# Principles of Mathematical Analysis

3^{rd}Edition

ISBN10: 007054235X

ISBN13: 9780070542358

Copyright: 1976

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ISBN10: 007054235X | ISBN13: 9780070542358

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The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

### Program Details

### Chapter 1: The Real and Complex Number Systems

#### Introduction

#### Ordered Sets

#### Fields

#### The Real Field

#### The Extended Real Number System

#### The Complex Field

#### Euclidean Spaces

#### Appendix

#### Exercises

### Chapter 2: Basic Topology

#### Finite, Countable, and Uncountable Sets

#### Metric Spaces

#### Compact Sets

#### Perfect Sets

#### Connected Sets

#### Exercises

### Chapter 3: Numerical Sequences and Series

#### Convergent Sequences

#### Subsequences

#### Cauchy Sequences

#### Upper and Lower Limits

#### Some Special Sequences

#### Series

#### Series of Nonnegative Terms

#### The Number *e*

#### The Root and Ratio Tests

#### Power Series

#### Summation by Parts

#### Absolute Convergence

#### Addition and Multiplication of Series

#### Rearrangements

#### Exercises

### Chapter 4: Continuity

#### Limits of Functions

#### Continuous Functions

#### Continuity and Compactness

#### Continuity and Connectedness

#### Discontinuities

#### Monotonic Functions

#### Infinite Limits and Limits at Infinity

#### Exercises

### Chapter 5: Differentiation

#### The Derivative of a Real Function

#### Mean Value Theorems

#### The Continuity of Derivatives

#### L'Hospital's Rule

#### Derivatives of Higher-Order

#### Taylor's Theorem

#### Differentiation of Vector-valued Functions

#### Exercises

### Chapter 6: The Riemann-Stieltjes Integral

#### Definition and Existence of the Integral

#### Properties of the Integral

#### Integration and Differentiation

#### Integration of Vector-valued Functions

#### Rectifiable Curves

#### Exercises

### Chapter 7: Sequences and Series of Functions

#### Discussion of Main Problem

#### Uniform Convergence

#### Uniform Convergence and Continuity

#### Uniform Convergence and Integration

#### Uniform Convergence and Differentiation

#### Equicontinuous Families of Functions

#### The Stone-Weierstrass Theorem

#### Exercises

### Chapter 8: Some Special Functions

#### Power Series

#### The Exponential and Logarithmic Functions

#### The Trigonometric Functions

#### The Algebraic Completeness of the Complex Field

#### Fourier Series

#### The Gamma Function

#### Exercises

### Chapter 9: Functions of Several Variables

#### Linear Transformations

#### Differentiation

#### The Contraction Principle

#### The Inverse Function Theorem

#### The Implicit Function Theorem

#### The Rank Theorem

#### Determinants

#### Derivatives of Higher Order

#### Differentiation of Integrals

#### Exercises

### Chapter 10: Integration of Differential Forms

#### Integration

#### Primitive Mappings

#### Partitions of Unity

#### Change of Variables

#### Differential Forms

#### Simplexes and Chains

#### Stokes' Theorem

#### Closed Forms and Exact Forms

#### Vector Analysis

#### Exercises

### Chapter 11: The Lebesgue Theory

#### Set Functions

#### Construction of the Lebesgue Measure

#### Measure Spaces

#### Measurable Functions

#### Simple Functions

#### Integration

#### Comparison with the Riemann Integral

#### Integration of Complex Functions

#### Functions of Class *L*^{2}

#### Exercises

### Bibliography

### List of Special Symbols

### Index

### Chapter 1: The Real and Complex Number Systems

#### Introduction

#### Ordered Sets

#### Fields

#### The Real Field

#### The Extended Real Number System

#### The Complex Field

#### Euclidean Spaces

#### Appendix

#### Exercises

### Chapter 2: Basic Topology

#### Finite, Countable, and Uncountable Sets

#### Metric Spaces

#### Compact Sets

#### Perfect Sets

#### Connected Sets

#### Exercises

### Chapter 3: Numerical Sequences and Series

#### Convergent Sequences

#### Subsequences

#### Cauchy Sequences

#### Upper and Lower Limits

#### Some Special Sequences

#### Series

#### Series of Nonnegative Terms

#### The Number *e*

#### The Root and Ratio Tests

#### Power Series

#### Summation by Parts

#### Absolute Convergence

#### Addition and Multiplication of Series

#### Rearrangements

#### Exercises

### Chapter 4: Continuity

#### Limits of Functions

#### Continuous Functions

#### Continuity and Compactness

#### Continuity and Connectedness

#### Discontinuities

#### Monotonic Functions

#### Infinite Limits and Limits at Infinity

#### Exercises

### Chapter 5: Differentiation

#### The Derivative of a Real Function

#### Mean Value Theorems

#### The Continuity of Derivatives

#### L'Hospital's Rule

#### Derivatives of Higher-Order

#### Taylor's Theorem

#### Differentiation of Vector-valued Functions

#### Exercises

### Chapter 6: The Riemann-Stieltjes Integral

#### Definition and Existence of the Integral

#### Properties of the Integral

#### Integration and Differentiation

#### Integration of Vector-valued Functions

#### Rectifiable Curves

#### Exercises

### Chapter 7: Sequences and Series of Functions

#### Discussion of Main Problem

#### Uniform Convergence

#### Uniform Convergence and Continuity

#### Uniform Convergence and Integration

#### Uniform Convergence and Differentiation

#### Equicontinuous Families of Functions

#### The Stone-Weierstrass Theorem

#### Exercises

### Chapter 8: Some Special Functions

#### Power Series

#### The Exponential and Logarithmic Functions

#### The Trigonometric Functions

#### The Algebraic Completeness of the Complex Field

#### Fourier Series

#### The Gamma Function

#### Exercises

### Chapter 9: Functions of Several Variables

#### Linear Transformations

#### Differentiation

#### The Contraction Principle

#### The Inverse Function Theorem

#### The Implicit Function Theorem

#### The Rank Theorem

#### Determinants

#### Derivatives of Higher Order

#### Differentiation of Integrals

#### Exercises

### Chapter 10: Integration of Differential Forms

#### Integration

#### Primitive Mappings

#### Partitions of Unity

#### Change of Variables

#### Differential Forms

#### Simplexes and Chains

#### Stokes' Theorem

#### Closed Forms and Exact Forms

#### Vector Analysis

#### Exercises

### Chapter 11: The Lebesgue Theory

#### Set Functions

#### Construction of the Lebesgue Measure

#### Measure Spaces

#### Measurable Functions

#### Simple Functions

#### Integration

#### Comparison with the Riemann Integral

#### Integration of Complex Functions

#### Functions of Class *L*^{2}

#### Exercises

### Bibliography

### List of Special Symbols

### Index

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