Weikard-advanced-calc-notes.pdf - ADVANCED CALCULUS Lecture...

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ADVANCED CALCULUS Lecture notes for MA 440/540 & 441/541 2015/16 Rudi Weikard 1 2 3 4 5 x - 2 - 1 1 log H x L Based on lecture notes by G. Stolz and G. Weinstein Version of September 3, 2016
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Contents First things first 1 0.1. The goals 1 0.2. The rules 1 0.3. Hints 2 0.4. The language of mathematics 3 Chapter 1. The real numbers 5 1.1. Field axioms 5 1.2. Order axioms 6 1.3. The induction principle 8 1.4. Counting and infinity 10 1.5. The least upper bound axiom 11 Chapter 2. Sequences and series 15 2.1. Sequences 15 2.2. Sums and the Σ-notation 18 2.3. Series 18 Chapter 3. A zoo of functions 23 Chapter 4. Continuity 27 4.1. Limits of functions 27 4.2. Continuous functions 28 4.3. The intermediate value theorem and some of its consequences 29 4.4. Uniform convergence and continuity 29 Chapter 5. Differentiation 33 5.1. Derivatives 33 5.2. The mean value theorem and Taylor’s theorem 35 5.3. Uniform convergence and differentiation 36 Chapter 6. Integration 39 6.1. Existence and uniqueness of integrals 39 6.2. Properties of integrals 41 6.3. The fundamental theorem of calculus 42 6.4. Integration of piecewise continuous functions 42 6.5. Uniform convergence and integration 43 Chapter 7. Special topics 45 7.1. Generalized limits 45 7.2. Trigonometric functions and their inverses 46 i
ii CONTENTS 7.3. Analytic geometry 48 7.4. Sets of measure zero and some consequences 49 Appendix A. Some set theory and logic 51 A.1. Elements of logic 51 A.2. Basics of set theory 52 A.3. Functions 53 A.4. The recursion theorem 54 Index 55
First things first 0.1. The goals Our goal in this class is threefold: (1) to obtain a body of knowledge in Advanced Calculus, the basis of the analysis of real-valued functions of one real variable; (2) to learn how to communicate ideas and facts in both a written and an oral form; (3) and, perhaps most importantly, to become acquainted with — indeed, to master — the process of creating mathematics. In conducting this class we shall try to model a mathematical community in which both collaboration and competition are prevalent. This community is — no, you are — on the verge of discovering the foundations for a number of rules and recipes which have been successfully in use for some time. In the process you will recreate a body of knowledge almost as if you were the first to discover it. However, as we have only nine months to do this rather than a century or two, there will be some help available to you, most prominently in the form of these notes which will delineate broadly a path in which discovery will (or could) proceed. In this course it is allowed and, in fact, required to criticize the person on the board for flaws or incomplete arguments (you are a scientific community). Criticism has to be leveled in a professional manner, in particular, it has to be free from any personal insults. At the same time you have to learn to accept criticism without taking it personally. By learning to stand up for your ideas (or to accept that you made a mistake) you may get something out of this course which is of value not only in mathematics.

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