This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. It can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting where lectures are organized around the presentation of problems and solutions. This book is appropriate for any student who has taken (or is concurrently taking) an introductory course in calculus. The book includes sixteen appendices that review some indispensable prerequisites on techniques of proof writing with special attention to the notation used the course.
John M. Erdman, Portland State University, Portland, OR
Preface For students: How to use this book
Chapter 1. Intervals 1.1. Distance and neighborhoods 1.2. Interior of a set
Chapter 2. Topology of the real line 2.1. Open subsets of \R 2.2. Closed subsets of \R
Chapter 3. Continuous functions from \R to \R 3.1. Continuity—as a local property 3.2. Continuity—as a global property 3.3. Functions defined on subsets of \R
Chapter 4. Sequences of real numbers 4.1. Convergence of sequences 4.2. Algebraic combinations of sequences 4.3. Sufficient condition for convergence 4.4. Subsequences
Chapter 5. Connectedness and the intermediate value theorem 5.1. Connected subsets of \R 5.2. Continuous images of connected sets 5.3. Homeomorphisms
Chapter 6. Compactness and the extreme value theorem 6.1. Compactness 6.2. Examples of compact subsets of \R 6.3. The extreme value theorem
Chapter 7. Limits of real valued functions 7.1. Definition 7.2. Continuity and limits
Chapter 8. Differentiation of real valued functions 8.1. The families \lobo𝑂 and \lobo𝑜 8.2. Tangency 8.3. Linear approximation 8.4. Differentiability
Chapter 9. Metric spaces 9.1. Definitions 9.2. Examples 9.3. Equivalent metrics
Chapter 10. Interiors, closures, and boundaries 10.1. Definitions and examples 10.2. Interior points 10.3. Accumulation points and closures
Chapter 11. The topology of metric spaces 11.1. Open and closed sets 11.2. The relative topology
Chapter 12. Sequences in metric spaces 12.1. Convergence of sequences 12.2. Sequential characterizations of topological properties 12.3. Products of metric spaces
Chapter 13. Uniform convergence 13.1. The uniform metric on the space of bounded functions 13.2. Pointwise convergence
Chapter 14. More on continuity and limits 14.1. Continuous functions 14.2. Maps into and from products 14.3. Limits
Chapter 15. Compact metric spaces 15.1. Definition and elementary properties 15.2. The extreme value theorem 15.3. Dini’s theorem
Chapter 16. Sequential characterization of compactness 16.1. Sequential compactness 16.2. Conditions equivalent to compactness 16.3. Products of compact spaces 16.4. The Heine–Borel theorem
Chapter 17. Connectedness 17.1. Connected spaces 17.2. Arcwise connected spaces
Chapter 18. Complete spaces 18.1. Cauchy sequences 18.2. Completeness 18.3. Completeness vs. compactness
Chapter 19. A fixed point theorem 19.1. The contractive mapping theorem 19.2. Application to integral equations
Chapter 20. Vector spaces 20.1. Definitions and examples 20.2. Linear combinations 20.3. Convex combinations
Chapter 21. Linearity 21.1. Linear transformations 21.2. The algebra of linear transformations 21.3. Matrices 21.4. Determinants 21.5. Matrix representations of linear transformations
Chapter 22. Norms 22.1. Norms on linear spaces 22.2. Norms induce metrics 22.3. Products 22.4. The space \fml𝐵(𝑆,𝑉)
Chapter 23. Continuity and linearity 23.1. Bounded linear transformations 23.2. The Stone–Weierstrass theorem 23.3. Banach spaces 23.4. Dual spaces and adjoints
Chapter 24. The Cauchy integral 24.1. Uniform continuity 24.2. The integral of step functions 24.3. The Cauchy integral
Chapter 25. Differential calculus 25.1. \lobo𝑂 and \lobo𝑜 functions 25.2. Tangency 25.3. Differentiation 25.4. Differentiation of curves 25.5. Directional derivatives 25.6. Functions mapping into product spaces
Chapter 26. Partial derivatives and iterated integrals 26.1. The mean value theorem(s) 26.2. Partial derivatives 26.3. Iterated integrals
Chapter 27. Computations in \Rⁿ 27.1. Inner products 27.2. The gradient 27.3. The Jacobian matrix 27.4. The chain rule
Chapter 28. Infinite series 28.1. Convergence of series 28.2. Series of positive scalars 28.3. Absolute convergence 28.4. Power series
Chapter 29. The implicit function theorem 29.1. The inverse function theorem 29.2. The implicit function theorem
Chapter 30. Higher order derivatives 30.1. Multilinear functions 30.2. Second order differentials 30.3. Higher order differentials
Appendix A. Quantifiers
Appendix B. Sets
Appendix C. Special subsets of \R
Appendix D. Logical connectives D.1. Disjunction and conjunction D.2. Implication D.3. Restricted quantifiers D.4. Negation
Appendix E. Writing mathematics E.1. Proving theorems E.2. Checklist for writing mathematics E.3. Fraktur and Greek alphabets
Appendix F. Set operations F.1. Unions F.2. Intersections F.3. Complements
Appendix G. Arithmetic G.1. The field axioms G.2. Uniqueness of identities G.3. Uniqueness of inverses G.4. Another consequence of uniqueness
Appendix H. Order properties of \R
Appendix I. Natural numbers and mathematical induction
Appendix J. Least upper bounds and greatest lower bounds J.1. Upper and lower bounds J.2. Least upper and greatest lower bounds J.3. The least upper bound axiom for \R J.4. The Archimedean property
Appendix K. Products, relations, and functions K.1. Cartesian products K.2. Relations K.3. Functions
Appendix L. Properties of functions L.1. Images and inverse images L.2. Composition of functions L.3. The identity function L.4. Diagrams L.5. Restrictions and extensions
Appendix M. Functions that have inverses M.1. Injections, surjections, and bijections M.2. Inverse functions
Appendix N. Products
Appendix O. Finite and infinite sets Appendix P. Countable and uncountable sets Bibliography