A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book’s material has been extensively classroom tested in the author’s two-semester undergraduate course on real analysis at The George Washington University.
The first part of the text presents the calculus of functions of one variable. This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions. It also includes optional sections on Stirling’s formula, functions of bounded variation, Riemann–Stieltjes integration, and other topics.
The second part focuses on functions of several variables. It introduces the topological ideas (such as compact and connected sets) needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in Rn.
The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises. A full solutions manual offers complete solutions to all exercises for qualifying instructors.
With clear proofs, detailed examples, and numerous exercises, this textbook gives a thorough treatment of the subject. It progresses from single variable to multivariable functions, providing a logical development of material that will prepare students for more advanced analysis-based courses.
Author(s): Hugo D. Junghenn
Publisher: A Chapman & Hall/CRC
Year: 2015
Language: English
Pages: xxiii,581
Cover
S Title
A COURSE IN REAL ANALYSIS
© 2015 by Taylor & Francis Group LLC
ISBN 978-1-4822-1928-9 (eBook - PDF)
Dedeication
Contents
Preface
List of Figures
List of Tables
List of Symbols
Part I: Functions of One Variable
Chapter 1: The Real Number System
1.1 From Natural Numbers to Real Numbers
1.2 Algebraic Properties of R
Exercises
1.3 Order Structure of R
Exercises
1.4 Completeness Property of R
Exercises
1.5 Mathematical Induction
Exercises
1.6 Euclidean Space
Exercises
Chapter 2: Numerical Sequences
2.1 Limits of Sequences
Exercises
2.2 Monotone Sequences
Exercises
2.3 Subsequences and Cauchy Sequences
Exercises
2.4 Limits Inferior and Superior
Exercises
Chapter 3: Limits and Continuity on R
3.1 Limit of a Function
Exercises
*3.2 Limits Inferior and Superior
Exercises
3.3 Continuous Functions
Exercises
3.4 Properties of Continuous Functions
Exercises
3.5 Uniform Continuity
Exercises
Chapter 4: Differentiation on R
4.1 Definition of Derivative and Examples
Exercises
4.2 The Mean Value Theorem
Exercises
*4.3 Convex Functions
4.4 Inverse Functions
Exercises
4.5 L’Hospital’s Rule
Exercises
4.6 Taylor’s Theorem on R
Exercises
*4.7 Newton’s Method
Exercises
Chapter 5: Riemann Integration on R
5.1 The Riemann–Darboux Integral
Exercises
5.2 Properties of the Integral
Exercises
5.3 Evaluation of the Integral
Exercises
*5.4 Stirling’s Formula
5.5 Integral Mean Value Theorems
Exercises
*5.6 Estimation of the Integral
5.7 Improper Integrals
Exercises
5.8 A Deeper Look at Riemann Integrability
Exercises
*5.9 Functions of Bounded Variation
*5.10 The Riemann–Stieltjes Integral
Chapter 6: Numerical Infinite Series
6.1 Definition and Examples
Exercises
6.2 Series with Nonnegative Terms
Exercises
6.3 More Refined Convergence Tests
Exercises
6.4 Absolute and Conditional Convergence
Exercises
*6.5 Double Sequences and Series
Exercises
Chapter 7: Sequences and Series of Functions
7.1 Convergence of Sequences of Functions
Exercises
7.2 Properties of the Limit Function
Exercises
7.3 Convergence of Series of Functions
Exercises
7.4 Power Series
Exercises
Part II: Functions of Several Variables
Chapter 8: Metric Spaces
8.1 Definitions and Examples
Exercises
8.2 Open and Closed Sets
Exercises
8.3 Closure, Interior, and Boundary
Exercises
8.4 Limits and Continuity
Exercises
8.5 Compact Sets
Exercises
*8.6 The Arzelà–Ascoli Theorem
Exercises
8.7 Connected Sets
Exercises
8.8 The Stone–Weierstrass Theorem
Exercises
*8.9 Baire’s Theorem
Exercises
Chapter 9: Differentiation on R^n
9.1 Definition of the Derivative
Exercises
9.2 Properties of the Differential
Exercises
9.3 Further Properties of the Differential
Exercises
9.4 Inverse Function Theorem
Exercises
9.5 Implicit Function Theorem
Exercises
9.6 Higher Order Partial Derivatives
Exercises
9.7 Higher Order Differentials and Taylor’s Theorem
Exercises
*9.8 Optimization
Exercises
Chapter 10: Lebesgue Measure on R^n
10.1 General Measure Theory
Exercises
10.2 Lebesgue Outer Measure
Exercises
10.3 Lebesgue Measure
Exercises
10.4 Borel Sets
Exercises
10.5 Measurable Functions
Exercises
Chapter 11: Lebesgue Integration on R^n
11.1 Riemann Integration on R^n
11.2 The Lebesgue Integral
Exercises
11.3 Convergence Theorems
Exercises
11.4 Connections with Riemann Integration
11.5 Iterated Integrals
Exercises
11.6 Change of Variables
Exercises
Chapter 12: Curves and Surfaces in R^n
12.1 Parameterized Curves
Exercises
12.2 Integration on Curves
Exercises
12.3 Parameterized Surfaces
Exercises
12.4 m-Dimensional Surfaces
Exercises
Chapter 13: Integration on Surfaces
13.1 Differential Forms
Exercises
13.2 Integrals on Parameterized Surfaces
Exercises
13.3 Partitions of Unity
13.4 Integration on Compact m-Surfaces
Exercises
13.5 The Fundamental Theorems of Calculus
Exercises
*13.6 Closed Forms in R^n
Part III: Appendices
Appendix A: Set Theory
Appendix B: Linear Algebra
Appendix C: Solutions to Selected Problems
Bibliography
Back Cover
