Advanced Calculus: An Introduction to Analysis

Description

Introduces analysis, presenting analytical proofs backed by geometric intuition and placing minimum reliance on geometric argument. This edition separates continuity and differentiation and expands coverage of integration to include discontinuous functions. The discussion of differentiation of a vector function of a vector variable has been modernized by defining the derivative to be the Jacobian matrix; and, the general form of the chain rule is given, as is the general form of the implicit transformation theorem. Table of contents PART I. CALCULUS OF ONE VARIABLE. Chapter 1. The Number System. Chapter 2. Functions, Sequences, and Limits. Chapter 3. Continuity and More Limits. Chapter 4. Differentiation. Chapter 5. Integration. Chapter 6. The Elementary Transcendental Functions. PART II. VECTOR CALCULUS. Chapter 7. Vectors and Curves. Chapter 8. Functions of Several Variables. Chapter 9. Limits and Continuity. Chapter 10. Differential Functions. Chapter 11. The Inversion Theorem. Chapter 12. Multiple Integrals. Chapter 13. Line and Surface Integrals. PART III. THEORY OF CONVERGENCE. Chapter 14. Infinite Series. Chapter 15. Sequence and Series of Functions. Chapter 16. Uniform Convergence. Chapter 17. The Taylor Series. Chapter 18. Improper Integrals. Chapter 19. Integral Representations of Functions. Chapter 20. Gamma and Beta Functions. Chapter 21. Laplace's Method and Stirling's Formula. Chapter 22. Fourier Series. Elementary Differentiation and Integration Formulas. Answers, Hints, and Solutions. Index.