- Jeffery Cooper
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.
Engineers and scientists who wish to see how careful mathematical reasoning can be used to solve applied problems; upper division students in Advanced Calculus
Hardbound, 688 Pages
Published: September 2004
Imprint: Academic Press
"This is a solid, well-written advanced calculus book that deserves to be on the shelves of mathematics department offices when faculty are selecting course resources." -J. Feroe, Vassar College, in CHOICE, JUNE 2005 "...this textbook is based on a very healthy philosophy that it is easier to learn mathematical analysis when it is intertwined with meaningful applications. The book is fun to read and, I am sure, will be fun to learn from." -Victor Roytburd, Rensselaer Polytechnic Institute, in SIAM REVIEW âIn my opinion the book by Cooper is a viable competitor to Strichartz...To summarize, this textbook is based on a very healthy philosophy that it is easier to learn mathematical analysis when it is intertwined with meaningful applications. The book is fun to read and, I am sure, will be fun to learn from.â â Victor Roytburd, Rensselaer Polytechnic Institute
- PrefacePart I:1. Foundations1.1 Ordered Fields1.2 Completeness1.3 Using Inequalities1.4 Induction1.5 Sets and Functions2. Sequences of Real Numbers2.1 Limits of Sequences2.2 Criteria for Convergence2.3 Cauchy Sequences3. Continuity3.1 Limits of Functions3.2 Continuous Functions3.3 Further Properties of Continuous Functions3.4 Golden-Section Search3.5 The Intermediate Value Theorem4. The Derivative4.1 The Derivative and Approximation4.2 The Mean Value Theorem4.3 The Cauchy Mean Value Theorem and lâHopitalâs Rule4.4 The Second Derivative Test5. Higher Derivatives and Polynomial Approximation5.1 Taylor Polynomials5.2 Numerical Differentiation5.3 Polynomial Inerpolation5.4 Convex Funtions6. Solving Equations in One Dimension6.1 Fixed Point Problems6.2 Computation with Functional Iteration6.3 Newtonâs Method7. 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 Integrals8. Series 8.1 Infinite Series 8.2 Sequences and Series of Functions 8.3 Power Series and Analytic FunctionsAppendix I I.1 The Logarithm Functions and Exponential Functions I.2 The Trigonometric FuntionsPart II:9. Convergence and Continuity in Rn 9.1 Norms 9.2 A Little Topology 9.3 Continuous Functions of Several Variables10. The Derivative in Rn 10.1 The Derivative and Approximation in Rn 10.2 Linear Transformations and Matrix Norms 10.3 Vector-Values Mappings11. Solving Systems of Equations 11.1 Linear Systems 11.2 The Contraction Mapping Theorem 11.3 Newtonâs Method 11.4 The Inverse Function Theorem 11.5 The Implicit Function Theorem 11.6 An Application in Mechanics12. 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 Problems13. 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 Economics14. 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 Probability15. Applications of Integration to Differential Equations 15.1 Interchanging Limits and Integrals 15.2 Approximation by Smooth Functions 15.3 Diffusion 15.4 Fluid FlowAppendix II A Matrix FactorizationSolutions to Selected ExercisesReferencesIndex