Elementary classical analysis / Jerrold E. Marsden, Michael J. Hoffman

Includes bibliographical references and index

Contents Preface Preface to the First Edition Introduction: Sets and Functions 1 The Real Line and Euclidean Space 1.1 Ordered Fields and the Number System 1.2 Completeness and the Real Number System 1.3 Least Upper Bounds 1.4 Cauchy Sequences 1.5 Cluster Points; lim inf and lim sup 1.6 Euclidean Space 1.7 Norms, Inner Products, and Metrics 1.8 The Complex Numbers 2 The Topology of Euclidean Space 2.1 Open Sets 2.2 Interior of a Set 2.3 Closed Sets 2.4 Accumulation Points 2.5 Closure of a Set 2.6 Boundary of a Set 2.7 Sequences 2.8 Completeness 2.9 Series of Real Numbers and Vectors 3 Compact and Connected Sets 3.1 Compactness 3.2 The Heine-Borel Theorem 3.3 Nested Set Property 3.4 Path-Connected Sets 3.5 Connected Sets 4 Continuous Mappings 4.1 Continuity 4.2 Images of Compact and Connected Sets 4.3 Operations on Continuous Mappings 4.4 The Boundedness of Continuous Functions on Compact Sets 4.5 The Intermediate Value Theorem 4.6 Uniform Continuity 4.7 Differentiation of Functions of One Variable 4.8 Integration of Functions of One Variable 5 Uniform Convergence 5.1 Pointwise and Uniform Convergence 5.2 The Weierstrass M Test 5.3 Integration and Differentiation of Series 5.4 The Elementary Functions 5.5 The Space of Continuous Functions 5.6 The Arzela-Ascoli Theorem 5.7 The Contraction Mapping Principle and Its Applications 5.8 The Stone-Weierstrass Theorem 5.9 The Dirichlet and Abel Tests 5.10 Power Series and Cesaro and Abel Summability 6 Differentiable Mappings 6.1 Definition of the Derivative 6.2 Matrix Representation 6.3 Continuity of Differentiable Mappings; Differentiable Paths 6.4 Conditions for Differentiability 6.5 The Chain Rule 6.6 Product Rule and Gradients 6.7 The Mean Value Theorem 6.8 Taylor's Theorem and Higher Derivatives 6.9 Maxima and Minima 7 The Inverse and Implicit Function Theorems and Related Topics 7.1 Inverse Function Theorem 7.2 Implicit Function Theorem 7.3 The Domain-Straightening Theorem 7.4 Further Consequences of the Implicit Function Theorem 7.5 An Existence Theorem for Ordinary Differential Equations 7.6 The Morse Lemma 7.7 Constrained Extrema and Lagrange Multipliers 8 Integration 8.1 Integrable Functions 8.2 Volume and Sets of Measure Zero 8.3 Lebesgue's Theorem 8.4 Properties of the Integral 8.5 Improper Integrals 8.6 Some Convergence Theorems 8.7 Introduction to Distributions 9 Fubini's Theorem and the Change of Variables Formula 9.2 Fubini's Theorem 9.3 Change of Variables Theorem 9.4 Polar Coordinates 9.5 Spherical and Cylindrical Coordinates 9.6 A Note on the Lebesgue Integral 9.7 Interchange of Limiting Operations 10 Fourier Analysis 10.1 Inner Product Spaces 10.2 Orthogonal Families of Functions 10.3 Completeness and Convergence Theorems 10.4 Functions of Bounded Variation and Fejer Theory (Optional) 10.5 Computation of Fourier Series 10.6 Further Convergence Theorems 10.7 Applications 10.8 Fourier Integrals 10.9 Quantum Mechanical Formalism Appendix A: Miscellaneous Exercises Appendix B: References and Suggestions for Further Study Appendix C: Answers and Suggestions for Selected Odd-Numbered Exercises Index