内容簡介
數學主要講述思想的方法，深入理解數學比掌握一大堆的定理、定義、問題和技術顯得更為重要。理論和定義共同作用。
圖書目錄
Preface
1 Preliminaries
1.1 The Logic of 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 Exercises
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 Expansion
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 Discon-tinuities
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 Eouicontinuity
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
8.2 Trigonometric Functions
8.3 Summary
9 Euclidean Space and Metric Spaces
9.1 Structures on Euclidean Space
9.2 Topology of Metric Spaces
9.3 Continuous Functions on Metric Spaces
9.4 Summary
10 Differential Calculus in Euclidean Space
10.1 The Differential
10.2 Higher Derivatives
10.3 Summary
11 Ordinary Differential Equations
11.1 Existence and Uniqueness
11.2 Other Methods of Solution
11.3 Vector Fields and Flows
11.4 Summary
12 Fourier Series
12.1 Origins of Fourier Series
12.2 Convergence of Fourier Series
12.3 Summary
13 Implicit Functions, Curves, and Surfaces
13.1 The Implicit Function Theorem
13.2 Curves and Surfaces
13.3 Maxima and Minima on Surfaces
13.4 Arc Length
13.5 Summary
14 The Lebesgue Integral
14.1 The Concept of Measure
14.2 Proof of Existence of Measures
14.3 The Integral
14.4 The Lebesgue Spaces L1 and L2
14.5 Summary
15 Multiple Integrals
15.1 Interchange of Integrals
15.2 Change of Variable in Multiple Integrals
15.3 Summary
序言
Do not ask permission to understand.
Do not wait for the word of authority.
Seize reason in your own hand.
With your own teeth savor the fruit.
Mathematics is more than a collection of theorems, definitions,problems and techniques; it is a way of thought. The same can be said about an individual branch of mathematics, such as analysis. Analysis has its roots in the work of Archimedes and other ancient Greek ge-ometers, who developed techniques to find areas, volumes, centers of gravity, arc lengths, and tangents to curves. In the seventeenth century these techniques were further developed, culminating in the invention of the calculus of Newton and Leibniz.
During the eighteenth centu-ry the calculus was fashioned into a tool of bold computational power and applied to diverse problems of practical and theoretical interest.At the same time the foundation of analysis——the logical justification for the success of the methods——was left in limbo. This had practical consequences: for example, Euler——the leading mathematician of the eighteenth century——developed all the techniques needed for the study of Fourier series, but he never carried out the project.
On the contrary,he argued in print against the possibility of representing functions as Fourier series, when this proposal was put forth by Daniel Bernoulli,and his argument was based on fundamental misconceptions concerning the nature of functions and infinite series.
In the nineteenth century, the problem of the foundation of anal-ysis was faced squarely and resolved. The theory that was developed forms most of the content of this book. We will describe it in its logical order, starting from the most basic concepts such as sets and numbers and building up to the more involved concepts of limits, continuity,derivative, and integral. The actual historical order of discovery was almost the reverse; much like peeling a cabbage, mathematicians be-gan with the outermost layers and worked their way inward.
Cauchy and Bolzano began the process in the 1820s by developing the theo-ry of functions without defining the real numbers. The first rigorous definition of the real number system came in the work of Dedekind,Weierstrass, and Heine in the 1860s. Set theory came later in the work of Cantor, Peano, and Frege.
The consequences of the nineteenth century foundational work were enormous and are still being felt today. Perhaps the least important consequence was the establishment of a logically valid explanation of the calculus. More important, with the clearing away of the concep-tual murk, new problems emerged with clarity and were developed into important theories. We will give some illustrations of these new nineteenth century discoveries in our discussions of differential equa-tions, Fourier series, higher dimensional calculus, and manifolds.
Most important of all, however, the nineteenth century foundational work paved the way for the work of the twentieth century. Analysis today is a subject of vast scope and beauty, ranging from the abstract to the concrete, characterized both by the bold computational power of the eighteenth century and the logical subtlety of the nineteenth century.Most of these developments are beyond the scope of this book or at best merely hinted at. Still, it is my hope that the reader, after hav-ing entered so deeply along the way of analysis, will be encouraged to continue the study.
My goal in writing this book is to communicate the mathematical ideas of the subject to the reader. I have tried to be generous with ex-planations. Perhaps there will be places where I belabor the obvious,nevertheless, I think there is enough truly challenging material here to inspire even the strongest students. On the other hand, there will inevitably be places where each reader will find difficulties in follow-ing the arguments.
When this happens, I suggest that you write your questions in the margins. Later, when you go over the material, you may find that you can answer the question. If not, be sure to ask your instructor or another student; often, it is a minor misunderstanding that causes confusion and can easily be cleared up. Sometimes, the in-herent difficulty of the material will demand considerable effort on your part to attain understanding. I hope you will not become frustrated in the process; it is something which all students of mathematics must confront. I believe that what you learn through a process of struggle is more likely to stick with you than what you learn without effort.
Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness,but seeing the overall strategy of the argument, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole.
It is a long-term process; in a sense,you cannot appreciate the significance of the first theorem until you have learned the last theorem. So please be sure to review old mate-rial; you may find the chapter summaries useful for this purpose. The mathematical ideas presented in this book are of fundamental impor-tance, and you are sure to encounter them again in further studies in both pure and applied mathematics. Learn them well and they will serve you well in the future. It may not be an easy task, but it is a worthy one.
