A Problem Text in Advanced Calculus

381 Pages·2009·2.65 MB·english

by  John M. Erdman

A ProblemText in Advanced Calculus
John M. Erdman
Portland State University
Version October 15, 2009
(cid:13)c 2005 John M. Erdman
E-mail address: [email protected]
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To Argentina

Contents
PREFACE xi
FOR STUDENTS: HOW TO USE THIS PROBLEMTEXT xv
Chapter 1. INTERVALS 1
1.1. DISTANCE AND NEIGHBORHOODS 1
1.2. INTERIOR OF A SET 2
Chapter 2. TOPOLOGY OF THE REAL LINE 5
2.1. OPEN SUBSETS OF R 5
2.2. CLOSED SUBSETS OF R 6
Chapter 3. CONTINUOUS FUNCTIONS FROM R TO R 9
3.1. CONTINUITY—AS A LOCAL PROPERTY 9
3.2. CONTINUITY—AS A GLOBAL PROPERTY 10
3.3. FUNCTIONS DEFINED ON SUBSETS OF R 13
Chapter 4. SEQUENCES OF REAL NUMBERS 17
4.1. CONVERGENCE OF SEQUENCES 17
4.2. ALGEBRAIC COMBINATIONS OF SEQUENCES 19
4.3. SUFFICIENT CONDITION FOR CONVERGENCE 20
4.4. SUBSEQUENCES 23
Chapter 5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM 27
5.1. CONNECTED SUBSETS OF R 27
5.2. CONTINUOUS IMAGES OF CONNECTED SETS 29
5.3. HOMEOMORPHISMS 30
Chapter 6. COMPACTNESS AND THE EXTREME VALUE THEOREM 33
6.1. COMPACTNESS 33
6.2. EXAMPLES OF COMPACT SUBSETS OF R 34
6.3. THE EXTREME VALUE THEOREM 36
Chapter 7. LIMITS OF REAL VALUED FUNCTIONS 39
7.1. DEFINITION 39
7.2. CONTINUITY AND LIMITS 40
Chapter 8. DIFFERENTIATION OF REAL VALUED FUNCTIONS 43
8.1. THE FAMILIES O AND o 43
8.2. TANGENCY 45
8.3. LINEAR APPROXIMATION 46
8.4. DIFFERENTIABILITY 47
Chapter 9. METRIC SPACES 51
9.1. DEFINITIONS 51
9.2. EXAMPLES 52
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vi CONTENTS
9.3. STRONGLY EQUIVALENT METRICS 55
Chapter 10. INTERIORS, CLOSURES, AND BOUNDARIES 57
10.1. DEFINITIONS AND EXAMPLES 57
10.2. INTERIOR POINTS 58
10.3. ACCUMULATION POINTS AND CLOSURES 58
Chapter 11. THE TOPOLOGY OF METRIC SPACES 61
11.1. OPEN AND CLOSED SETS 61
11.2. THE RELATIVE TOPOLOGY 63
Chapter 12. SEQUENCES IN METRIC SPACES 65
12.1. CONVERGENCE OF SEQUENCES 65
12.2. SEQUENTIAL CHARACTERIZATIONS OF TOPOLOGICAL PROPERTIES 65
12.3. PRODUCTS OF METRIC SPACES 66
Chapter 13. UNIFORM CONVERGENCE 69
13.1. THE UNIFORM METRIC ON THE SPACE OF BOUNDED FUNCTIONS 69
13.2. POINTWISE CONVERGENCE 70
Chapter 14. MORE ON CONTINUITY AND LIMITS 73
14.1. CONTINUOUS FUNCTIONS 73
14.2. MAPS INTO AND FROM PRODUCTS 77
14.3. LIMITS 79
Chapter 15. COMPACT METRIC SPACES 83
15.1. DEFINITION AND ELEMENTARY PROPERTIES 83
15.2. THE EXTREME VALUE THEOREM 84
15.3. DINI’S THEOREM 85
Chapter 16. SEQUENTIAL CHARACTERIZATION OF COMPACTNESS 87
