目錄
Preface
1 Preliminaries
1.1 The Logic cf Quantifiers
1.1.1 Rules of Quantifiers
1.1.2 Examples
1.1.3 Exercises
1.2 Infinite Sets
1.2.1 Countable Sets
1.2.2 Uncountable Sets
1.2.3 Exexcises
1.3 Proofs
1.3.1 How to Discover Proofs
1.3.2 How to Understand Proofs
1.4 The Rational Number System
1.5 The Axiom of Choice*
2 Construction of the Real Number System
2.1 Cauchy Sequences
2.1.1 Motivation
2.1.2 The Definition
2.1.3 Exercises
2.2 The Reals as an Ordered Field
2.2.1 Defining Arithmetic
2.2.2 The Field Axioms
2.2.3 Order
2.2.4 Exercises
2.3 Limits and Completeness
2.3.1 Proof of Completeness
2.3.2 Square Roots
2.3.3 Exercises
2.4 Other Versions and Visions
2.4.1 Infinite Decimal Expansions
2.4.2 Dedekind Cuts*
2.4.3 Non-Standard Analysis*
2.4.4 Constructive Analysis*
2.4.5 Exercises
2.5 Summary
3 Topology of the Real Line
3.1 The Theory of Limits
3.1.1 Limits, Sups, and Infs
3.1.2 Limit Points
3.1.3 Exercises
3.2 Open Sets and Closed Sets
3.2.1 Open Sets
3.2.2 Closed Sets
3.2.3 Exercises
3.3 Compact Sets
3.3.1 Exercises
3.4 Summary
4 Continuous Functions
4.1 Concepts of Continuity
4.1.1 Definitions
4.1.2 Limits of Functions and Limits of Sequences
4.1.3 Inverse Images of Open Sets
4.1.4 Related Definitions
4.1.5 Exercises
4.2 Properties of Continuous Functions
4.2.1 Basic Properties
4.2.2 Continuous Functions on Compact Domains
4.2.3 Monotone Functions
4.2.4 Exercises
4.3 Summary
5 Differential Calculus
5.1 Concepts of the Derivative
5.1.1 Equivalent Definitions
5.1.2 Continuity and Continuous Differentiability
5.1.3 Exercises
5.2 Properties of the Derivative
5.2.1 Local Properties
5,2.2 Intermediate Value and Mean Value Theorems
5.2.3 Global Properties
5.2.4 Exercises
5.3 The Calculus of Derivatives
5.3 ,1 Product and Quotient Rules
5.3.2 The Chain Rule
5.3.3 Inverse Function Theorem
5.3.4 Exercises
5.4 Higher Derivatives and Taylor's Theorem
5.4.1 Interpretations of the Second Derivative
5.4.2 Taylor's Theorem
5.4.3 L'HSpital's Rule*
5.4.4 Lagrange Remainder Formula*
5.4.5 Orders of Zeros*
5.4.6 Exercises
5,5 Summary
6 Integral Calculus
6.1 Integrals of Continuous Functions
6.1.1 Existence of the Integral
6.1.2 Fundamental Theorems of Calculus
6.1.3 Useful Integration Formulas
6.1.4 Numerical Integration
6.1.5 Exercises
6.2 The Riemann Integral
6.2.1 Definition of the Integral
6.2.2 Elementary Properties of the Integral
6.2.3 Functions with a Countable Number of Discontinuities*
6.2.4 Exercises
6.3 Improper Integrals*
6.3.1 Definitions and Examples
6.3.2 Exercises
6.4 Summary
7 Sequences and Series of Functions
7.1 Complex Numbers
7.1.1 Basic Properties of C
7.1.2 Complex-Valued Functions
7.1.3 Exercises
7.2 Numerical Series and Sequences
7.2.1 Convergence and Absolute Convergence
7.2.2 Rearrangements
7.2.3 Summation by Parts*
7.2.4 Exercises
7.3 Uniform Convergence
7.3.1 Uniform Limits and Continuity
7.3.2 Integration and Differentiation of Limits
7.3.3 Unrestricted Convergence*
7.3.4 Exercises
7.4 Power Series
7.4.1 The Radius of Convergence
7.4.2 Analytic Continuation
7.4.3 Analytic Functions on Complex Domains*
7.4.4 Closure Properties of Analytic Functions*
7.4.5 Exercises
7.5 Approximation by Polynomials
7.5.1 Lagrange Interpolation
7.5.2 Convolutions and Approximate Identities
7.5.3 The Weierstrass Approximation Theorem
7.5.4 Approximating Derivatives
7.5.5 Exercises
7.6 Equicontinuity
7.6.1 The Definition of Equicontinuity
7.6.2 The Arzela-Ascoli Theorem
7.6.3 Exercises
7.7 Summary
8 Transcendental Functions
8.1 The Exponential and Logarithm
8.1.1 Five Equivalent Definitions
8.1.2 Exponential Glue and Blip Functions
8.1.3 Functions with Prescribed Taylor Expansions*
8.1.4 Exercises
8.2 Trigonometric Functions
8.2.1 Definition of Sine and Cosine
