**Pure and Applied Undergraduate Texts**

Volume: 32;
2018;
360 pp;
Hardcover

MSC: Primary 00;

**Print ISBN: 978-1-4704-4246-0
Product Code: AMSTEXT/32**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

**Electronic ISBN: 978-1-4704-4762-5
Product Code: AMSTEXT/32.E**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

#### Supplemental Materials

# A Problems Based Course in Advanced Calculus

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*John M. Erdman*

This textbook is suitable for a course in advanced calculus
that promotes active learning through problem solving. It can be used
as a base for a Moore method or inquiry based class, or as a guide in
a traditional classroom setting where lectures are organized around the
presentation of problems and solutions. This book is appropriate for
any student who has taken (or is concurrently taking) an introductory
course in calculus. The book includes sixteen appendices that review
some indispensable prerequisites on techniques of proof writing with
special attention to the notation used the course.

A solutions
manual is freely available electronically.

#### Readership

Undergraduate students interested in introduction to proofs.

#### Reviews & Endorsements

[This] book is a piece of excellent mathematical writing. The readers—student or teacher— will enjoy precise and clear exposition and careful editing, as well as fine language sparking with a decent dose of humor.

-- Piotr Sworowski, Mathematical Reviews

Learning from this book might be a challenging and time-consuming task, but the reader will be rewarded by a deep understanding of advanced calculus.

-- Antonín Slavik, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## A Problems Based Course in Advanced Calculus

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xiii14
- For students: How to use this book xvii18
- Chapter 1. Intervals 122
- Chapter 2. Topology of the real line 526
- Chapter 3. Continuous functions from \R to \R 1132
- Chapter 4. Sequences of real numbers 2142
- Chapter 5. Connectedness and the intermediate value theorem 3354
- Chapter 6. Compactness and the extreme value theorem 3960
- Chapter 7. Limits of real valued functions 4566
- Chapter 8. Differentiation of real valued functions 4970
- Chapter 9. Metric spaces 5980
- Chapter 10. Interiors, closures, and boundaries 6788
- Chapter 11. The topology of metric spaces 7192
- Chapter 12. Sequences in metric spaces 7798
- Chapter 13. Uniform convergence 81102
- Chapter 14. More on continuity and limits 85106
- Chapter 15. Compact metric spaces 99120
- Chapter 16. Sequential characterization of compactness 103124
- Chapter 17. Connectedness 109130
- Chapter 18. Complete spaces 113134
- Chapter 19. A fixed point theorem 117138
- Chapter 20. Vector spaces 125146
- Chapter 21. Linearity 135156
- Chapter 22. Norms 153174
- Chapter 23. Continuity and linearity 163184
- Chapter 24. The Cauchy integral 175196
- Chapter 25. Differential calculus 189210
- Chapter 26. Partial derivatives and iterated integrals 203224
- Chapter 27. Computations in \Rⁿ 217238
- Chapter 28. Infinite series 233254
- Chapter 29. The implicit function theorem 251272
- Chapter 30. Higher order derivatives 265286
- Appendix A. Quantifiers 277298
- Appendix B. Sets 279300
- Appendix C. Special subsets of \R 283304
- Appendix D. Logical connectives 285306
- Appendix E. Writing mathematics 291312
- Appendix F. Set operations 297318
- Appendix G. Arithmetic 303324
- Appendix H. Order properties of \R 309330
- Appendix I. Natural numbers and mathematical induction 313334
- Appendix J. Least upper bounds and greatest lower bounds 317338
- Appendix K. Products, relations, and functions 323344
- Appendix L. Properties of functions 327348
- Appendix M. Functions that have inverses 331352
- Appendix N. Products 337358
- Appendix O. Finite and infinite sets 339360
- Appendix P. Countable and uncountable sets 343364
- Bibliography 347368
- Index 349370
- Back Cover Back Cover1384