space is then defined to be the set of equivalence classes of modulo the equivalence relation . Note that because the target space N need not possess any algebraic structure, also need not have such a structure. This is, in fact, a sharp contrast with the case of Euclidean spaces. If is a smooth function defined near p, then we define the k-jet of f at p, , to be the equivalence class of f modulo . Multijets John Mather introduced the notion of multijet. Loosely speaking, a multijet is a finite list of jets over different base-points. Mather proved the multijet transversality theorem, which he used in his study of stable mappings. Jets of sections Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection . Then sections of E are smooth functions such that is the identity automorphism of M. The jet of a section s over a neighborhood of a point p is just the jet of this smooth function from M to E at p. The space of jets of sections at p is denoted by . Although this notation can lead to confusion with the more general jet spaces of functions between two manifolds, the context typically eliminates any such ambiguity. Unlike jets of functions from a manifold to another manifold, the space of jets of sections at p carries the structure of a vector space inherited from the vector space structure on the sections themselves. As p varies over M, the jet spaces form a vector bundle over M, the k-th-order jet bundle of E, denoted by Jk(E). Example: The first-order jet bundle of the tangent bundle. We work in local coordinates at a point and use the Einstein notation. Consider a vector field in a neighborhood of p in M. The 1-jet of v is obtained by taking the first-order Taylor polynomial of the coefficients of the vector field: In the x coordinates, the 1-jet at a point can be identified with a list of real numbers . In the same way that a tangent vector at a point can be identified with the list (vi), subject to a certain transformation law under coordinate transitions, we have to know how the list is affected by a transition. So consider the transformation law in passing to another coordinate system yi. Let wk be the coefficients of the vector field