Mathematical Analysis

Transcription

1 Mathematical Analysis A Concise Introduction Bernd S. W. Schroder Louisiana Tech University Program of Mathematics and Statistics Ruston, LA 31CENTENNIAL BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication

2 Contents Table of Contents Preface v xi Part I: Analysis of Functions of a Single Real Variable 1 The Real Numbers Field Axioms Order Axioms Lowest Upper and Greatest Lower Bounds Natural Numbers, Integers, and Rational Numbers Recursion, Induction, Summations, and Products 17 2 Sequences of Real Numbers Limits Limit Laws Cauchy Sequences Bounded Sequences Infinite Limits 44 3 Continuous Functions Limits of Functions Limit Laws One-Sided Limits and Infinite Limits Continuity Properties of Continuous Functions Limits at Infinity 69 4 Differentiable Functions Differentiability Differentiation Rules Rolle's Theorem and the Mean Value Theorem 80

3 vi Contents 5 The Riemann Integral I Riemann Sums and the Integral Uniform Continuity and Integrability of Continuous Functions The Fundamental Theorem of Calculus The Darboux Integral 97 6 Series of Real Numbers I Series as a Vehicle To Define Infinite Sums Absolute Convergence and Unconditional Convergence Some Set Theory The Algebra of Sets Countable Sets Uncountable Sets The Riemann Integral II Outer Lebesgue Measure Lebesgue's Criterion for Riemann Integrability More Integral Theorems Improper Riemann Integrals The Lebesgue Integral Lebesgue Measurable Sets Lebesgue Measurable Functions Lebesgue Integration Lebesgue Integrals versus Riemann Integrals Series of Real Numbers II Limits Superior and Inferior The Root Test and the Ratio Test Power Series Sequences of Functions Notions of Convergence Uniform Convergence Transcendental Functions The Exponential Function Sine and Cosine L'Hopital's Rule Numerical Methods Approximation with Taylor Polynomials Newton's Method Numerical Integration 214

4 Contents vii Part II: Analysis in Abstract Spaces 14 Integration on Measure Spaces Measure Spaces Outer Measures Measurable Functions Integration of Measurable Functions Monotone and Dominated Convergence Convergence in Mean, in Measure, and Almost Everywhere Product a-algebras Product Measures and Fubini's Theorem The Abstract Venues for Analysis Abstraction I: Vector Spaces Representation of Elements: Bases and Dimension Identification of Spaces: Isomorphism Abstraction II: Inner Product Spaces Nicer Representations: Orthonormal Sets Abstraction III: Normed Spaces Abstraction IV: Metric Spaces LP Spaces Another Number Field: Complex Numbers The Topology of Metric Spaces Convergence of Sequences Completeness Continuous Functions Open and Closed Sets Compactness The Normed Topology of R d Dense Subspaces Connectedness Locally Compact Spaces Differentiation in Normed Spaces Continuous Linear Functions Matrix Representation of Linear Functions Differentiability The Mean Value Theorem How Partial Derivatives Fit In Multilinear Functions (Tensors) Higher Derivatives The Implicit Function Theorem 380

5 viii Contents 18 Measure, Topology, and Differentiation Lebesgue Measurable Sets in R d C and Approximation of Integrable Functions Tensor Algebra and Determinants Multidimensional Substitution Introduction to Differential Geometry Manifolds Tangent Spaces and Differentiable Functions Differential Forms, Integrals Over the Unit Cube A>Forms and Integrals Over A>Chains Integration on Manifolds Stokes' Theorem Hilbert Spaces Orthonormal Bases Fourier Series The Riesz Representation Theorem 475 Part III: Applied Analysis 21 Physics Background Harmonic Oscillators Heat and Diffusion Separation of Variables, Fourier Series, and Ordinary Differential Equations Maxwell's Equations The Navier Stokes Equation for the Conservation of Mass Ordinary Differential Equations Banach Space Valued Differential Equations An Existence and Uniqueness Theorem Linear Differential Equations The Finite Element Method Ritz-Galerkin Approximation Weakly Differentiable Functions Sobolev Spaces Elliptic Differential Operators Finite Elements 536 Conclusion and Outlook 544

6 Contents ix Appendices A Logic 545 A.I Statements 545 A.2 Negations 546 B Set Theory 547 B.I The Zermelo-Fraenkel Axioms 547 B.2 Relations and Functions 548 C Natural Numbers, Integers, and Rational Numbers 549 C.I The Natural Numbers 549 C.2 The Integers 550 C.3 The Rational Numbers 550 Bibliography 551 Index 553