Working Analysis

Working Analysis

1st Edition - September 21, 2004

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  • Author: Jeffery Cooper
  • Hardcover ISBN: 9780121876043

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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


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



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.

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