Calculus

Description

"Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis." Preface Contents Part I - Prologue 1. Basic Properties of Numbers 2. Numbers of Various Sorts Part II - Foundations 3. Functions Appendix. Ordered Pairs 4. Graphs Appendix 1. Vectors Appendix 2. The Conic Sections Appendix 3. Polar Coordinates 5. Limits 6. Continuous Functions 7. Three Hard Theorems 8. Least Upper Bounds Appendix. Uniform Continuity Part III - Derivatives and Integrals 9. Derivatives 10. Differentiation 11. Significance of the Derivative Appendix. Convexity and Concavity 12. Inverse Functions Appendix. Parametric Representation of Curves 13. Integrals Appendix. Riemann Sums 14. The Fundamental Theorem of Calculus 15. The Trigonometric Functions *16. π is Irrational *17. Planetary Motion 18. The Logarithm and Exponential Functions 19. Integration in Elementary Terms Appendix. The Cosmopolitan Integral Part IV - Infinite Sequences and Infinite Series 20. Approximation by Polynomial Functions *21. e is Transcendental 22. Infinite Sequences 23. Infinite Series 24. Uniform Convergence and Power Series 25. Complex Numbers 26. Complex Functions 27. Complex Power Series Part V - Epilogue 28. Fields 29. Construction of the Real Numbers 30. Uniqueness of the Real Numbers Suggested Reading Answers (to selected problems) Glossary of Symbols Index