Spivak M. Calculus

3rd edition. — Cambridge University Press, 2006. — 682 p.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.Prologne
Basic Properties of Numbers
Numbers of Various Sorts
Foundations
Functions
Appendix. Ordered Pairs
Graphs
Aþþendix 1. Vectors
Appendix 2. The Cone Sections
Appendix 3. Polar Coordinates
Limits
Continuous Functions
Three Hard Theorems
Least Upper Bounds
Appendix. Unform Continuity
Derivatives and Integrals
Derivatives
Differentiation
Significance of the Derivative
Appendix. Convexity and Concavity
Inverse Functions
Appendx. Parametric Representation of Curves
Integrals
Appendix. Riemann Sums
The Fundamental Theorem of Calculus
The Trigonometric Functions
*Pi is Irrational
*Planetary Motion
The Logarithm and Exponential Functions
Integration in Elementary Terms
Appendix. The Cosmopolitan Integral
Infinite Sequences and Infinite Series
Approximation by Polynomial Functions
*e is Transcendental
Infinite Sequences
Infinite Series
Uniform Convergence and Power Series
Complex Numbers
Complex Functions
Complex Power Series $
Epilogue
Fields
Construction of the Real Numbers
Uniqueness of the Real Numbers
Suggested Reading
Answers (to selected problems)
Glossary of Symbols
Index