Explorations in Analysis, Topology, and Dynamics: An Introduction to Abstract Mathematics (Pure and Applied Undergraduate Texts)

This book is an introduction to the theory of calculus in the style of inquiry-based learning. The text guides students through the process of making mathematical ideas rigorous, from investigations and problems to definitions and proofs. The format allows for various levels of rigor as negotiated between instructor and students, and the text can be of use in a theoretically oriented calculus course or an analysis course that develops rigor gradually. Material on topology (e.g., of higher dimensional Euclidean spaces) and discrete dynamical systems can be used as excursions within a study of analysis or as a more central component of a course. The themes of bisection, iteration, and nested intervals form a common thread throughout the text. The book is intended for students who have studied some calculus and want to gain a deeper understanding of the subject through an inquiry-based approach. Cover Title Page Preface Chapter 1. Real Numbers and Sequences 1.1. What are the real numbers? 1.2. A first look at sequences 1.3. A first look at series 1.4. Properties of sequences and series 1.5. Subsequences and the Bolzano–Weierstrass Theorem 1.6. Sequences and convergence in higher dimensions Chapter 2. An Introduction to Point-Set Topology and Continuity 2.1. The closure of a set; closed sets and open sets 2.2. Compact sets 2.3. Continuous functions 2.4. Continuous images of compact sets 2.5. Continuous images of intervals 2.6. Continuous mappings into bR ^{m} Chapter 3. Differential Calculus 3.1. The derivative 3.2. Using derivatives to find maxima and minima 3.3. The Mean Value Theorem 3.4. Rules for differentiation 3.5. Taylor’s Theorem 3.6. Power series expansions of a few common functions Chapter 4. Integral Calculus 4.1. The Riemann integral 4.2. The First Fundamental Theorem of Calculus (a telescoping sum) 4.3. Change of variables for integration 4.4. Averages and the Mean Value Theorem for Integrals 4.5. Accumulation and the Second Fundamental Theorem of Calculus 4.6. Methods for finding antiderivatives Chapter 5. Discrete Dynamical Systems 5.1. Iterating functions and types of orbits 5.2. The logistic map, modeling, and bifurcations 5.3. The doubling map and chaos 5.4. The tent map and fractals 5.5. The rotation map and Benford’s Law 5.6. The billiard map and phase space Chapter 6. Iterating Algorithms and Representations of Real Numbers 6.1. Iterating algorithms 6.2. Decimals and binaries 6.3. The Gauss map and continued fractions 6.4. The Euclidean algorithm and inscribing squares in rectangles Appendix A. Definitions, Proofs, and Mathematical Language A.1. Writing mathematics A.2. Writing definitions A.3. Writing proofs Appendix B. Sets and Functions between Sets B.1. The language of set theory B.2. Functions between sets Appendix C. Graphs Appendix D. Hints to Selected Problems Index Titles in Series Back Cover Offers an introduction to the theory of calculus in the style of inquiry-based learning. The text guides students through the process of making mathematical ideas rigorous, from investigations and problems to definitions and proofs. The format allows for various levels of rigour as negotiated between instructor and students.