分析基础:微积分理论 《分析基础:微积分理论》旨在为对严格的证明不熟悉的读者,提供一本学习标准积分教程的后继教程。对于学习高等分析教程,复变量、微分方程、傅里叶分析、数值分析、多变量微积分以及统计的人来说读《分析基础:微积分理论》是很有必要的。同样也可以作为将来中学老师的参考教科书。 有关实线的概念《分析基础:微积分理论》研究不多,许多抽象概念,如矩阵空间和有序系,在《分析基础:微积分理论》中都避免讨论;最小上界的性质在《分析基础:微积分理论》中拿来即用,实线的有序性贯穿于书的始终;数字序列的透彻讲解作为学习标准微积分的基础;选修部分可以让学生学习矩阵空间和Riemann-Stieltjes积分。 Preface 1 Introduction 1 The Set N of Natural Numbers 2 The Set Q of Rational Numbers 3 The Set R of Real Numbers 4 The Completeness Axiom 5 The Symbols+∞and-∞ 6 *A Development of R 2 Sequences 7 Limits of Sequences 8 A Discussion about Proofs 9 Limit Theorems for Sequences 10 Monotone Sequences and Cauchy Sequences 11 Subsequences 12 lira sup's and lim inf's 13 *Some Topological Concepts in Metric Spaces 14 Series 15 Alternating Series and Integral Tests 16 *Decimal Expansions of Peal Numbers 3 Continuity 17 Continuous Functions 18 Properties of Continuous Functions 19 Uniform Continuity 20 Limits of Functions 21 *More on Metric Spaces: Continuity 22 *More on Metric Spaces: Connectedness 4 Sequences and Series of Functions 23 Power Series 24 Uniform Convergence 25 More on Uniform Convergence 26 Differentiation and Integration of Power Series 27 *Weierstrass's Approximation Theorem Differentiation 28 Basic Properties of the Derivative 29 The Mean Value Theorem 30 *UHospital's Rule 31 Taylor's Theorem 5 Integration 32 The Riemann Integral 33 Properties of the Riemann Integral 34 Fundamental Theorem of Calculus 35 *Riemann-Stieltjes Integrals 36 *Improper Integrals 37 *A Discussion of Exponents and Logarithms Appendix on Set Notation Selected Hints and Answers References 8ymhola Index Index