# Working Analysis

## 1st Edition

**Authors:**Jeffery Cooper

**Hardcover ISBN:**9780121876043

**eBook ISBN:**9780080575254

**Imprint:**Academic Press

**Published Date:**21st September 2004

**Page Count:**688

## Description

The text is for a two semester course in advanced calculus. It develops the basic ideas of calculus rigorously but with an eye to showing how mathematics connects with other areas of science and engineering. In particular, effective numerical computation is developed as an important aspect of mathematical analysis.

## Key Features

- Maintains a rigorous presentation of the main ideas of advanced calculus, interspersed with applications that show how to analyze real problems
- Includes a wide range of examples and exercises drawn from mechanics, biology, chemical engineering and economics
- Describes links to numerical analysis and provides opportunities for computation; some MATLAB codes are available on the author's webpage
- Enhanced by an informal and lively writing style

## Readership

Engineers and scientists who wish to see how careful mathematical reasoning can be used to solve applied problems; upper division students in Advanced Calculus

## Table of Contents

Preface Part I:

Foundations 1.1 Ordered Fields 1.2 Completeness 1.3 Using Inequalities 1.4 Induction 1.5 Sets and Functions

Sequences of Real Numbers 2.1 Limits of Sequences 2.2 Criteria for Convergence 2.3 Cauchy Sequences

Continuity 3.1 Limits of Functions 3.2 Continuous Functions 3.3 Further Properties of Continuous Functions 3.4 Golden-Section Search 3.5 The Intermediate Value Theorem

The Derivative 4.1 The Derivative and Approximation 4.2 The Mean Value Theorem 4.3 The Cauchy Mean Value Theorem and l’Hopital’s Rule 4.4 The Second Derivative Test

Higher Derivatives and Polynomial Approximation 5.1 Taylor Polynomials 5.2 Numerical Differentiation 5.3 Polynomial Inerpolation 5.4 Convex Funtions

Solving Equations in One Dimension 6.1 Fixed Point Problems 6.2 Computation with Functional Iteration 6.3 Newton’s Method

Integration 7.1 The Definition of the Integral 7.2 Properties of the Integral 7.3 The Fundamental Theorem of Calculus and Further Properties of the Integral 7.4 Numerical Methods of Integration 7.5 Improper Integrals

Series 8.1 Infinite Series 8.2 Sequences and Series of Functions 8.3 Power Series and Analytic Functions

Appendix I I.1 The Logarithm Functions and Exponential Functions I.2 The Trigonometric Funtions

Part II:

Convergence and Continuity in Rn 9.1 Norms 9.2 A Little Topology 9.3 Continuous Functions of Several Variables

The Derivative in Rn 10.1 The Derivative and Approximation in Rn 10.2 Linear Transformations and Matrix Norms 10.3 Vector-Values Mappings

Solving Systems of Equations 11.1 Linear Systems 11.2 The Contraction Mapping Theorem 11.3 Newton’s Method 11.4 The Inverse Function Theorem 11.5 The Implicit Function Theorem 11.6 An Application in Mechanics

Quadratic Approximation and Optimization 12.1 Higher Derivatives and Quadratic Approximation 12.2 Convex Functions 12.3 Potentials and Dynamical Systems 12.4 The Method of Steepest Descent 12.5 Conjugate Gradient Methods 12.6 Some Optimization Problems

Constrained Optimization 13.1 Lagrange Multipliers 13.2 Dependence on Parameters and Second-order Conditions 13.3 Constrained Optimization with Inequalities 13.4 Applications in Economics

Integration in Rn 14.1 Integration Over Generalized Rectangles 14.2 Integration Over Jordan Domains 14.3 Numerical Methods 14.4 Change of Variable in Multiple Integrals 14.5 Applications of the Change of Variable Theorem 14.6 Improper Integrals in Several Variables 14.7 Applications in Probability

Applications of Integration to Differential Equations 15.1 Interchanging Limits and Integrals 15.2 Approximation by Smooth Functions 15.3 Diffusion 15.4 Fluid Flow

Appendix II A Matrix Factorization

Solutions to Selected Exercises

References

Index

## Details

- No. of pages:
- 688

- Language:
- English

- Copyright:
- © Academic Press 2005

- Published:
- 21st September 2004

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080575254

- Hardcover ISBN:
- 9780121876043

## About the Author

### Jeffery Cooper

### Affiliations and Expertise

University of Maryland, U.S.A.

## Reviews

"This is a solid, well-written advanced calculus book that deserves to be on the shelves of mathematics department offices when faculty are selecting course resources." -J. Feroe, Vassar College, in CHOICE, JUNE 2005 "...this textbook is based on a very healthy philosophy that it is easier to learn mathematical analysis when it is intertwined with meaningful applications. The book is fun to read and, I am sure, will be fun to learn from." -Victor Roytburd, Rensselaer Polytechnic Institute, in SIAM REVIEW “In my opinion the book by Cooper is a viable competitor to Strichartz...To summarize, this textbook is based on a very healthy philosophy that it is easier to learn mathematical analysis when it is intertwined with meaningful applications. The book is fun to read and, I am sure, will be fun to learn from.” – Victor Roytburd, Rensselaer Polytechnic Institute