Advanced Calculus

Third Edition

Demonstrating analytical and numerical techniques for attacking problems in the application of mathematics, this well-organized, clearly written text presents the logical relationship and fundamental notations of analysis. Buck discusses analysis not solely as a tool, but as a subject in its own right. This skill-building volume familiarizes students with the language, concepts, and standard theorems of analysis, preparing them to read the mathematical literature on their own. The text revisits certain portions of elementary calculus and gives a systematic, modern approach to the differential and integral calculus of functions and transformations in several variables, including an introduction to the theory of differential forms. The material is structured to benefit those students whose interests lean toward either research in mathematics or its applications.

“Buck’s text is a classic and a gem of an advanced calculus text. In recent decades there has been a proliferation of Real Analysis/Advanced Calculus texts but Buck’s text is unique in the combination of its perspicuity of exposition and the way it is able to communicate very abstract ideas in unusually concrete, computational, and intuitive terms. Its strength is its ability to deconstruct what are often presented as very abstract ideas in other texts while at the same time not oversimplifying the ideas. Its emphasis on computational examples and special cases (e.g., the implicit function theorem) makes for a much more natural transition for students who are used to thinking about analysis in computational terms from elementary calculus and differential equations, linear algebra, into the realms of advanced modern analysis. I’ve never seen a text that does this as well as Buck.” — Steven Deckelman, *University of Wisconsin, Stout*

“A classic.” — Li Guo, *Rutgers University*

“Like the earlier editions, the book is well written and nicely organized. Students find it to be readable, which is not the case with many mathematics texts.” — Thomas J. Smith, *Manhattan College*

“First and foremost, I commend you for providing a classic book at a reasonable student price!” — Paul Loya, *Binghamton University*

“A very high-quality text. Excellent explanations, good examples, and the problem sets are at the correct level of challenge. I’ve used earlier editions several times for a full-year course, much to the beneficial development of the students.” — Daniel D. Bonar, *Denison University*

“The excellent examples help students get to the heart of mathematical concepts for better understanding of real analysis.” — Rahim G. Karimpour, *Southern Illinois University*

“Buck’s book is a classic and is appropriate for our advanced calculus course. It is an excellent treatment!” — J. David Logan, *University of Nebraska*

“A classic introduction to a difficult subject. The new edition reads better than the previous editions’ already-good presentations.” — Javier Trigos, *California State University, Bakersfield*

1. Sets and Functions

2. Continuity

3. Differentiation

4. Integration

5. Series

6. Uniform Convergence

7. Differentiation of Transformations

8. Applications to Geometry and Analysis

9. Differential Geometry and Vector Calculus

10. Numerical Methods

Appendix 1. Logic and Set Theory

Appendix 2. Foundations of the Real Number System

Appendix 3. Linear Algebra

Appendix 4. Applications of Mathematics

Appendix 5. Introduction to Complex Analysis

Appendix 6. Further Topics in Real Analysis