PART I. ADVANCED CALCULUS IN ONE VARIABLE.
1. Real Numbers and Limits of Sequences.
2. Continuous Functions.
3. Rieman Integral.
4. The Derivative.
5. Infinite Series.
PART II. ADVANCED TOPICS IN ONE VARIABLE.
6. Fourier Series.
7. The Riemann-Stieltjes Integral.
PART III. ADVANCED CALCULUS IN SEVERAL VARIABLES.
8. Euclidean Space.
9. Continuous Functions on Euclidean Space.
10. The Derivative in Euclidean Space.
11. Riemann Integration in Euclidean Space.
Appendix A. Set Theory.
“The book is well-suited for students who have had some basic calculus and linear algebra, as an intermediate step before beginning more advanced topics as measure theory, functional analysis, and the theory of differential equations.” (Bull Belg Math Soc, 1 July 2010)
"This is an excellent book, well worth considering for a textbook for an undergraduate analysis course." (MAA Reviews July, 2008)
- Provides a rigorous approach to proofs, fundamental theorems, and the foundations of calculus
- Highlights the connections and interplay between calculus and linear algebra, emphasizing the concepts of a vector space, a linear transformation (including a linear functional), a norm, and a scalar product
- Offers a 'Test Yourself' section at the end of every chapter, which is a sample hour test with solutions to aid the study of readers
- Gradually guides the reader from the study of topology of the real line to the beginning theorems and concepts of graduate analysis, expressed from a modern viewpoint
- Features the system of real numbers as a Cauchy-complete Archimedean ordered field, and the traditional theorems of advanced calculus are presented.
- Employs a multi-part outline approach and provides broad hints to guide students through the more substantial proof exercises. Solutions to most of the non-proof exercises are also provided.