Elementary Classical Analysis

Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics. Designed for courses in advanced calculus and introductory real analysis • Focuses primarily on analysis in Euclidean space with a view toward applications • Written to appeal to students in engineering and the physical sciences as well as pure mathematics • More material on variable calculus • Expanded treatment of metric spaces • Detailed coverage of the foundations of the real number system Table of Contents 1. Introduction: Sets and Functions Supplement on the Axioms of Set Theory 2. The Real Line and Euclidean Space Ordered Fields and the Number Systems Completeness and the Real Number System Least Upper Bounds Cauchy Sequences Cluster Points: lim inf and lim sup Euclidean Space Norms, Inner Products, and Metrics The Complex Numbers 3. Topology of Euclidean Space Open Sets Interior of a Set Closed Sets Accumulation Points Closure of a Set Boundary of a Set Sequences Completeness Series of Real Numbers and Vectors 4. Compact and Connected Sets Compacted-ness The Heine-Borel Theorem Nested Set Property Path-Connected Sets Connected Sets 5. Continuous Mappings Continuity Images of Compact and Connected Sets Operations on Continuous Mappings The Boundedness of Continuous Functions of Compact Sets The Intermediate Value Theorem Uniform Continuity Differentiation of Functions of One Variable Integration of Functions of One Variable 6. Uniform Convergence Pointwise and Uniform Convergence The Weierstrass M Test Integration and Differentiation of Series The Elementary Functions The Space of Continuous Functions The Arzela-Ascoli Theorem The Contraction Mapping Principle and Its Applications The Stone-Weierstrass Theorem The Dirichlet and Abel Tests Power Series and Cesaro and Abel Summability 7. Differentiable Mappings Definition of the Derivative Matrix Representation Continuity of Differentiable Mappings; Differentiable Paths Conditions for Differentiability The Chain Rule Product Rule and Gradients The Mean Value Theorem Taylor's Theorem and Higher Derivatives Maxima and Minima 8. The Inverse and Implicit Function Theorems and Related Topics Inverse Function Theorem Implicit Function Theorem The Domain-Straightening Theorem Further Consequences of the Implicit Function Theorem An Existence Theorem for Ordinary Differential Equations The Morse Lemma Constrained Extrema and Lagrange Multipliers 9. Integration Integrable Functions Volume and Sets of Measure Zero Lebesgue's Theorem Properties of the Integral Improper Integrals Some Convergence Theorems Introduction to Distributions 10. Fubini's Theorem and the Change of Variables Formula Introduction Fubini's Theorem Change of Variables Theorem Polar Coordinates Spherical Coordinates and Cylindrical Coordinates A Note on the Lebesgue Integral Interchange of Limiting Operations 11. Fourier Analysis Inner Product Spaces Orthogonal Families of Functions Completeness and Convergence Theorems Functions of Bounded Variation and Fejér Theory (Optional) Computation of Fourier Series Further Convergence Theorems Applications Fourier Integrals Quantum Mechanical Formalism Appendices Notes on the Axioms of Set Theory Miscellaneous Problems Suggestions for Further Study Answers to Selected Problems Index