ADVANCED CALCULUS

Lecture notes for

MA 440/540 & 441/541

Rudi Weikard12345x-2-11logHxLBased on lecture notes by G. Stolz and G. Weinstein

Version of August 3, 2022

?2022. This manuscript version is made available under the CC-BY-NC-SA 4.0 license

First things ifirst iii

The goalsiii

The rulesiii

Hintsiv

The language of mathematics v

Chapter 1. The real numbers 1

1.1. Field axioms 1

1.2. Order axioms 2

1.3. The induction principle 4

1.4. Counting and inifinity 6

1.5. The least upper bound axiom 7

Chapter 2. Sequences and series 11

2.1. Sequences 11

2.2. Sums and the Σ-notation 14

2.3. Series 14

Chapter 3. A zoo of functions 19

Chapter 4. Continuity 23

4.1. Limits of functions 23

4.2. Continuous functions 24

4.3. The intermediate value theorem and some of its consequences 25

4.4. Uniform convergence and continuity 25

Chapter 5. Diffferentiation 29

5.1. Derivatives 29

5.2. The mean value theorem and Taylor's theorem 31

5.3. Uniform convergence and diffferentiation 32

Chapter 6. Integration 35

6.1. Existence and uniqueness of integrals 35

6.2. Properties of integrals 37

6.3. The fundamental theorem of calculus 38

6.4. Integration of piecewise continuous functions 38

6.5. Uniform convergence and integration 39

Chapter 7. Special topics 41

7.1. Generalized limits 41

7.2. Trigonometric functions and their inverses 42

i ii CONTENTS

7.3. Analytic geometry 44

7.4. Sets of measure zero and some consequences 45

Appendix A. Some set theory and logic 47

A.1. Elements of logic 47

A.2. Basics of set theory 48

A.3. Functions 49

A.4. The recursion theorem 50

Index51

First things ifirst

The goals

Our goal in this class is threefold:

(1) to obtain a b odyof kno wledgein Adv ancedCalculus, the basis of the analysis of real-valued functions of one real variable; (2) to learn ho wto c ommunicateideas an dfacts in b otha written and an oral form; (3) and, p erhapsmost imp ortantly,to b ecomeacquain tedwith - ind eed,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 ifirst 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 lflaws or incomplete arguments (you are a scientiific 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.

The rules

The following rules, based on intellectual and academic honesty, will be in force. (1) Ev erybodywill ha vethe opp ortunityto presen tpro ofsof theorems. Y ouwill ha ve the proof written out on paper and present it with the help of a document camera. (2) The audience (including the instructor) ma yc hallengea statemen tm adei nthe course of the proof at any point.quotesdbs_dbs2.pdfusesText_3

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