Working Analysis
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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.
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:
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
Product details
- No. of pages: 688
- Language: English
- Copyright: © Academic Press 2004
- Published: September 21, 2004
- Imprint: Academic Press
- Hardcover ISBN: 9780121876043
About the Author
Jeffery Cooper
Affiliations and Expertise
University of Maryland, U.S.A.