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.

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. Integartion: Further Topics
8. Approximation by Taylor Polynomials
9. Sequences and Series of Functions
10. The Euclidean Space
11. Continuity, Compactness, and Connectedness
12. Metric Spaces
13. Differentiating Functions of Several Variables
14. Local Approximation of Real-Valued Functions
15. Approximating Nonlinear Mappings by Linear Mappings
16. Images and Inverses: The Inverse Function Theorem
17. The Implicit Function Theorem and its Applications
18. Integarting Functions of Several Variables
19. Iterated Integartion and Changes of Variables
20. Line and Surface Integrals