Jr+1(π) can be written as a linear combination with smooth coefficients of the basic contact forms|I| is known as the order of the contact form . Note that contact forms on Jr+1(π) have orders at most r. Contact forms provide a characterization of those local sections of πr+1 which are prolongations of sections of π. Let ψ ∈ ΓW(πr+1), then ψ = jr+1σ where σ ∈ ΓW(π) if and only if Vector fields A general vector field on the total space E, coordinated by , is A vector field is called horizontal, meaning that all the vertical coefficients vanish, if = 0. A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρi = 0. For fixed (x, u), we identify having coordinates (x, u, ρi, φα), with an element in the fiber TxuE of TE over (x, u) in E, called a tangent vector in TE. A section is called a vector field on E with and ψ in Γ(TE). The jet bundle Jr(π) is coordinated by . For fixed (x, u, w), identify having coordinates with an element in the fiber of TJr(π) over (x, u, w) ∈ Jr(π), called a tangent vector in TJr(π). Here, are real-valued functions on Jr(π). A section is a vector field on Jr(π), and we say Partial differential equations Let (E, π, M) be a fiber bundle. An r-th order partial differential equation on π is a closed embedded submanifold S of the jet manifold Jr(π). A solution is a local section σ ∈ ΓW(π) satisfying , for all p in M. Consider an example of a first order partial differential equation. Example Let π be the trivial bundle (R2 × R, pr1, R2) with global coordinates (x1, x2, u1). Then the map F : J1(π) → R defined by gives rise to the differential equation which can be written The particular has first prolongation given by and is a solution of this differential equation, because and so for every p ∈ R2. Jet prolongation A local diffeomorphism ψ : Jr(π) → Jr(π) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θ is any contact form on Jr(π), then ψ*θ is also a contact form. The flow generated by a vector field Vr on the jet space Jr(π) forms a one-parameter group of contact transformations if and only if the