respect to V2 is Hence, for to preserve the contact ideal, we require And so the second prolongation of V to a vector field on J2(π) is Note that the first prolongation of V can be recovered by omitting the second derivative terms in V2, or by projecting back to J1(π). Infinite jet spaces The inverse limit of the sequence of projections gives rise to the infinite jet space J∞(π). A point 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∞ maps into p. Just by thinking in terms of coordinates, J∞(π) appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J∞(π), not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections of manifolds is the sequence of injections of commutative algebras. Let's denote simply by . Take now the direct limit of the 's. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object J∞(π). Observe that , being born as a direct limit, carries an additional structure: it is a filtered commutative algebra. Roughly speaking, a concrete element will always belong to some , so it is a smooth function on the finite-dimensional manifold Jk(π) in the usual sense. Infinitely prolonged PDEs Given a k-th order system of PDEs E ⊆ Jk(π), the collection I(E) of vanishing on E smooth functions on J∞(π) is an ideal in the algebra , and hence in the direct limit too. Enhance I(E) by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal I of which is now closed under the operation of taking total derivative. The submanifold E(∞) of J∞(π) cut out by I is called the infinite prolongation of E. Geometrically, E(∞) is the manifold of formal solutions of E. A point of E(∞) can be easily seen to be represented by a section σ whose k-jet's graph is tangent to E at the point 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