Explorations in Analysis, Topology, and Dynamics: An Introduction to Abstract Mathematics

ISBN-13: 9781470452704

Alejandro A. Uribe;Daniel A. Visscher

Opis

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 rigour 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.

Spis treści

• Real numbers and sequences
• An introduction to point-set topology and continuity
• Differential calculus
• Integral calculus
• Discrete dynamical systems
• Iterating algorithms and represenations of real numbers
• Appendix A: Definitions, proofs, and mathematical language
• Appendix B: Sets and functions between sets
• Appendix C: Graphs
• Appendix D: Hints to selected problems
• Index.