Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 477ing differential equation,˙ F (t) = Γ(F (t)),with initial position v, is the map D ∞ × V → V given by [Kock (1981);Kock and Reyes (2003)](t, v) ↦→ e t·Γ (v), (3.299)where the r.h.s here means the sum of the following ‘series’ (which has onlyfinitely many non–vanishing terms, since t is assumed nilpotent):v + tΓ(v) + t22! Γ2 (v) + t33! Γ3 (v) + . . .where Γ 2 (v) means Γ(Γ(v)), etc.There is an analogous result for second order differential equations ofthe form¨F (t) = Γ(F (t)).The formal solution of this second order differential equation with initialposition v and initial velocity w, is given by [Kock (1981); Kock and Reyes(2003)]F (t) = v + t · w + t2 t3 t4Γ(v) + Γ(w) +2! 3! 4! Γ2 (v) + t55! Γ2 (w) + ....Also, given f : R → V , where V is a Euclidean vector space, and giveng : R → R. Then for any a, b ∈ R,∫ baf(g(x)) · g ′ (x) dx =∫ g(b)g(a)f(u) du.Linear maps between Euclidean vector spaces preserve differentiation andintegration of functions R → V ; we shall explicitly need the following particularassertion: Let F : V → W be a linear map between Euclidean vectorspaces. Then for any f : R → V ,F (∫ baf(t) dt) =∫ baF (f(t)) dt.