such uses. Let be the vector space of germs of smooth functions at a point p in . Let be the ideal consisting of germs of functions that vanish at p. (This is the maximal ideal for the local ring .) Then the ideal consists of all function germs that vanish to order k at p. We may now define the jet space at p by If is a smooth function, we may define the k-jet of f at p as the element of by setting This is a more general construction. For an -space , let be the stalk of the structure sheaf at and let be the maximal ideal of the local ring . The kth jet space at is defined to be the ring ( is the product of ideals). Taylor's theorem Regardless of the definition, Taylor's theorem establishes a canonical isomorphism of vector spaces between and . So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism. Jet spaces from a point to a point We have defined the space of jets at a point . The subspace of this consisting of jets of functions f such that f(p) = q is denoted by Jets of functions between two manifolds If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an invariant fashion. Jets do not transform as tensors. Instead, jets of functions between two manifolds belong to a jet bundle. Jets of functions from the real line to a manifold Suppose that M is a smooth manifold containing a point p. We shall define the jets of curves through p, by which we henceforth mean smooth functions such that f(0) = p. Define an equivalence relation as follows. Let f and g be a pair of curves through p. We will then say that f and g are equivalent to order k at p if there is some neighborhood U of p, such that, for every smooth function , . Note that these jets are well-defined since the composite functions and are just mappings from the real line to itself. This equivalence relation is sometimes called that of k-th-order contact between curves