at p. We now define the k-jet of a curve f through p to be the equivalence class of f under , denoted or . The k-th-order jet space is then the set of k-jets at p. As p varies over M, forms a fibre bundle over M: the k-th-order tangent bundle, often denoted in the literature by TkM (although this notation occasionally can lead to confusion). In the case k=1, then the first-order tangent bundle is the usual tangent bundle: T1M = TM. To prove that TkM is in fact a fibre bundle, it is instructive to examine the properties of in local coordinates. Let (xi)= (x1,...,xn) be a local coordinate system for M in a neighborhood U of p. Abusing notation slightly, we may regard (xi) as a local diffeomorphism . Claim. Two curves f and g through p are equivalent modulo if and only if . Indeed, the only if part is clear, since each of the n functions x1,...,xn is a smooth function from M to . So by the definition of the equivalence relation , two equivalent curves must have . Conversely, suppose that ; is a smooth real-valued function on M in a neighborhood of p. Since every smooth function has a local coordinate expression, we may express ; as a function in the coordinates. Specifically, if q is a point of M near p, then for some smooth real-valued function ψ of n real variables. Hence, for two curves f and g through p, we have The chain rule now establishes the if part of the claim. For instance, if f and g are functions of the real variable t , then which is equal to the same expression when evaluated against g instead of f, recalling that f(0)=g(0)=p and f and g are in k-th-order contact in the coordinate system (xi). Hence the ostensible fibre bundle TkM admits a local trivialization in each coordinate neighborhood. At this point, in order to prove that this ostensible fibre bundle is in fact a fibre bundle, it suffices to establish that it has non-singular transition functions under a change of coordinates. Let be a different coordinate system and let be the associated change of coordinates diffeomorphism of Euclidean space to itself. By means of an affine transformation of , we may assume without loss of generality that ρ(0)=0. With this assumption, it suffices to prove