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The Way of Analysis, Revised Edition

Author(s): Robert S. Strichartz, Cornell University, Ithaca, New York
Details:
  • ISBN-13: 9780763714970
  • ISBN-10:0763714976
  • Paperback    739 pages      © 2000
Price: $287.95 US List
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The Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral.  The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries. Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings.

Features & Benefits

Flexible enough to be used in one- or two-term course with or without Lebesgue integral.
Strong motivation with theorems and examples are introduced in a context that makes them seem natural.
Extensive chapter summaries provide a outline for review purposes.
Three chapters on applications are included to show how the abstract ideas are used.

Applicable Courses

The Way of Analysis is intended for a one- or two-semester real analysis course, including or not including an introduction to Lebesgue integration, at the undergraduate or beginning graduate level. 

  • Introduction to Real Analysis  
  • Real Analysis I & II 
  • Principles of Analysis  
  • Applied Real Analysis
Preface:

1. Preliminaries
1.1 The Logic of Quantifiers
1.2 Infinite Sets
1.3 Proofs
1.4 The Rational Number System
1.5 The Axiom of Choice*

2. Construction of the Real Number System
2.1 Cauchy Sequences
2.2 The Reals as an Ordered Field
2.3 Limits and Completeness
2.4 Other Versions and Visions
2.5 Summary

3. Topology of the Real Line
3.1 The Theory of Limits
3.2 Open Sets and Closed Sets
3.3 Compact Sets
3.4 Summary

4. Continuous Functions
4.1 Concepts of Continuity
4.2 Properties of Continuous Functions
4.3 Summary

5. Differential Calculus
5.1 Concepts of the Derivative
5.2 Properties of the Derivative
5.3 The Calculus of Derivatives
5.4 Higher Derivatives and Taylor's Theorem
5.5 Summary

6. Integral Calculus
6.1 Integrals of Continuous Functions
6.2 The Riemann Integral
6.3 Improper Integrals*
6.4 Summary

7. Sequences and Series of Functions
7.1 Complex Numbers
7.2 Numerical Series and Sequences
7.3 Uniform Convergence
7.4 Power Series
7.5 Approximation by Polynomials
7.6 Equicontinuity
7.7Summary

8. Transcendental Functions
8.1 The Exponential and Logarithm
8.2 Trigonometric Functions
8.3 Summary

9. Euclidean Space and Metric Spaces
9.1 Structures on Euclidean Space
9.2 Topology of Metric Spaces
9.3 Continuous Functions on Metric Spaces
9.4 Summary

10. Differential Calculus in Euclidean Space
10.1 The Differential
10.2 Higher Derivatives
10.3 Summary

11. Ordinary Differential Equations
11.1 Existence and Uniqueness
11.2 Other Methods of Solution*
11.3 Vector Fields and Flows*
11.4 Summary

12. Fourier Series
12.1 Origins of Fourier Series
12.2 Convergence of Fourier Series
12.3 Summary

13. Implicit Functions, Curves, and Surfaces
13.1 The Implicit Function Theorem
13.2 Curves and Surfaces}
13.3 Maxima and Minima on Surfaces
13.4 Arc Length
13.5 Summary

14. The Lebesgue Integral
14.1 The Concept of Measure
14.2 Proof of Existence of Measures*
14.3 The Integral
14.4 The Lebesgue Spaces L1 and L2
14.5 Summary

15. Multiple Integral
15.1 Interchange of Integrals
15.2 Change of Variable in Multiple Integrals
15.3 Summary

Index

Robert S. Strichartz-Cornell University, Ithaca, New York

Robert S. Strichartz, Cornell University
Received his Ph.D. (1966) from Princeton University and is currently teaches mathematics at Cornell University. Research interests cover a wide range of topics in analysis, including harmonic analysis, partial differential equations, analysis on Lie groups and manifolds, integral geometry, wavelets and fractals. Robert's early work using methods of harmonic analysis to obtain fundamental estimates for linear wave equations has played an important role in recent developments in the theory of nonlinear wave equations.  His work on fractals began with the study of self-similar measures and their Fourier transforms. More recently his have been concentrating on a theory of differential equations on fractals created by Jun Kigami. Much of this work has been done in collaboration with undergraduate students through a summer Research Experiences for Undergraduates (REU) program at Cornell that he directs. Robert wrote an expository article  Analysis On Fractals, Notices of the AMS 46 (1999), 1199 - 1208  explaining the basic ideas in this subject area and the connections with other areas of mathematics.