Advanced calculus
Author(s)
Bibliographic Information
Advanced calculus
（The Sally series, . Pure and applied undergraduate texts ; 5）
American Mathematical Society, [2009?]
2nd ed
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Note
Originally published as Advanced calculus, 2nd ed.: Belmont, CA : Thomson Brooks/Cole, c2006
Includes index
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"