Editing Jet (mathematics) (section)

==Jets at a point in Euclidean space: rigorous definitions==

===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 analysis|complex domains]], to [[p-adic analysis]], and to other areas of analysis.

Let <math>C^\infty({\mathbb R}^n,{\mathbb R}^m)</math> be the [[vector space]] of [[smooth function]]s <math>f:{\mathbb R}^n\rightarrow {\mathbb R}^m</math>. Let ''k'' be a non-negative integer, and let ''p'' be a point of <math>{\mathbb R}^n</math>. We define an [[equivalence relation]] <math>E_p^k</math> 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 derivative]]s agree at ''p'' up to (and including) their ''k''-th-order derivatives. In short,<math>f \sim g \,\!</math> iff <math> f-g = 0 </math> to ''k''-th order.

The '''''k''-th-order jet space''' of <math>C^\infty({\mathbb R}^n,{\mathbb R}^m)</math> at ''p'' is defined to be the set of equivalence classes of <math>E^k_p</math>, and is denoted by <math>J^k_p({\mathbb R}^n,{\mathbb R}^m)</math>.

The '''''k''-th-order jet''' at ''p'' of a smooth function <math>f\in C^\infty({\mathbb R}^n,{\mathbb R}^m)</math> is defined to be the equivalence class of ''f'' in <math>J^k_p({\mathbb R}^n,{\mathbb R}^m)</math>.

===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.

Let <math>C_p^\infty({\mathbb R}^n,{\mathbb R}^m)</math> be the [[vector space]] of [[germ (mathematics)|germs]] of [[smooth function]]s <math>f:{\mathbb R}^n\rightarrow {\mathbb R}^m</math> at a point ''p'' in <math>{\mathbb R}^n</math>. Let <math>{\mathfrak m}_p</math> be the ideal consisting of germs of functions that vanish at ''p''. (This is the [[maximal ideal]] for the [[local ring]] <math>C_p^\infty({\mathbb R}^n,{\mathbb R}^m)</math>.) Then the ideal <math>{\mathfrak m}_p^{k+1}</math> consists of all function germs that vanish to order ''k'' at ''p''. We may now define the '''jet space''' at ''p'' by

:<math>J^k_p({\mathbb R}^n,{\mathbb R}^m)=C_p^\infty({\mathbb R}^n,{\mathbb R}^m)/{\mathfrak m}_p^{k+1}</math>

If <math>f:{\mathbb R}^n\rightarrow {\mathbb R}^m</math> is a smooth function, we may define the ''k''-jet of ''f'' at ''p'' as the element of <math>J^k_p({\mathbb R}^n,{\mathbb R}^m)</math> by setting

:<math>J^k_pf=f \pmod {{\mathfrak m}_p^{k+1}}</math>
This is a more general construction. For an [[Locally ringed space|<math>\mathbb{F}</math>-space]] <math>M</math>, let <math>\mathcal{F}_p</math> be the [[Stalk (sheaf)|stalk]] of the [[structure sheaf]] at <math>p</math> and let <math>{\mathfrak m}_p</math> be the [[maximal ideal]] of the [[local ring]] <math>\mathcal{F}_p</math>. The kth jet space at <math>p</math> is defined to be the ring <math>J^k_p(M)=\mathcal{F}_p/{\mathfrak m}_p^{k+1}</math>(<math>{\mathfrak m}_p^{k+1}</math> is the [[Ideal (ring theory)#Ideal operations|product of ideals]]).

===Taylor's theorem===
Regardless of the definition, Taylor's theorem establishes a canonical isomorphism of vector spaces between <math>J^k_p({\mathbb R}^n,{\mathbb R}^m)</math> and <math>{\mathbb R}^m[z_1, \dotsc, z_n]/(z_1, \dotsc, z_n)^{k+1}</math>. So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism.

===Jet spaces from a point to a point===
We have defined the space <math>J^k_p({\mathbb R}^n,{\mathbb R}^m)</math> of jets at a point <math>p\in {\mathbb R}^n</math>. The subspace of this consisting of jets of functions ''f'' such that ''f''(''p'')&nbsp;=&nbsp;''q'' is denoted by
:<math>J^k_p({\mathbb R}^n,{\mathbb R}^m)_q=\left\{J^kf\in J^k_p({\mathbb R}^n,{\mathbb R}^m) \mid f(p) = q \right\}</math>