This book is based on an honors course in advanced calculus that we gave in the and 10 develop the differential and integral calculus on manifolds, while
LYNN H LOOMIS and SHLOMO STERNBERG Department of Mathematics, Harvard University ADVANCED CALCULUS REVISED EDITION JONES AND
AHLFORS: Complex AnalysIs BUCK: Advanced Calculus BUSACKER AND SAATY: Finite Graphs and Networks CHENEY: Introduction to Approximation
Advanced Calculus of Real-Valued Functions of a Real Variable and Vector- Valued Functions of a Vector Variable Houghton Mifflin Company · Boston
2 jan 2012 · Advanced Calculus course 10in my first set of advanced calculus notes (2010) I used the term function Ask if interested, I think I have a pdf
(1) to obtain a body of knowledge in Advanced Calculus, the basis of the analysis of real-valued 1 http://www math uab edu/~rudi/teaching/algebra pdf
Advanced calculus : a differential fonns approach I Harold M Edwards -- [3rd ed ] p em Includes index ISBN 978-1-4612-6688-4 (alk paper) I Calculus
"advanced calculus" to turn into courses on the foundations of analysis Students typically start with a year and a half of calculus that emphasizes computations
BY EDWIN BIDWELL WILSON, Ph D It is probable that almost every teacher of advanced calculus feels the need of a text suited to present conditions and
of mathematics at any given time There are a variety of needs to be served by a textbook in advanced calculus In planning this edition of our book, we have
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![[PDF] ADVANCED CALCULUS Rudi Weikard - UAB](https://pdfprof.com/Listes/17/25099-17Weikard-advanced-calc-notes.pdf.pdf.jpg "[PDF] ADVANCED CALCULUS Rudi Weikard - UAB")
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
Contents
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
[PDF] ADVANCED CALCULUS - Harvard Mathematics Department