16.1. SEQUENTIAL COMPACTNESS 87
16.2. CONDITIONS EQUIVALENT TO COMPACTNESS 88
16.3. PRODUCTS OF COMPACT SPACES 89
16.4. THE HEINE-BOREL THEOREM 90
Chapter 17. CONNECTEDNESS 93
17.1. CONNECTED SPACES 93
17.2. ARCWISE CONNECTED SPACES 94
Chapter 18. COMPLETE SPACES 97
18.1. CAUCHY SEQUENCES 97
18.2. COMPLETENESS 97
18.3. COMPLETENESS VS. COMPACTNESS 98
Chapter 19. APPLICATIONS OF A FIXED POINT THEOREM 101
19.1. THE CONTRACTIVE MAPPING THEOREM 101
19.2. APPLICATION TO INTEGRAL EQUATIONS 105
Chapter 20. VECTOR SPACES 107
20.1. DEFINITIONS AND EXAMPLES 107
20.2. LINEAR COMBINATIONS 111
20.3. CONVEX COMBINATIONS 112
Chapter 21. LINEARITY 115
21.1. LINEAR TRANSFORMATIONS 115
CONTENTS vii
21.2. THE ALGEBRA OF LINEAR TRANSFORMATIONS 119
21.3. MATRICES 120
21.4. DETERMINANTS 124
21.5. MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS 125
Chapter 22. NORMS 129
22.1. NORMS ON LINEAR SPACES 129
22.2. NORMS INDUCE METRICS 130
22.3. PRODUCTS 131
22.4. THE SPACE B(S,V) 134
Chapter 23. CONTINUITY AND LINEARITY 137
23.1. BOUNDED LINEAR TRANSFORMATIONS 137
23.2. THE STONE-WEIERSTRASS THEOREM 141
23.3. BANACH SPACES 143
23.4. DUAL SPACES AND ADJOINTS 144
Chapter 24. THE CAUCHY INTEGRAL 145
24.1. UNIFORM CONTINUITY 145
24.2. THE INTEGRAL OF STEP FUNCTIONS 147
24.3. THE CAUCHY INTEGRAL 150
Chapter 25. DIFFERENTIAL CALCULUS 157
25.1. O AND o FUNCTIONS 157
25.2. TANGENCY 159
25.3. DIFFERENTIATION 160
25.4. DIFFERENTIATION OF CURVES 163
25.5. DIRECTIONAL DERIVATIVES 165
25.6. FUNCTIONS MAPPING INTO PRODUCT SPACES 166
Chapter 26. PARTIAL DERIVATIVES AND ITERATED INTEGRALS 169
26.1. THE MEAN VALUE THEOREM(S) 169
26.2. PARTIAL DERIVATIVES 173
26.3. ITERATED INTEGRALS 177
Chapter 27. COMPUTATIONS IN Rn 181
27.1. INNER PRODUCTS 181
27.2. THE GRADIENT 183
27.3. THE JACOBIAN MATRIX 187
27.4. THE CHAIN RULE 189
Chapter 28. INFINITE SERIES 195
28.1. CONVERGENCE OF SERIES 195
28.2. SERIES OF POSITIVE SCALARS 200
28.3. ABSOLUTE CONVERGENCE 200
28.4. POWER SERIES 202
Chapter 29. THE IMPLICIT FUNCTION THEOREM 209
29.1. THE INVERSE FUNCTION THEOREM 209
29.2. THE IMPLICIT FUNCTION THEOREM 213
Appendix A. QUANTIFIERS 219
Appendix B. SETS 221
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Appendix C. SPECIAL SUBSETS OF R 225
Appendix D. LOGICAL CONNECTIVES 227
D.1. DISJUNCTION AND CONJUNCTION 227
D.2. IMPLICATION 228
D.3. RESTRICTED QUANTIFIERS 230
D.4. NEGATION 230
Appendix E. WRITING MATHEMATICS 233
E.1. PROVING THEOREMS 233
E.2. CHECKLIST FOR WRITING MATHEMATICS 234
E.3. FRAKTUR AND GREEK ALPHABETS 236
Appendix F. SET OPERATIONS 237
