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Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
Focuses primarily on analysis in Euclidean space with a view toward applications
Written to appeal to students in engineering and the physical sciences as well as pure mathematics
More material on variable calculus
Expanded treatment of metric spaces
Detailed coverage of the foundations of the real number system
1. Introduction: Sets and Functions
Supplement on the Axioms of Set Theory
2. The Real Line and Euclidean Space
Ordered Fields and the Number Systems
Completeness and the Real Number System
Least Upper Bounds
Cauchy Sequences
Cluster Points: lim inf and lim sup
Euclidean Space
Norms, Inner Products, and Metrics
The Complex Numbers
3. Topology of Euclidean Space
Open Sets
Interior of a Set
Closed Sets
Accumulation Points
Closure of a Set
Boundary of a Set
Sequences
Completeness
Series of Real Numbers and Vectors
4. Compact and Connected Sets
Compacted-ness
The Heine-Borel Theorem
Nested Set Property
Path-Connected Sets
Connected Sets
5. Continuous Mappings
Continuity
Images of Compact and Connected Sets
Operations on Continuous Mappings
The Boundedness of Continuous Functions of Compact Sets
The Intermediate Value Theorem
Uniform Continuity
Differentiation of Functions of One Variable
Integration of Functions of One Variable
6. Uniform Convergence
Pointwise and Uniform Convergence
The Weierstrass M Test
Integration and Differentiation of Series
The Elementary Functions
The Space of Continuous Functions
The Arzela-Ascoli Theorem
The Contraction Mapping Principle and Its Applications
The Stone-Weierstrass Theorem
The Dirichlet and Abel Tests
Power Series and Cesaro and Abel Summability
7. Differentiable Mappings
Definition of the Derivative
Matrix Representation
Continuity of Differentiable Mappings; Differentiable Paths
Conditions for Differentiability
The Chain Rule
Product Rule and Gradients
The Mean Value Theorem
Taylor's Theorem and Higher Derivatives
Maxima and Minima
8. The Inverse and Implicit Function Theorems and Related Topics
Inverse Function Theorem
Implicit Function Theorem
The Domain-Straightening Theorem
Further Consequences of the
Implicit Function Theorem
An Existence Theorem for Ordinary Differential Equations
The Morse Lemma
Constrained Extrema and Lagrange Multipliers
9. Integration
Integrable Functions
Volume and Sets of Measure Zero
Lebesgue's Theorem
Properties of the Integral
Improper Integrals
Some Convergence Theorems
Introduction to Distributions
10. Fubini's Theorem and the Change of Variables Formula
Introduction
Fubini's Theorem
Change of Variables Theorem
Polar Coordinates
Spherical Coordinates and Cylindrical Coordinates
A Note on the Lebesgue Integral
Interchange of Limiting Operations
11. Fourier Analysis
Inner Product Spaces
Orthogonal Families of Functions
Completeness and Convergence Theorems
Functions of Bounded Variation and Fejér Theory (Optional)
Computation of Fourier Series
Further Convergence Theorems
Applications
Fourier Integrals
Quantum Mechanical Formalism
Miscellaneous Exercises
References
Answers to Selected Odd-Numbered Exercises
Index
Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
Focuses primarily on analysis in Euclidean space with a view toward applications
Written to appeal to students in engineering and the physical sciences as well as pure mathematics
More material on variable calculus
Expanded treatment of metric spaces
Detailed coverage of the foundations of the real number system
Table of Contents
1. Introduction: Sets and Functions
Supplement on the Axioms of Set Theory
2. The Real Line and Euclidean Space
Ordered Fields and the Number Systems
Completeness and the Real Number System
Least Upper Bounds
Cauchy Sequences
Cluster Points: lim inf and lim sup
Euclidean Space
Norms, Inner Products, and Metrics
The Complex Numbers
3. Topology of Euclidean Space
Open Sets
Interior of a Set
Closed Sets
Accumulation Points
Closure of a Set
Boundary of a Set
Sequences
Completeness
Series of Real Numbers and Vectors
4. Compact and Connected Sets
Compacted-ness
The Heine-Borel Theorem
Nested Set Property
Path-Connected Sets
Connected Sets
5. Continuous Mappings
Continuity
Images of Compact and Connected Sets
Operations on Continuous Mappings
The Boundedness of Continuous Functions of Compact Sets
The Intermediate Value Theorem
Uniform Continuity
Differentiation of Functions of One Variable
Integration of Functions of One Variable
6. Uniform Convergence
Pointwise and Uniform Convergence
The Weierstrass M Test
Integration and Differentiation of Series
The Elementary Functions
The Space of Continuous Functions
The Arzela-Ascoli Theorem
The Contraction Mapping Principle and Its Applications
The Stone-Weierstrass Theorem
The Dirichlet and Abel Tests
Power Series and Cesaro and Abel Summability
7. Differentiable Mappings
Definition of the Derivative
Matrix Representation
Continuity of Differentiable Mappings; Differentiable Paths
Conditions for Differentiability
The Chain Rule
Product Rule and Gradients
The Mean Value Theorem
Taylor's Theorem and Higher Derivatives
Maxima and Minima
8. The Inverse and Implicit Function Theorems and Related Topics
Inverse Function Theorem
Implicit Function Theorem
The Domain-Straightening Theorem
Further Consequences of the
Implicit Function Theorem
An Existence Theorem for Ordinary Differential Equations
The Morse Lemma
Constrained Extrema and Lagrange Multipliers
9. Integration
Integrable Functions
Volume and Sets of Measure Zero
Lebesgue's Theorem
Properties of the Integral
Improper Integrals
Some Convergence Theorems
Introduction to Distributions
10. Fubini's Theorem and the Change of Variables Formula
Introduction
Fubini's Theorem
Change of Variables Theorem
Polar Coordinates
Spherical Coordinates and Cylindrical Coordinates
A Note on the Lebesgue Integral
Interchange of Limiting Operations
11. Fourier Analysis
Inner Product Spaces
Orthogonal Families of Functions
Completeness and Convergence Theorems
Functions of Bounded Variation and Fejér Theory (Optional)
Computation of Fourier Series
Further Convergence Theorems
Applications
Fourier Integrals
Quantum Mechanical Formalism
Miscellaneous Exercises
References
Answers to Selected Odd-Numbered Exercises
Index