Calculus on Manifolds

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
····························································  
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.

Required tech() This repository is Safari–centric and designed on a 13" MacBook running OS X El Capitan. Given the simplex design of the page–frame (HTML_CSS, MathJax) versus the complex embedding of the carousel⁴ main–frame (JS, PHP, c++), a browser–universal repository seems exessive and idyllic at this stage.

Copyrighting dynamic arrangements of digitized mathematical content seems intuitively paradoxical and superfluous, but – in fact, the inverse can immediately be shown to be true. By embedding the appropriate license file in each clone and_or fork of each version of the source, a well–defined attribution ecology is formed within the repository's geneological record. Furthermore, by compiling the metadata of each version>clone>fork, one can analize the structural distance of that copy from the source. Therefore, [repo + metadata(repo) + License]::authenticate::autr(s_code).