Chapter 1. Tools for Analysis
Chapter 2. Convergent Sequences
Chapter 3. Continuous Functions
Chapter 4. Differentiation
Chapter 5. Elementary Functions as Solutions of Differential Equations
Chapter 6. Integration: Two Fundamental Theorems
Chapter 7. Integration: Further Topics
Chapter 8. Approximation By Taylor Polynomials
Chapter 9. Sequences And Series Of Functions
Chapter 10. The Euclidean Space
Chapter 11. Continuity, Compactness, And Connectednes
Chapter 12. Metric Spaces
Chapter 13. Differentiating Functions Of Several Variables
Chapter 14. Local Approximation Of Real-Valued Functions
Chapter 15. Approximating Nonlinear Mappings By Linear Mappings
Chapter 16. Images And Inverses: The Inverse Function Theorem
Chapter 17. Integrating Functions Of Several Variables
Chapter 18. Iterated Integration And Changes Of Variables
Chapter 19. Line And Surface Integrals

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.---Summary provided by publisher