A Contemporary Approach to Classical Theorems of Formal Calculus
MSC2010_LCCCN_ISBN: 57-01_66-10910_9780805390216
Luqman Malik
IMU P-RNK: 1A (adv. baccalaureate)
Series: Foundations in Mathematical Analysis
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Distribution license: Hilix Research and Technologies, Inc.®
CONTENTS
Preface
Part 1
Prologue
1. Basic Properties of Numbers 3
2. Numbers of Various Sorts 21
Part 2
Foundations
3. Functions 39
Appendix. Ordered Pairs 54
4. Graphs 56
Appendix 1. Vectors 75
Appendix 2. The Conic Sections 80
Appendix 3. Polar Coordinates 84
5. Limits 90
6. Continuous Functions 113
7. Three Hard Theorems 120
8. Least Upper Bounds 131
Appendix. Uniform Continuity 142
Part 3
Derivatives and Integrals
9. Derivatives 147
10. Differentiation 166
11. Significance of the Derivative 185
Appendix. Convexity and Concavity 216
12. Inverse Functions 227
Appendix. Parametric Representation of Curves 241
13. Integrals 250
Appendix. Riemann Sums 279
14. The Fundamental Theorem of Calculus 282
15. The Trigonometric Functions 300
16. π is Irrational 321 ⚐
17. Planetary Motion 327 ⚐
18. The Logarithm and Exponentail Functions 336
19. Integration in Elementary Terms 359
Appendix. The Cosmopolitan Integral 397
Part 4
Infinite Sequences and Infinite Series
20. Approximation by Polynomial Functions 405
21. e is Transcendental 435 ⚐
22. Infinite Sequences 445
23. Infinite Series 464
24. Uniform Convergence and Power Series 491
25. Complex Numbers 517
26. Complex Functions 532
27. Complex Power Series 546
Part 5
Epilogue
28. Fields 571
29. Construction of the Real Numbers 578
30. Uniqueness of the Real Numbers 591
Suggested reading 599
Answers (to Selected Problems) 609
Glossery of Symbols 655
Index 659
This is the second installment in Spivak's analysis sequence. A strong footing in Calculus, A Hitchhiker's Guide to Calculus, and the present text should prepare the serious student of mathematical analysis for Spivak's five–volume Comprehensive introduction to Differential Geometry. Mathematicians use to polorize over the supremecy of pure versus applied mathematics; that paradigm has all but vanished. Today, it is a question of algebra versus analysis. If one finds themself pivoting between the two, I have no doubt that an honest attempt at Spivak will settle the matter for them.
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