A Problems Based Course in Advanced Calculus (Pure and Applied Undergraduate Texts) (Pure and Applied Undergraduate Texts, 32)

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. Cover......Page 1 Title page......Page 4 Contents......Page 8 Preface......Page 14 For students: How to use this book......Page 18 1.1. Distance and neighborhoods......Page 22 1.2. Interior of a set......Page 24 2.1. Open subsets of \R......Page 26 2.2. Closed subsets of \R......Page 28 3.1. Continuity—as a local property......Page 32 3.2. Continuity—as a global property......Page 34 3.3. Functions defined on subsets of \R......Page 37 4.1. Convergence of sequences......Page 42 4.2. Algebraic combinations of sequences......Page 45 4.3. Sufficient condition for convergence......Page 46 4.4. Subsequences......Page 50 5.1. Connected subsets of \R......Page 54 5.2. Continuous images of connected sets......Page 56 5.3. Homeomorphisms......Page 58 Chapter 6. Compactness and the extreme value theorem......Page 60 6.1. Compactness......Page 61 6.2. Examples of compact subsets of \R......Page 62 6.3. The extreme value theorem......Page 64 7.1. Definition......Page 66 7.2. Continuity and limits......Page 67 8.1. The families \lobo and \lobo......Page 70 8.2. Tangency......Page 73 8.3. Linear approximation......Page 74 8.4. Differentiability......Page 76 Chapter 9. Metric spaces......Page 80 9.2. Examples......Page 81 9.3. Equivalent metrics......Page 85 10.1. Definitions and examples......Page 88 10.2. Interior points......Page 89 10.3. Accumulation points and closures......Page 90 11.1. Open and closed sets......Page 92 11.2. The relative topology......Page 95 12.1. Convergence of sequences......Page 98 12.2. Sequential characterizations of topological properties......Page 99 12.3. Products of metric spaces......Page 100 13.1. The uniform metric on the space of bounded functions......Page 102 13.2. Pointwise convergence......Page 104 14.1. Continuous functions......Page 106 14.2. Maps into and from products......Page 112 14.3. Limits......Page 114 15.1. Definition and elementary properties......Page 120 15.2. The extreme value theorem......Page 122 15.3. Dini’s theorem......Page 123 16.1. Sequential compactness......Page 124 16.2. Conditions equivalent to compactness......Page 126 16.3. Products of compact spaces......Page 127 16.4. The Heine–Borel theorem......Page 128 17.1. Connected spaces......Page 130 17.2. Arcwise connected spaces......Page 132 18.1. Cauchy sequences......Page 134 18.2. Completeness......Page 135 18.3. Completeness vs. compactness......Page 136 19.1. The contractive mapping theorem......Page 138 19.2. Application to integral equations......Page 143 20.1. Definitions and examples......Page 146 20.2. Linear combinations......Page 151 20.3. Convex combinations......Page 153 21.1. Linear transformations......Page 156 21.2. The algebra of linear transformations......Page 160 21.3. Matrices......Page 163 21.4. Determinants......Page 167 21.5. Matrix representations of linear transformations......Page 169 22.1. Norms on linear spaces......Page 174 22.2. Norms induce metrics......Page 176 22.3. Products......Page 177 22.4. The space \fml(,)......Page 181 23.1. Bounded linear transformations......Page 184 23.2. The Stone–Weierstrass theorem......Page 189 23.3. Banach spaces......Page 192 23.4. Dual spaces and adjoints......Page 193 24.1. Uniform continuity......Page 196 24.2. The integral of step functions......Page 199 24.3. The Cauchy integral......Page 202 25.1. \lobo and \lobo functions......Page 210 25.2. Tangency......Page 213 25.3. Differentiation......Page 214 25.4. Differentiation of curves......Page 218 25.5. Directional derivatives......Page 220 25.6. Functions mapping into product spaces......Page 222 26.1. The mean value theorem(s)......Page 224 26.2. Partial derivatives......Page 230 26.3. Iterated integrals......Page 235 27.1. Inner products......Page 238 27.2. The gradient......Page 241 27.3. The Jacobian matrix......Page 246 27.4. The chain rule......Page 247 Chapter 28. Infinite series......Page 254 28.1. Convergence of series......Page 255 28.2. Series of positive scalars......Page 260 28.3. Absolute convergence......Page 261 28.4. Power series......Page 262 Chapter 29. The implicit function theorem......Page 272 29.1. The inverse function theorem......Page 273 29.2. The implicit function theorem......Page 277 30.1. Multilinear functions......Page 286 30.2. Second order differentials......Page 292 30.3. Higher order differentials......Page 297 Appendix A. Quantifiers......Page 298 Appendix B. Sets......Page 300 Appendix C. Special subsets of \R......Page 304 D.1. Disjunction and conjunction......Page 306 D.2. Implication......Page 308 D.3. Restricted quantifiers......Page 309 D.4. Negation......Page 310 E.1. Proving theorems......Page 312 E.2. Checklist for writing mathematics......Page 313 E.3. Fraktur and Greek alphabets......Page 316 F.1. Unions......Page 318 F.2. Intersections......Page 320 F.3. Complements......Page 322 G.1. The field axioms......Page 324 G.3. Uniqueness of inverses......Page 326 G.4. Another consequence of uniqueness......Page 327 Appendix H. Order properties of \R......Page 330 Appendix I. Natural numbers and mathematical induction......Page 334 J.1. Upper and lower bounds......Page 338 J.2. Least upper and greatest lower bounds......Page 339 J.3. The least upper bound axiom for \R......Page 341 J.4. The Archimedean property......Page 342 K.1. Cartesian products......Page 344 K.2. Relations......Page 345 K.3. Functions......Page 346 L.1. Images and inverse images......Page 348 L.2. Composition of functions......Page 349 L.4. Diagrams......Page 350 L.5. Restrictions and extensions......Page 351 M.1. Injections, surjections, and bijections......Page 352 M.2. Inverse functions......Page 355 Appendix N. Products......Page 358 Appendix O. Finite and infinite sets......Page 360 Appendix P. Countable and uncountable sets......Page 364 Bibliography......Page 368 Index......Page 370 Back Cover......Page 384