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Working Analysis
1st Edition - September 21, 2004
Author: Jeffery Cooper
Language: English
Hardback ISBN:9780121876043
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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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.
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