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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.
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