Ivancevic_Applied-Diff-Geom

798 Applied Differential Geometry: A Modern Introductionmaps (see Figure 5.1),f 1 , f 2 : M → Nare said to be k−tangent (or tangent of order k, or have a kth order contact)at a point x on a domain manifold M, denoted by f 1 ∼ f 2 , ifff 1 (x) = f 2 (x) called 0 − tangent,∂ x f 1 (x) = ∂ x f 2 (x), called 1 − tangent,∂ xx f 1 (x) = ∂ xx f 2 (x), called 2 − tangent,... etc. to the order k.Fig. 5.1 An intuitive geometrical picture behind the k−jet concept, based on the ideaof higher–order tangency or contact (see text for explanation).In this way defined k−tangency is an equivalence relation, i.e.,f 1 ∼ f 2 ⇒ f 2 ∼ f 1 , f 1 ∼ f 2 ∼ f 3 ⇒ f 1 ∼ f 3 , f 1 ∼ f 1.Now a k−jet (or, a jet of order k), denoted by j k xf, of a smooth mapf : M → N at a point x ∈ M (see Figure 5.1), is defined as an equivalenceclass of k−tangent maps at x,j k xf = {f ′ : f ′ is k − tangent to f at x}.The point x is called the source and the point f(x) is the target of thek−jet j k xf.We choose local coordinates on M and N in the neighborhood of thepoints x and f(x), respectively. Then the k−jet j k xf of any map close tof, at any point close to x, can be given by its Taylor–series expansion at