a coordinate system on J r(π). Let (U, u) be an adapted coordinate chart on E, where u = (xi, uα). The induced coordinate chart (Ur, ur) on J r(π) is defined by where and the functions known as the derivative coordinates: Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (U r, u r) is a finite-dimensional C∞ atlas on J r(π). Jet bundles Since the atlas on each defines a manifold, the triples , and all define fibered manifolds. In particular, if is a fiber bundle, the triple defines the r-th jet bundle of π. If W ⊂ M is an open submanifold, then If p ∈ M, then the fiber is denoted . Let σ be a local section of π with domain W ⊂ M. The r-th jet prolongation of σ is the map defined by Note that , so really is a section. In local coordinates, is given by We identify with . Algebraic-geometric perspective An independently motivated construction of the sheaf of sections is given.Consider a diagonal map , where the smooth manifold is a locally ringed space by for each open . Let be the ideal sheaf of , equivalently let be the sheaf of smooth germs which vanish on for all . The pullback of the quotient sheaf from to by is the sheaf of k-jets. The direct limit of the sequence of injections given by the canonical inclusions of sheaves, gives rise to the infinite jet sheaf . Observe that by the direct limit construction it is a filtered ring. Example If π is the trivial bundle (M × R, pr1, M), then there is a canonical diffeomorphism between the first jet bundle and T*M × R. To construct this diffeomorphism, for each σ in write . Then, whenever p ∈ MConsequently, the mapping is well-defined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if (xi, u) are coordinates on M × R, where u = idR is the identity coordinate, then the derivative coordinates ui on J1(π) correspond to the coordinates ∂i on T*M. Likewise, if π is the trivial bundle (R × M, pr1, R), then there exists a canonical diffeomorphism between and R × TM. Contact structure The space Jr(π) carries a natural distribution, that is, a sub-bundle of the tangent bundle TJr(π)), called