Spivak M. Calculus

4th ed. — Publish or Perish, 2008. — 703 p. — ISBN: 978-0-914098-91-1.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 honors students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.
This edition differs from the third mainly in the inclusion of additional problems, as well as a complete update of the Suggested Reading, together with some changes of exposition, mainly in Chapters 5 and 20.Prologue
Basic Properties of Numbers
Numbers of Various Sorts
Foundations
Functions
Graphs
Limits
Continuous Functions
Three Hard Theorems
Last Upper Bounds
Derivatives and Integrals
Derivatives
Differentiation
Significance of the Derivative
Inverse Functions
Integrals
The Fundamental Theorem of Calculus
The Trigonometric Functions
π is Irrational
Planetary Motion
The Logarithm and Exponential Functions
Integration in Elementary Terms
Infinite Sequences and Infinite Series
Approximation by Polynomial Functions
e is Transcendental
Infinite Sequences
Infinite Series
Uniform Convergence and Prower 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