F.1. UNIONS 237
F.2. INTERSECTIONS 239
F.3. COMPLEMENTS 240
Appendix G. ARITHMETIC 243
G.1. THE FIELD AXIOMS 243
G.2. UNIQUENESS OF IDENTITIES 244
G.3. UNIQUENESS OF INVERSES 245
G.4. ANOTHER CONSEQUENCE OF UNIQUENESS 245
Appendix H. ORDER PROPERTIES OF R 247
Appendix I. NATURAL NUMBERS AND MATHEMATICAL INDUCTION 249
Appendix J. LEAST UPPER BOUNDS AND GREATEST LOWER BOUNDS 253
J.1. UPPER AND LOWER BOUNDS 253
J.2. LEAST UPPER AND GREATEST LOWER BOUNDS 253
J.3. THE LEAST UPPER BOUND AXIOM FOR R 255
J.4. THE ARCHIMEDEAN PROPERTY 256
Appendix K. PRODUCTS, RELATIONS, AND FUNCTIONS 259
K.1. CARTESIAN PRODUCTS 259
K.2. RELATIONS 260
K.3. FUNCTIONS 260
Appendix L. PROPERTIES OF FUNCTIONS 263
L.1. IMAGES AND INVERSE IMAGES 263
L.2. COMPOSITION OF FUNCTIONS 264
L.3. The IDENTITY FUNCTION 264
L.4. DIAGRAMS 265
L.5. RESTRICTIONS AND EXTENSIONS 265
Appendix M. FUNCTIONS WHICH HAVE INVERSES 267
M.1. INJECTIONS, SURJECTIONS, AND BIJECTIONS 267
M.2. INVERSE FUNCTIONS 269
Appendix N. PRODUCTS 271
Appendix O. FINITE AND INFINITE SETS 273
Appendix P. COUNTABLE AND UNCOUNTABLE SETS 277
Appendix Q. SOLUTIONS TO EXERCISES 281
CONTENTS ix
Q.1. Exercises in chapter 01 281
Q.2. Exercises in chapter 02 281
Q.3. Exercises in chapter 03 282
Q.4. Exercises in chapter 04 284
Q.5. Exercises in chapter 05 285
Q.6. Exercises in chapter 06 286
Q.7. Exercises in chapter 07 286
Q.8. Exercises in chapter 08 287
Q.9. Exercises in chapter 09 289
Q.10. Exercises in chapter 10 290
Q.11. Exercises in chapter 11 290
Q.12. Exercises in chapter 12 291
Q.13. Exercises in chapter 13 291
Q.14. Exercises in chapter 14 293
Q.15. Exercises in chapter 15 295
Q.16. Exercises in chapter 16 295
Q.17. Exercises in chapter 17 296
Q.18. Exercises in chapter 18 298
Q.19. Exercises in chapter 19 300
Q.20. Exercises in chapter 20 302
Q.21. Exercises in chapter 21 304
Q.22. Exercises in chapter 22 309
Q.23. Exercises in chapter 23 311
Q.24. Exercises in chapter 24 314
Q.25. Exercises in chapter 25 318
Q.26. Exercises in chapter 26 322
Q.27. Exercises in chapter 27 327
Q.28. Exercises in chapter 28 332
Q.29. Exercises in chapter 29 339
Q.30. Exercises in appendix D 342
Q.31. Exercises in appendix F 343
Q.32. Exercises in appendix G 344
Q.33. Exercises in appendix H 345
Q.34. Exercises in appendix I 345
Q.35. Exercises in appendix J 346
Q.36. Exercises in appendix K 347
Q.37. Exercises in appendix L 348
Q.38. Exercises in appendix M 349
Q.39. Exercises in appendix N 351
Q.40. Exercises in appendix O 351
Q.41. Exercises in appendix P 352
Bibliography 355
Index 357