汤姆·M. 阿波斯托尔(Tom M. Apostol)是加州理工学院数学系荣誉教授。他于1946年在华盛顿大学西雅图分校获得数学硕士学位，于1948年在加州大学伯克利分校获得数学博士学位。
目錄：
Chapter 1 The Real and Complex Number Systems1.1 Introduction 11.2 The field axioms . 11.3 The order axioms 21.4 Geometric representation of real numbers 31.5 Intervals 31.6 Integers 41.7 The unique factorization theorem for integers 41.8 Rational numbers 61.9 Irrational numbers 71.10 Upper bounds， maximum element， least upper bound(supremum) . 81.11 The completeness axiom 91.12 Some properties of the supremum 91.13 Properties of the integers deduced from the completeness axiom 101.14 The Archimedean property of the real-number system . 101.15 Rational numbers with finite decimal representation 111.16 Finite decimal approximations to real numbers 111.17 Infinite decimal representation of real numbers . 121.18 Absolute values and the triangle inequality 121.19 The Cauchy—Schwarz inequality 131.20 Plus and minus infinity and the extended real number system R* 141.21 Complex numbers 151.22 Geometric representation of complex numbers 171.23 The imaginary unit 181.24 Absolute value of a complex number . 181.25 Impossibility of ordering the complex numbers . 191.26 Complex exponentials 191.27 Further properties of complex exponentials 201.28 The argument of a complex number . 201.29 Integral powers and roots of complex numbers . 211.30 Complex logarithms 221.31 Complex powers 231.32 Complex sines and cosines 241.33 Infinity and the extended complex plane C* 24Exercises 25Chapter 2 Some Basic Notions of Set Theory2.1 Introductiou 322.2 Notations 322.3 Ordered pairs 332.4 Cartesian product of two sets 332.5 Relations and functions 342.6 Further terminology concerning functions 352.7 One-to-one functions and inverses 362.8 Composite functions 372.9 Sequences. 382.10 Similar (equinumerous) sets 382.11 Finite and infinite sets 392.12 Countable and uncountable sets 392.13 Uncountability of the real-number system 422.14 Set algebra 432.15 Countable collections of countable sets Exercises 43Chapter 3 Elements of Point Set Topology3.1 Introduction 473.2 Euclidean space Rt 473.3 Open balls and open sets in R* 493.4 The structure of open sets in RH 503.5 Closed sets . 523.6 Adhèrent points. Accumulation points 523.7 Closed sets and adhèrent points 533.8 The Bolzano—Weierstrass theorem 543.9 The Cantor intersection theorem 563.10 The Lindel?f covering theorem 563.11 The Heine—Borel covering theorem 583.12 Compactness in R‘ 593.13 Metric spaces 603.14 Point set topology in metric spaces 613.15 Compact subsets of a metric space 633.16 Boundary of a setExercises 65Chaqter 4 Limits and Continuity 4.1 Introduction 704.2 Convergent sequences in a metric space 724.3 Cauchy sequences 744.4 Complete metric spaces . 744.5 Limit of a function 764.6 Limits of complex-valued functions 4.7 Limits of vector-valued functions 774.8 Continuous functions 784.9 Continuity of composite functions.4.10 Continuous complex-valued and vector-valued functions 794.11 Examples of continuous functions 804.12 Continuity and inverse images of open or closed sets 804.13 Functions continuous on compact sets 814.14 Topolo$ical mappings (homeomorphisms) 824.15 Bolzano’s theorem 844.16 Connectedness 844.17 Components of a metric space . 864.18 Arcwise connectedness 874.19 Uniform continuity 884.20 Uniform continuity and compact sets 904.21 Fixed-point theorem for contractions 914.22 Discontinuities of real-valued functions 924.23 Monotonic functions 94Exercises 95Chapter 5 DerJvatives 5.1Introduction 1045.2 Definition of derivative .1045.3 Derivatives and continuity 1055.4 Algebra of derivatives1065.5 The chain rule 1065.6 One-si
內容試閱：
从目录可以看出，本书是在“高等微积分”的水平上阐述数学分析中的论题．编写本书的目的在于展现这门学科，所以要求叙述忠实于原貌、精确严密，包含最进展，同时又不过于学究气．本书提供了从初等微积分向实变函数论及复变函数论等高等课程的一种过渡，并介绍了一些涉及现代分析的抽象理论．与第1版相比，第2版的主要更新表现在以下方面：在考虑一般的度量空间以及n维欧氏空间时介绍点集拓扑；增加了关于勒贝格积分的两章；删去了曲线积分、向量分析和曲面积分方面的内容；重排了某些章的顺序；完全重写了很多节；增加了若干新的练习．勒贝格积分由Riesz-Nagy方法引入，此方法直接着眼于函数及其积分，而不依赖于测度论．为了适应大学本科水平的教学，在介绍勒贝格积分时，进行了简化、延伸和调整．本书第1版曾被用于从本科一年级到研究生一年级各种水平的数学课程，既用作教科书，又用作补充参考书．第2版保持了这种灵活性．例如，第1章至第5章及第12章和第13章可用于单变量或多变量函数的微分学课程，第6章至第11章及第14章和第15章可用于积分论的课程．也可以按其他方式进行多种组合，教师则可以参考下一页的图示根据自己的需要，选择适当的章节，图中显示了各章之间的逻辑依赖关系．我要向不厌其烦地就第1版写信给我的许多人表示感谢，他们的评论和建议有助于我进行修订．特别要感谢Charalambos Aliprantis博士，他细心地阅读了第2版的全部手稿并提出了许多有益的建议，还提供了一些新的练习．最后，向加州理工学院的学生们表示由衷的感谢，是他们对数学的热情激发了我编著此书．T. M. A．1973年9月于帕萨迪纳
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