Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 799x, with coefficients up to degree k. Therefore, in a fixed coordinate chart,a k−jet can be identified with the collection of Taylor coefficients up todegree k.The set of all k−jets of smooth maps from M to N is called the k−jetspace and denoted by J k (M, N). It has a natural smooth manifold structure.Also, a map from a k−jet space J k (M, N) to a smooth manifold Mor N is called a jet bundle (we will make this notion more precise later).For example, consider a simple function f : X → Y, x ↦→ y = f(x),mapping the X−axis into the Y −axis. In this case, M = X is a domainand N = Y is a codomain. A 0−jet at a point x ∈ X is given by its graph(x, f(x)). A 1−jet is given by a triple (x, f(x), f ′ (x)), a 2−jet is givenby a quadruple (x, f(x), f ′ (x), f ′′ (x)), and so on up to the order k (wheref ′ (x) = df(x)dx, etc.). The set of all k−jets from X to Y is called the k−jetspace J k (X, Y ).Fig. 5.2 Common spaces associated with a function f on a smooth manifold M (seetext for explanation).In case of a function of two variables, f(x, y), the common spaces relatedto f, including its 1–jet j 1 f, are depicted in Figure 5.2 (see [Omohundro(1986)]). Recall that a hypersurface is a codimension–1 submanifold. Givena sample function f(x, y) = x 2 + y 2 in M = R 2 , then: (a) shows its graphas a hypersurface in R × M; (b) shows its level sets in M; (c) shows its