內容大鋼
R.庫朗、F.約翰著的《微積分和數學分析引論(第2卷)(英文版)》共分為2卷三冊，內容以及形式上有如下三個特點：一是引導讀者直達本學科的核心內容；二是注重應用，指導讀者靈活運用所掌握的知識；三是突出了直覺思維在數學學習中的作用。作者不掩飾難點以使得該學科貌似簡單，而是通過揭示概念之間的內在聯繫和直觀背景努力幫助那些對這門學科真正感興趣的讀者。
本書為第2分冊，第一章主要圍繞著一元函數展開討論，二、三、四章分別介紹了微積分的基本概念、運算及其在物理和幾何中的應用，隨後講述了泰勒展開式、數值方法、數項級數、函數項級數、三角級數，最後介紹了一些與振動有關的類型簡單的微分方程。本書各章均提供了大量的例題和習題，其中一部分有相當的難度，但絕大部分是對正文內容的補充。
目錄
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
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