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By Jeffery Cooper, University of Maryland, U.S.A.
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.
Audience
Engineers and scientists who wish to see how careful mathematical reasoning can be used to solve applied problems; upper division students in Advanced Calculus
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
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