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Elementary Classical Analysis

by: Marsden, Jerrold E.; Hoffman, Michael J.

Elementary Classical Analysis

by: Marsden, Jerrold E.; Hoffman, Michael J.

ISBN 13:
9780716728863
ISBN 10:
0716728869
Edition: 3rd
Format: Paperback
Copyright: 01/01/2024
Publisher: W. H. Freeman

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Summary

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.

Preface
Preface to the First Edition
Introduction: Sets and Functions	p. 1
The Real Line and Euclidean Space	p. 25
Ordered Fields and the Number System
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
The Topology of Euclidean Space	p. 103
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
Compact and Connected Sets	p. 151
Compactness
The Heine-Borel Theorem
Nested Set Property
Path-Connected Sets
Connected Sets
Continuous Mappings	p. 177
Continuity
Images of Compact and Connected Sets
Operations on Continuous Mappings
The Boundedness of Continuous Functions on Compact Sets
The Intermediate Value Theorem
Uniform Continuity
Differentiation of Functions of One Variable
Integration of Functions of One Variable
Uniform Convergence	p. 237
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
Differentiable Mappings	p. 327
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
The Inverse and Implicit Function Theorems and Related Topics	p. 391
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
Integration	p. 445
Integrable Functions
Volume and Sets of Measure Zero
Lebesgue's Theorem
Properties of the Integral
Improper Integrals
Some Convergence Theorems
Introduction to Distributions
Fubini's Theorem and the Change of Variables Formula	p. 497
Fubini's Theorem
Change of Variables Theorem
Polar Coordinates
Spherical and Cylindrical Coordinates
A Note on the Lebesgue Integral
Interchange of Limiting Operations
Fourier Analysis	p. 543
Inner Product Spaces
Orthogonal Families of Functions
Completeness and Convergence Theorems
Functions of Bounded Variation and Fejer Theory (Optional)
Computation of Fourier Series
Further Convergence Theorems
Applications
Fourier Integrals
Quantum Mechanical Formalism
Appendix A: Miscellaneous Exercises	p. 663
Appendix B: References and Suggestions for Further Study	p. 677
Appendix C: Answers and Suggestions for Selected Odd-Numbered Exercises	p. 683
Index	p. 729
Table of Contents provided by Blackwell. All Rights Reserved.

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Elementary Classical Analysis

Elementary Classical Analysis

9780716728863

0716728869

Summary

Table of Contents

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