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This textbook presents all the basics for the first two years of a course in mathematical analysis, from the natural numbers to Stokes-Cartan Theorem.
The main novelty which distinguishes this book is the choice of introducing the Kurzweil-Henstock integral from the very beginning. Although this approach requires a small additional effort by the student, it will be compensated by a substantial advantage in the development of the theory, and later on when learning about more advanced topics.
The text guides the reader with clarity in the discovery of the many different subjects, providing all necessary tools – no preliminaries are needed. Both students and their instructors will benefit from this book and its novel approach, turning their course in mathematical analysis into a gratifying and successful experience.
This book brings together the classical topics of mathematical analysis normally taught in the first two years of a university course. It is the outcome of the lessons I have been teaching for many years in the undergraduate courses in mathematics, physics, and engineering at my university. Many excellent books on mathematical analysis have already been written, so a natural question to ask is: Why write another book on this subject? I will try to provide a brief answer to that question.
The main novelty of this book lies in the treatment of the theory of the integral. Kurzweil and Henstock’s theory is presented here in Chaps. 7, 9, and 11. Compared to Riemann’s theory, it requires modest additional effort from the student, but that effort will be repaid with significant benefits. Consider that it includes Lebesgue’s theory itself, since a function is integrable according to Kurzweil and Henstock if and only if it is integrable according to Lebesgue, together with its absolute value. In this theory, the Fundamental Theorem turns out to be very general and natural, and it finds its generalization in Taylor’s formula with an integral remainder, requiring only essential hypotheses. Moreover, the improper integral happens to be a normal integral.
Despite the modest additional effort required, no demands for preliminary knowledge about the integral will be made on the student for the purpose of reading this book. Students will be guided as they construct all the necessary mathematical tools, starting from the very beginning.
Part I The Basics of Mathematical Analysis
1 Sets of Numbers and Metric Spaces
2 Continuity
3 Limits
4 Compactness and Completeness
5 Exponential and Circular Functions
Part II Differential and Integral Calculus in R
6 The Derivative
7 The Integral
Part III Further Developments
8 Numerical Series and Series of Functions
9 More on the Integral
Part IV Differential and Integral Calculus in RN
10 The Differential
11 The Integral
12 Differential Forms