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warner-chapter1

# warner-chapter1 - After establishing sonte notational...

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After establishing sonte notational conventions vvhich will be used throughout the book. we rvill begin w'ith the notion of a differentiable manifold. These are spaces which are iocally like Euclidean space and which have enough structure so that the basic concepts of calculus can be carried over. In this first chapter we shall primarily be concerned with the analogs and implications fcrr manifoldsof the fundamental theorems of differential calculus. Later, in Chapter 4, we shall consider the theory of integration on manifolds. From the notion of directional derivative in Euclidean space we will obtain the notion of a tangent vector to a diffe:rentiable manifold. We will study mappings between manifolds and the effect that mappinqs have on tangent vectors. We niil investrgate the implications fbr mappings of manifolds of the classical inverse and implicit iunction theorems.We will seethat the fundantental existence and uniqur:ness theorems for ordinary differential equations translate into existence and uniqueness statements for integral curves of vector {ields. The chapter closes with the Frobenius theorem. which pertains to the existence and Lrniqr:eness of integrai manitblds of rnvolutive distributions on ntanrtblds. PRELIMINARIES 1.1 Some Basic Notation and Terminology Throughout this textwe rvill describe sets either b,v listings of their elem,:nts. for example ta, . . . . - a,,\- or by expressions of the form {,r:P}. which denote the set of all .r satislying property P. The expression a e I meansthata isanelentelt of the set L If a set Aisasubset of a setB (that is, ce .Brvhenever aeA). we u'rite A - B. If A c= B andB c: A,then.4 equals,B, denoted A: B. The negations of e. c and : &r€denoted by f, f , and I respectivelv. A set y' is apropersubset of,B if A c B but A * B. 2

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We will denote the empty stztby b . We will often denote a collection \un: ae A\ of sets u" indexecl by the set,4 simply by {u,\ tf explicit mention of the index set is not necessaiy. The union of the sets in the collection {(J,: a e A} will be denoted l) (1, or simply l) (1,. Similarly, their Preliminaries q€A (Jo or simply n t/" (J,: fra: a belongs to sonre U"). (J o: {a: a beiongs to everYU"}' g , f: f-L(B o C) --, D g(-f (u)) for every a €f-1(B n C)' For notational -not exclude the case in which f-'(B A C): g ' mappings/and g, we shall consider their conrposition with the understanding that the domain of g "/may intersection will be denoted defined by g"[email protected]): convenience, we shall That is, given anY two g "-f as being defined, well be the emptY set. The cartesian Product A x (a, b) of points a e A and b cartesian product .f x S of the o f . 4 x B i n t o C x D ' We shall denote the identitY ntaP A diagram of maPs such as B of ttro sets A crnd B ls the set of all pairs e n. If f: A - C and g" B - 'D., then the ntaps f oia g is the map (a, bS r--.
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