微积分和数学分析引论(第1卷影印版)

Chapter 1 Introduction 1.1 The Continuum of Numbers a. The System of Natural Numbers and Its Extension. Counting and Measuring b. Real Numbers and Nested Intervals c. Decimal Fractions. Bases Other Than Ten d. Definition of Neighborhood e. Inequalities 1.2 The Concept of Function a. Mapping-Graph b. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function c. Graphical Representation. Monotonic Functions d. Continuity e. The Intermediate Value Theorem. Inverse Functions 1.3 The Elementary Functions a. Rational Functions b. Algebraic Functions c. Trigonometric Functions d. The Exponential Function and the Logarithm e. Compound Functions,Symbolic Products, Inverse Functions 1.4 Sequences 1.5 Mathematical Induction 1.6 The Limit of a Sequence 1.7 Further Discussion of the Concept of Limit a. Definition of Convergence and Divergence b. Rational Operations with Limits c. Intrinsic Convergence Tests. Monotone Sequences d. Infinite Series and the Summation Symbol e. The Number e f. The Number r as a Limit 1.8 The Concept of Limit for Functions of a Continuous Variable a. Some Remarks about the Elementary Functions Supplements S.1 Limits and the Number Concept a. The Rational Numbers b. Real Numbers Determined by Nested Sequences of Rational Intervals c. Order, Limits, and Arithmetic Operations for Real Numbers d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria e. Least Upper Bound and Greatest Lower Bound f. Denumerability of the Rational Numbers S.2 Theorems on Continuous Functions S.3 Polar Coordinates S.4 Remarks on Complex Numbers PROBLEMS Chapter 2 The Fundamental Ideas of the Integral and Differential Calculus Chapter 3 The Techniques of Calculus Chapter 4 Applications in Physics and Geometry Chapter 5 Taylor' s Expansion Chapter 6 Numerical Methods Chapter 7 Infinite Sums and Products Chapter 8 Trigonometric Series Chapter 9 Differential Equations for the Simplest Types of Vibration List of Biographical Dates Index