Working Analysis (Hardcover)

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.

 

Table of Contents:

Preface

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