that is an invertible transformation under jet composition. (See also jet groups.) But since ρ is a diffeomorphism, is a smooth mapping as well. Hence, which proves that is non-singular. Furthermore, it is smooth, although we do not prove that fact here. Intuitively, this means that we can express the jet of a curve through p in terms of its Taylor series in local coordinates on M. Examples in local coordinates: As indicated previously, the 1-jet of a curve through p is a tangent vector. A tangent vector at p is a first-order differential operator acting on smooth real-valued functions at p. In local coordinates, every tangent vector has the form Given such a tangent vector v, let f be the curve given in the xi coordinate system by . If φ is a smooth function in a neighborhood of p with φ(p) = 0, then is a smooth real-valued function of one variable whose 1-jet is given by which proves that one can naturally identify tangent vectors at a point with the 1-jets of curves through that point. The space of 2-jets of curves through a point. In a local coordinate system xi centered at a point p, we can express the second-order Taylor polynomial of a curve f(t) through p by So in the x coordinate system, the 2-jet of a curve through p is identified with a list of real numbers . As with the tangent vectors (1-jets of curves) at a point, 2-jets of curves obey a transformation law upon application of the coordinate transition functions. Let (yi) be another coordinate system. By the chain rule, Hence, the transformation law is given by evaluating these two expressions at t = 0. Note that the transformation law for 2-jets is second-order in the coordinate transition functions. Jets of functions from a manifold to a manifold We are now prepared to define the jet of a function from a manifold to a manifold. Suppose that M and N are two smooth manifolds. Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows. Two maps f and g are said to be equivalent if, for every curve γ through p (recall that by our conventions this is a mapping such that ), we have on some neighborhood of 0. The jet