8.2.2 Relationship Between Sines, Cosines, and Com-plex Exponentials
8.2.3 Exercises
8.3 Summary
9 Euclidean Space and Metric Spaces
9.1 Structures on Euclidean Space
9.1.1 Vector Space and Metric Space
9.1.2 Norm and Inner Product
9.1.3 The Complex Case
9.1.4 Exercises
9.2 Topology of Metric Spaces
9.2.1 Open Sets
9.2.2 Limits and Closed Sets
9.2.3 Completeness
9.2.4 Compactness
9.2.5 Exercises
9.3 Continuous Functions on Metric Spaces
9.3.1 Three Equivalent Definitions
9.3.2 Continuous Functions on Compact Domains
9.3.3 Connectedness
9.3.4 The Contractive Mapping Principle
9.3.5 The Stone-Weierstrass Theorem*
9.3.6 Nowhere Differentiable Functions, and Worse*
9.3.7 Exercises
9.4 Summary
10 Differential Calculus in Euclidean Space
10.1 The Differential
10.1.1 Definition of Differentiability
10.1.2 Partial Derivatives
10.1.3 The Chain Rule
10.1.4 Differentiation cf Integrals
10.1.5 Exercises
10.2 Higher Derivatives
10.2.1 Equality cf Mixed Partials
10.2.2 Local Extrema
10.2.3 Taylor Expansions
10.2.4 Exercises
10.3 Summary
11 Ordinary Differential Equations
11.1 Existence and Uniqueness
11.1.1 Motivation
11.1.2 Picard Iteration
11.1.3 Linear Equations
11.1.4 Local Existence and Uniqueness*
11.1 ,5 Higher Order Equations*
11.1.6 Exercises
11.2 Other Methods of Solution*
11.2.1 Difference Equation Approximation
11.2.2 Peano Existence Theorem
11.2.3 Power-Series Solutiovs
11.2 ,4 Exercises
11.3 Vector Fields and Flows*
11.3.1 Integral Curves
11.3.2 Hamiltonian Mechanics
11.3.3 First-Order Linear P.D.E.'s
11.3.4 Exercises
11.4 Summary
12 Fourier Series
12.1 Origins of Fourier Series
12.1.1 Fourier Series Solutions of P.D.E.'s
12.1.2 Spectral Theory
12.1.3 Harmonic Analysis
12.1.4 ExeIcises
12.2 Convergence of Fourier Series
12.2.1 Uniform Convergence for Ci Functions
12.2.2 Summability of Fourier Series
12.2.3 Convergence in the Mean
12.2.4 Divergence and Gibb's Phenomenon*
12.2.5 Solution of the Heat Equation*
12.2.6 Exercises
12.3 Summary
13 Implicit Functions, Curves, and Surfaces
13.1 The Implicit Function Theorem
13.1.1 Statement of the Theorem
13.1.2 The Proof*
13.1.3 Exercises
13.2 Curves and Surfaces
13.2.1 Motivation and Examples
13.2.2 Immersions and Embeddings
13.2.3 Parametric Description of Surfaces
13.2.4 Implicit Description of Surfaces
13.2.5 Exercises
13.3 Maxima and Minima on Surfaces
13.3.1 Lagrange Multipliers
13.3.2 A Second Derivative Test*
13.3.3 Exercises
13.4 Arc Length
13.4.1 Rectifiable Curves
13.4.2 The Integral Formula for Arc Length
13.4.3 Arc Length Parameterization*
13.4.4 Exercises
13.5 Summary
14 The Lebesgue Integral
14.1 The Concept of Measure
14.1.1 Motivation
14.1.2 Properties of Length
14.1.3 Measurable Sets
14.1.4 Basic Properties of Measures
14.1.5 A Formula for Lebesgue Measure
14.1.6 Other Examples of Measures
14.1.7 Exercises
14.2 Proof of Existence of Measures*
14.2.1 Outer Measures
14.2.2 Metric Outer Measure
14.2.3 Hausdorff Measures*
14.2.4 Exercises
14.3 The Integral
14.3.1 Non-negative Measurable Functions
14.3.2 The Monotone Convergence Theorem
14.3.3 Integrable Functions
14.3.4 Almost Everywhere
14.3.5 Exercises
14.4 The Lebesgue Spaces L1 and L2
14.4.1 L1 as a Banach Space
14.4.2 L2 as a Hilbert Space
14.4.3 Fourier Series for L2 Functions
14.4.4 Exercises
14.5 Summary
15 Multiple Integrals
15.1 Interchange of Integrals
15.1.1 Integrals of Continuous Functions
15.1.2 Fubini's Theorem
15.1.3 The Monotone Class Lemma*
15.1.4 Exercises
15.2 Change of Variable in Multiple Integrals
15.2.1 Determinants and Volume
15.2.2 The Jacobian Factor*
15.2.3 Polar Coordinates
15.2.4 Change of Variable for Lebesgue Integrals*
15.2.5 Exercises
15.3 Summary
Index
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