Ivancevic_Applied-Diff-Geom

822 Applied Differential Geometry: A Modern Introduction5.5 Higher–Order Jet SpacesThe notion of 1– and 2–jet spaces is naturally extended to higher–order jetspaces. The k−jet space J k (X, Y ) of a fibre bundle Y → X is defined asthe disjoint union of the equivalence classes j k xs of sections s i : X → Y ofthe fibre bundle Y → X, identified by their values and the values of the firstk terms of their Taylor–series expansion at points x i in the domain (base)manifold X. J k (X, Y ) is a smooth manifold with the adapted coordinates(x α , y i α k ...α 1), wherey i α k···α 1(j k xs) = ∂ αk · · · ∂ α1 s i (x),(0 ≤ k ≤ k).The transformation law of these coordinates readsy ′ iα+α k ...α 1= ∂xµ∂ ′ x α d µy ′i α k ...α 1, (5.57)where α + α k . . . α 1 = (αα k . . . α 1 ) and∑d α = ∂ α + yα+α i k ...α 1∂ α k...α 1i = ∂ α + yα∂ i i + yαµ∂ i µ i + · · ·|α k ...α 1|