Advanced Calculus

Patrick M. Fitzpatrick (Author)

9780821847916

Hardback, published 28 February 2009

590 pages, illustrations
25.9 x 18.2 x 3.5 cm, 1.195 kg

This is a well-written and well-structured book with clearly explained proofs and a good supply of exercises, some of them are quite challenging. It is this reviewer's opinion that the volume should be an excellent and useful tool for undergraduate students." - Teodora-Liliana Radulescu, Zentralblatt MATH

Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables. Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material. Supplemented with numerous exercises, ""Advanced Calculus"" is a perfect book for undergraduate students of analysis.

• Preface
• Preliminaries
• 1. Tools for Analysis
• 2. Convergent Sequences
• 3. Continuous Functions
• 4. Differentiation
• 5. Elementary Functions as Solutions of Differential Equations
• 6. Integration: Two Fundamental Theorems
• 7. Integration: Further Topics
• 8. Approximation by Taylor Polynomials
• 9. Sequences and Series of Functions
• 10. The Euclidean Space Rn
• 11. Continuity, Compactness, and Connectedness
• 12. Metric Spaces
• 13. Differentiating Functions of Several Variables
• 14. Local Approximation of Real-Valued Functions
• 15. Approximating Nonlinear Mapping by Linear Mappings
• 16. Images and Inverses: The Inverse Function Theorem
• 17. The Implicit Function Theorem and its Applications
• 18. Integrating Functions of Several Variables
• 19. Iterated Integration and Changes of Variables
• 20. Line and Surface Integrals
• Appendix A: Consequences of the Field and Positivity Axioms
• Appendix B: Linear Algebra
• Index