Editing Jet bundle (section)

==Infinite jet spaces==
The [[inverse limit]] of the sequence of projections <math>\pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi)</math> gives rise to the '''infinite jet space''' ''J<sup>∞</sup>(π)''. A point <math>j_p^\infty(\sigma)</math> is the equivalence class of sections of π that have  the same ''k''-jet in ''p'' as σ  for all values of ''k''. The natural projection π<sub>∞</sub> maps <math>j_p^\infty(\sigma)</math> into ''p''.

Just by thinking in terms of coordinates, ''J<sup>∞</sup>(π)'' appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on ''J<sup>∞</sup>(π)'', not relying on differentiable charts, is given by the [[differential calculus over commutative algebras]]. Dual to the sequence of projections <math>\pi_{k+1,k}: J^{k+1}(\pi) \to J^k(\pi)</math> of manifolds is the sequence of injections <math>\pi_{k+1,k}^*: C^\infty(J^{k}(\pi)) \to C^\infty\left(J^{k+1}(\pi)\right)</math> of commutative algebras. Let's denote <math>C^\infty(J^{k}(\pi))</math> simply by <math>\mathcal{F}_k(\pi)</math>. Take now the [[direct limit]] <math>\mathcal{F}(\pi)</math> of the <math>\mathcal{F}_k(\pi)</math>'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object ''J<sup>∞</sup>(π)''. Observe that <math>\mathcal{F}(\pi)</math>, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element <math>\varphi\in\mathcal{F}(\pi)</math> will always belong to some <math>\mathcal{F}_k(\pi)</math>, so it is a smooth function on the finite-dimensional manifold ''J<sup>k</sup>''(π) in the usual sense.

===Infinitely prolonged PDEs===
Given a ''k''-th order system of PDEs ''E'' ⊆ ''J<sup>k</sup>(π)'', the collection ''I(E)'' of vanishing on ''E'' smooth functions on ''J<sup>∞</sup>(π)'' is an [[ideal (ring theory)|ideal]] in the algebra <math>\mathcal{F}_k(\pi)</math>, and hence in the direct limit <math>\mathcal{F}(\pi)</math> too.

Enhance ''I(E)'' by adding all the possible compositions of [[total derivative]]s applied to all its elements. This way we get a new ideal ''I'' of <math>\mathcal{F}(\pi)</math> which is now closed under the operation of taking total derivative. The submanifold ''E''<sub>(∞)</sub> of ''J''<sup>∞</sup>(π) cut out by ''I'' is called the '''infinite prolongation''' of ''E''.

Geometrically, ''E''<sub>(∞)</sub> is the manifold of '''formal solutions''' of ''E''. A point <math>j_p^\infty(\sigma)</math> of ''E''<sub>(∞)</sub> can be easily seen to be represented by a section σ whose ''k''-jet's graph is tangent to ''E'' at the point <math>j_p^k(\sigma)</math> with arbitrarily high order of tangency.

Analytically, if ''E'' is given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a point ''p'' that make vanish the [[Taylor series]] of <math>\varphi\circ j^k(\sigma)</math> at the point ''p''.

Most importantly, the closure properties of ''I'' imply that ''E''<sub>(∞)</sub> is tangent to the '''infinite-order contact structure''' <math>\mathcal{C}</math> on ''J<sup>∞</sup>(π)'', so that by restricting <math>\mathcal{C}</math> to ''E''<sub>(∞)</sub> one gets the [[diffiety]] <math>(E_{(\infty)}, \mathcal{C}|_{E_{(\infty)}})</math>, and can study the associated [[Diffiety#Vinogradov sequence|Vinogradov (C-spectral) sequence]].