Introduction to Real Analysis
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Language
English
ISBN
Unknown
Trinity University
Digital Commons @ Trinity
12-1-2013
Introduction to Real Analysis
William F. Trench
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Table of Contents
Preface
Chapter 1 The Real Numbers
Section 1.1 The Real Number System
Section 1.2 Mathematical Induction
Section 1.3 The Real Line
Chapter 2 Differential Calculus of Functions of One Variable
Section 2.1 Functions and Limits
Section 2.2 Continuity
Differentiable Functions of One Variable
Section 2.4 L'Hospital's Rule
Section 2.5 Taylor's Theorem
Chapter 3 Integral Calculus of Functions of One
Section 3.1 Definition of the Integral
Section 3.2 Existence of the Integral
Section 3.3 Properties of the Integral
Section 3.4 Improper Integrals
Section 3.5 A More Advanced Look at the Existence of the Proper Riemann Integral
Chapter 4 Infinite Sequences and Series
Section 4.1 Sequences of Real Numbers
Section 4.2 Earlier Topics Revisited with Sequences
Section 4.3 Infinite Series of Constants
Section 4.4 Sequences and Series of Functions
Section 4.5 Power Series
Chapter 5 Real-Valued Functions of Several Varables
Section 5.1 Structure of R-.4n
Section 5.2 Continuous Real-Valued Functions of n Variables
Section 5.3 Partial Derivatives and the Differential
Section 5.4 The Chain Rule and Taylor's Theorem
Chapter 6 Vector-Valued Functions of Several Variables
Section 6.1 Linear Trensformations and Matrices
Section 6.2 Continuity and Differentiability of Transformationds
Section 6.3 The Inverse Function Theroem
Section 6.4 The Implicit Function Theorem
Chapter 7 Integrals of Functions of Several Variables
Section 7.1 Definition and Existence of the Multiple Integral
Section 7.2 Iterated Integrals and Mutiple Integrals
Section 7.3 Change of Variables in Multiple Integrals
Chapter 8 Metric Spaces
Section 8.1 Introduction to Metric Spaces
Section 8.2 Compact Sets in a Metric Space
Section 8.3 Continuous Functions on Metric Spaces
Answers to Selected
Index
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