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Additional resources for Asymptotic Expansions: Their Derivation and Interpretation

Example text

Now we compute the Jacobian matrix doF. P2 + y2,P 3+y3,---,P n + yn)|^^0 = ei for i = 2, . . , n, where ( e i,. . , Cn) is the canonical basis of R” . Hence, doF = (/(p )e 2 ■■■Cn) is a lower triangular matrix with nonzero entries in the diagonal (because by hypothesis the first component of /(p ) is nonzero). Thus, doF is nonsingular and F is indeed a coordinate change in a neighborhood of zero. Now let (xi (t) , . . , Xfi(t)) = ^t(p “by) be a solution of the equation x' = /( x ) , where q = {0,q2,qz, ...

For simplicity of the exposition, we consider only times t > to. The case when t < to can be treated in an analogous manner. We define a function y : 7 —>■R” by y{t) = x i { t ) - X 2 {t). 56) that ||y(t)|| < Ik i-a : 2 ||-I- [ \\f{s,xi{s)) - f{s,X2 {s))\\ds Jto <\\x i - X 2\\+ L [ ||y(s)||(is, Jto for some constant L (because / is locally Lipschitz in x). Letting ^i(t) = l|y(t)ll. 55) for these values of t. 2. S m ooth d ependence on the initial conditions. In this section we show that for a function / of class the solutions are also of class on the initial conditions.

73. 73, although one can always complete it to a maximal one. Each map ipi: Ui M is called a chart or a coordinate system. Any differentiable structure induces a topology on M ; namely, a set A c M is said to be open if c R” is an open set for every i G /. Now we introduce the notion of a manifold. 74. 6. Equations on manifolds b) when equipped with the corresponding induced topology, the set M is a Hausdorff topological space with countable basis. We recall that a topological space is said to be Hausdorff if distinct points have disjoint open neighborhoods, and that it is said to have a count­ able basis if any open set can be written as a union of elements of a given countable family of open sets.

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