The Book
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.
The Author(s)
Patrick M. Fitzpatrick, University of Maryland, College Park, Maryland, USA
Table of Contents
- Tools for Analysis
- Convergent Sequences
- Continuous Functions
- Differentiation
- Elementary Functions as solutions of Differential Equations
- Integration: Two Fundamental Theorems
- Integartion: Further Topics
- Approximation by Taylor Polynomials
- Sequences and Series of Functions
- The Euclidean Space
- Continuity, Compactness, and Connectedness
- Metric Spaces
- Differentiating Functions of Several Variables
- Local Approximation of Real-Valued Functions
- Approximating Nonlinear Mappings by Linear Mappings
- Images and Inverses: The Inverse Function Theorem
- The Implicit Function Theorem and its Applications
- Integarting Functions of Several Variables
- Iterated Integartion and Changes of Variables
- Line and Surface Integrals
Consequences of the Field and Positivity Axioms
Linear Algebra
Index