Let M and N be smooth manifolds. The first jet bundle J^1(M,N) consists of equivalence classes j_p^1 f of smooth maps f: M → N with identical value and differential at p ∈ M.

For f: R → R, J^1(R,R) is identified with triples (x,y,p), where x = p, y = f(x), and p = df/dx evaluated at x.

Design

Definition 2.1 (First Jet Bundle)

Let M and N be smooth manifolds. The first jet bundle J^1(M,N) consists of equivalence classes j_p^1 f of smooth maps f: M → N with identical value and differential at p ∈ M.

Definition 2.2 (Jet of a Function on R)

For f: R → R, J^1(R,R) is identified with triples (x,y,p), where x = p, y = f(x), and p = df/dx evaluated at x.