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)(英文版)》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。
    本书为第2分册，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. Conver-
      gence, 1  b. Sets of points in the
      plane, 3 c. The boundary of a set.
      Closed and open sets, 6 d. Closure
      as set of limit points, 9 e. Points
      and sets of points in space, 9
  1.2  Functions of Several Independent
      Variables
      a. Functions and their domains, 11
      b. The simplest types of func-
      tions, 12  c. Geometrical representa-
      tion of functions, 13
  1.3  Continuity
      a. Definition, 17 b. The concept of
      limit of a function of several vari-
      ables, 19 c. The order to which a
      function vanishes, 22
  1.4  The Partial Derivatives of a
      Function
      a. Definition. Geometrical
      representation, 26 b. Examples,
      32 c. Continuity and the
     existence of partial derivatives, 34
      d. Change of the order of
      differentiation, 36
  1.5  The Differential of a Function
      and Its Geometrical Meaning
      a. The concept of differentia-
      bility, 40 b. Directional
      derivatives, 43 c. Geometric
      interpretation of differentiability,
      The tangent plane, 46  d. The total
      differential of a function, 49 e.
      Application to the calculus of
      errors, 52
  1.6  Functions of Functions (Com-
      pound Functions) and the
      Introduction of New In-
      dependent Variables
      a. Compound functions. The chain
      rule, 53 b. Examples, 59  c.
      Change of independent variables, 60
  1.7 The Mean Value Theorem and
      Taylor's Theorem for Functions
      of Several Variables
      a. Preliminary remarks about
      approximation by polynomials, 64
      b. The mean value theorem, 66
      c. Taylor's theorem for several in-
      dependent variables, 68
  1.8  Integrals of a Function Depend-
      ing on a Parameter
      a. Examples and definitions, 71
      b. Continuity and differentiability
      of an integral with respect to the
      parameter, 74 c. Interchange of
      integrations. Smoothing of
      functions, 80
  1.9  Differentials and Line Integrals
      a. Linear differential forms, 82
  ……
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
List of Biographical Dates
Index