By W. Abikoff
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Example text
This function depends in a C“ way on xo. In addition, a depends on xo,but can be chosen independent of xo over any compact subset of D. (c) Ifo(t), a 5 t I 6 , is an integral curve of A’, so is thecurve o,(t)= a(t +c), a-c I tI b - c, obtained by translating the time” parametrization of 0. 2). 2) does not contain t explicitly on the righthand side (that is, it is a so-called autonomous system). 2) so that we can obtain integral curves defined over maximal domains of t. For example, start off with an integral curve o(t), 0 I t 2 a , , with o(0) = xo.
3, which finishes the proof. 5. Implicit Function Theorem for Mappings 33 Finally we remark that all these different versions of the implicit function theorem may be intuitively summarized by saying that arbitrary C“ mappings satisfying maximal rank conditions behave locally just as linear mappings of vector spaces. Thus there is a good technical reason why a thorough knowledge of linear algebra is one of the most important prerequisites for the study of differential geometry! Exercises 1. Suppose 4 : M + N is a maximal rank mapping of manifolds (that is, &(Mp) = N 4 ( p )for all p E M ) .
Thus the system of order (n - 1) can be solved first, and then x l ( t ) can be found by “ quadrature,” that is, by an integration. The order of the differential equations defining the integral curves of Y has been essentially reduced by 1 . ” These observations constitute Lie’s main contribution to the classical problem of solving differential equations in the plane. If dY - P ( X l Y ) dx Q ~ Y ) is such a differential equation, the solution curves, when written in parametric form, are the integral curves of Lie observed that all the classical tricks for “solving” this equation by quadrature were associated, in the way we described above, with a one-parameter group of transformations in the plane.