微积分和数学分析引论 第1卷
作者：（美）Richard Courant，Fritz John
出版时间： 2018年版
内容简介
本书共分为2卷三册，内容以及形式上有如下三个特点：一是引导者直达本学科的核心内容；二是注重应用，指导读者灵活运用所掌握的知识；三是突出了直觉思维在数学学习中的作用。作者不掩饰难点以使得该学科貌似简单，而是通过揭示概念之间的内在联系和直观背景努力帮助那些对这门学科真正感兴趣的读者。本书第一章主要围绕着一元函数展开讨论，二、三、四章分别介绍了微积分的基本概念、运算及其在物理和几何中的应用，随后讲述了泰勒展开式、数值方法、数项级数、函数项级数、三角级数，最后介绍了一些与振动有关的类型简单的微分方程。本书各章均提供了大量的例题和习题，其中一部分有相当的难度，但绝大部分是对正文内容的补充。
目录
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
