Ebook: Advanced Calculus: Of Real-Valued Functions of a Real Variable and Vector-Valued Functions of a Vector Variable

In a calculus course, the student is expected to acquire a great number of
techniques and problem-solving devices and to put them to practical use. The
goal of an advanced calculus course is to put the calculus material into proper
perspective and to ask the question, "Why does calculus work?" In an
overwhelming number of cases, the answer is simple: "Because the real numbers have
the least-upper-bound property." This text is what I hope to be an informative
and entertaining documentation of this answer. At the same time, I have tried
to prepare the reader for topics beyond advanced calculus such as topology,
theory of functions, real variables and measure theory, functional analysis,
integration on manifolds, and last but not least, what is generally referred to as
applied mathematics. To avoid misunderstandings, let it be stressed that this
text is not an introduction to any of these subjects but rather a preparation
for them.


Contents:

Preface vii
1 Numbers 1
2 Functions 58
3 The Derivative 120
4 The Riemann Integral 141
5 The Euclidean n-Space 197
6 Vector-Valued Functions of a Vector Variable 226
7 Sequences of Functions 265
8 Linear Functions 296
9 The Derivative of a Vector-Valued Function of a Vector Variable 312
10 Nonlinear Functions 359
11 Multiple Integrals 392
12 Transformation of Integrals 438
13 Line and Surface Integrals 470
14 Infinite Series 566
Appendix 1 629
Appendix 2 631
Index 663