Continuous Functions of Vector Variables

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Springer Science & Business Media, Jul 31, 2002 - Mathematics - 210 pages
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This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.
 

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Contents

II
1
III
3
IV
7
V
13
VI
18
VII
23
VIII
33
IX
37
XIX
90
XX
95
XXI
100
XXII
105
XXIII
111
XXIV
119
XXV
124
XXVI
129

X
42
XI
48
XII
55
XIII
58
XIV
65
XV
73
XVI
78
XVII
85
XXVII
133
XXVIII
140
XXIX
149
XXX
155
XXXI
203
XXXII
205
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