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Request a sales quote*Working Analysis* 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.### Jeffery Cooper

- 1st Edition - September 21, 2004
- Author: Jeffery Cooper
- Language: English
- Hardback ISBN:9 7 8 - 0 - 1 2 - 1 8 7 6 0 4 - 3

Working Analysis 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 o… Read more

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Immediately download your ebook while waiting for your print delivery. No promo code is needed.

- 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

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

Preface

Part I:

1. Foundations

1.1 Ordered Fields

1.2 Completeness

1.3 Using Inequalities

1.4 Induction

1.5 Sets and Functions

2. Sequences of Real Numbers

2.1 Limits of Sequences

2.2 Criteria for Convergence

2.3 Cauchy Sequences

3. 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

4. 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

5. Higher Derivatives and Polynomial Approximation

5.1 Taylor Polynomials

5.2 Numerical Differentiation

5.3 Polynomial Inerpolation

5.4 Convex Funtions

6. Solving Equations in One Dimension

6.1 Fixed Point Problems

6.2 Computation with Functional Iteration

6.3 Newton’s Method

7. 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

8. 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:

9. Convergence and Continuity in Rn

9.1 Norms

9.2 A Little Topology

9.3 Continuous Functions of Several Variables

10. The Derivative in Rn

10.1 The Derivative and Approximation in Rn

10.2 Linear Transformations and Matrix Norms

10.3 Vector-Values Mappings

11. 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

12. 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

13. 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

14. 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

15. 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

Part I:

1. Foundations

1.1 Ordered Fields

1.2 Completeness

1.3 Using Inequalities

1.4 Induction

1.5 Sets and Functions

2. Sequences of Real Numbers

2.1 Limits of Sequences

2.2 Criteria for Convergence

2.3 Cauchy Sequences

3. 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

4. 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

5. Higher Derivatives and Polynomial Approximation

5.1 Taylor Polynomials

5.2 Numerical Differentiation

5.3 Polynomial Inerpolation

5.4 Convex Funtions

6. Solving Equations in One Dimension

6.1 Fixed Point Problems

6.2 Computation with Functional Iteration

6.3 Newton’s Method

7. 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

8. 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:

9. Convergence and Continuity in Rn

9.1 Norms

9.2 A Little Topology

9.3 Continuous Functions of Several Variables

10. The Derivative in Rn

10.1 The Derivative and Approximation in Rn

10.2 Linear Transformations and Matrix Norms

10.3 Vector-Values Mappings

11. 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

12. 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

13. 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

14. 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

15. 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

- No. of pages: 688
- Language: English
- Edition: 1
- Published: September 21, 2004
- Imprint: Academic Press
- Hardback ISBN: 9780121876043

JC

Affiliations and expertise

University of Maryland, U.S.A.