Advanced Calculus: An Introduction to Linear Analysis
April 2008, ©2008
Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.
Following an introduction dedicated to writing proofs, the book is divided into three parts:
Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals.
Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics.
Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals.
Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a "Test Yourself" section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory.
Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, Advanced Calculus is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.
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.
Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Assistant Chair of the Department of Mathematics at Louisiana State University, where he is also Director of Graduate Studies in Mathematics. Dr. Richardson's research interests include harmonic analysis and representation theory.
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.
“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)
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