Editing Jet (mathematics) (section)

===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]]).