functions with their Taylor polynomials.

Example.

Jets of order zero are identified with the values of functions. First jet is completely determined by the value $f(a)$ and the differential $\rd f(a)$.

Jets of maps between manifolds

For smooth maps between two smooth manifolds $f:M\to N$ one can define the $k$-equivalence relation near a compact $K\Subset M$ in the same way as before, using local coordinates. The equivalence classes are called jets of smooth maps.

Notation

There are several "standard" notation systems for jets. One of the most popular is as follows. For a given point $a\in M$ the space of $k$-jets at $a$ is denored by $J^k_a(M,N)$. The ensemble of jets at all points may be denoted then $J^k(M,N)=\bigsqcup_{a\in M}J^k_a (M,N)$. The jet of a function $f$ at $a$ may be denoted by $j^k_a f$, $j^k f(a)$, $f^{(k)}(a)$ or by variety of other ways.

Remark

For a fixed point $a$ the jet space $J^k_a(M,\R)$ has a natural affine structure independent on the local coordinates on $M$ near $a$ (which, in particular, allows to introduce the class of linear differential operators). The set $J^k(M,\R)$ has only the structure of a smooth manifold.

References

[1] P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[2] W. Guillemin, "Stable mappings and their singularities" , Springer (1973) MR0341518 Zbl 0294.58004
[3] T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) MR0501079 Zbl 0382.58006
How to Cite This Entry:
Jet. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jet&oldid=25769
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article