导语

  

内容提要

  

    R.库朗、F.约翰著的《微积分和数学分析引论(第2卷第1分册)(英文版)》内容以及形式上有如下三个特点：一是引领读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。

Chapter 1  Functions of Several Variables and Their Derivatives
  1.1  Points and Points Sets in the Plane and in Space
    a.Sequences of points. Convergence
    b.Sets of points in the plane
    c.The boundary of a set.Closed and open sets
    d.Closure as set of limit points
    e.Pointsand sets of points in space
  1.2  Functions of Several Independent Variables
    a.Functions and their domains
    b.The simplest types of functions
    c.Geometrical representation of functions
  1.3  Continuity
    a.Definition
    b.The concept of limit of a function of several variable
    c.The order to which a function vanishes
  1.4  The Partial Derivatives of a Function
    a.Definition. Geometrical representation
    b.Examples,
    c.Continuity and the existence of partial derivatives
    d.Change of the order of differentiation
  1.5  The Differential of a Function and Its Geometrical Meaning
    a.The concept of differentiability
    b.Directional derivatives
    c.Geometricinterpretation of differentiability,The tangent plane
    d.The total differential of a function
    e.Application to the calculus of errors
  1.6  Functions of Functions (Compound Functions) and the Introduction of New Independent Variables
    a.Compound functions. The chain rule
    b.Examples
    c.Change of independent variables
  1.7  The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables
    a.Preliminary remarks about approximation by polynomials
    b.The mean value theorem
    c.Taylor's theorem for several independent variables
  1.8  Integrals of a Function Depending on a Parameter
    a.Examples and definitions, 71
    b.Continuity and differentiability of an integral with respect to the parameter
    c.Interchange of integrations. Smoothing of functions
  1.9  Differentials and Line Integrals
    a.Linear differential forms
Chapter 2  Vectors, Matrices, Linear Transformations
Chapter 3  Developments and Applications of the Differential Calculus
Chapter 4  Multiple Integrals
Chapter 5  Relations Between Surface and Volume Integrals
Chapter 6  Differential Equations
Chapter 7  Calculus of Variations
Chapter 8  Functions of a Complex Variable