Calculus, Volume 1

An introduction to the Calculus, with an excellent balance between theory and technique. Integration is treated before differentiation--this is a departure from most modern texts, but it is historically correct, and it is the best way to establish the true connection between the integral and the derivative. Proofs of all the important theorems are given, generally preceded by geometric or intuitive discussion. This Second Edition introduces the mean-value theorems and their applications earlier in the text, incorporates a treatment of linear algebra, and contains many new and easier exercises. As in the first edition, an interesting historical introduction precedes each important new concept.

I. Introduction

Part 1. Historical Introduction

Part 2. Some Basic Concepts of the theory of sets

Part 3. A set of Axioms for the Real-Number System

Part 4. Mathematical Induction, Summation Notation,and Related Topics

1. The Concepts of Integral Calculus

2. Some Applications of Integration

3. Continuous Functions

4. Differential Calculus

5. The Relation Between Integration and Differentiation

6. The Logarithm,the Exponential,and the Inverse Trigonmetric Functions

7. Polynomial Approximations to Functions

8. Introduction to Differential Equations

9. Complex Numbers

10. Sequences, Infinite Series, Improper Integrals

11. Sequences and Series of functions

12. Vector algebra

13. Applications of Vector Algebra to Analytic Geometry

14. Calculus of Vector Valued Functions

15. Linear Spaces

16. Linear Transformations and Matrices

Answers to exercises 617

Index 657

TOM M. APOSTOL, Emeritus Professor at the California Institute of Technology, is the author of several highly regarded texts on calculus, analysis, and number theory, and is Director of Project MATHEMATICS!, a series of computer-animated mathematics videotapes.