微積分和數(shù)學分析引論（第1卷）

內容概要

During the latter part of the seventeenth century the new mathe-matical analysis emerged as the dominating force in mathematics.    It is characterized by the amazingly successful operation with infinite processes or limits. Two of these processes, differentiation and inte- gration, became the core of the systematic Differential and Integral Calculus, often simply called "Calculus," basic for all of analysis.    The importance of the new discoveries and methods was immediately felt and caused profound intellectual excitement. Yet, to gain mastery of the powerful art appeared at first a formidable task, for the avail-able publications were scanty, unsystematic, and often lacking in clarity. Thus, it was fortunate indeed for mathematics and science in general that leaders in the new movement soon recognized the vital need for writing textbooks aimed at making the subject ac-cessible to a public much larger than the very small intellectual elite of the early days. One of the greatest mathematicians of modern times,Leonard Euler, established in introductory books a firm tradition and these books of the eighteenth century have remained sources of inspira-tion until today, even though much progress has been made in the clarification and simplification of the material.    After Euler, one author after the other adhered to the separation of differential calculus from integral calculus, thereby obscuring a keypoint, the reciprocity between differentiation and integration. Only in1927 when the first edition of R. Courant's German Vorlesungen iiber Differential und Integrairechnung, appeared in the Springer-Verlagwas this separation eliminated and the calculus presented as a unifiedsubject.

作者簡介

作者：(美國)庫蘭特

書籍目錄

Chapter 1 Introduction  1.1  The Continum of Numbers  1.2  The Concept of Function  1.3  The Elementary Functions  1.4  Sequences  1.5  Mathematical Induction  1.6  The Limit of a Sequence  1.7  Further Discussion of the Concept of Limit  1.8  The Concept of Limit for Functions of a Continuous VariableSupplements  S.1  Limits and the Number Concept  S.2  Theorems on Continuous Functions  S.3  Polar Coordinates  S.4  Remarks on Complex Numbers  PROBLEMSChapter 2  The Fundamental Ideas of the Integral and Differential Calculus  2.1  The Integral  2.2  Elementary Examples of Integration  2.3  Fundamental Rules of Integration  2.4  The Integral as a Function of the Upper Limit (Indefinite Integral)  2.5  Lograithm Defined by an Integral  2.6  Exponential Function and Powers  2.7  The Integral of an Arbitrary Power of   2.8  The Derivative  2.9  The Integral,the Primitive Function,and the Fundamental Theorems of the Calculus   PROBLEMSChapter 3  The Technipues of CalculusChapter 4  Applications in Physics and GeometryChapter 5  Taylor's ExpansionChapter 6  Numerical MethodsChapter 7  Infinite Sums and ProductsChapter 8  Trigonometric SeriesChapter 9  Differential Epuations for the Simplest Types of VibrationList of Biograpical DatesIndex

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本書在內容以及形式上有如下三個特點：一是引領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(shù)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(lián)系和直觀背景努力幫助那些對這門學科真正感興趣的讀者。本書第一章主要圍繞著一元函數(shù)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，隨后講述了泰勒展開式、數(shù)值方法、數(shù)項級數(shù)、函數(shù)項級數(shù)、三角級數(shù)，最后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?shù)碾y度，但絕大部分是對內容的補充。另外，本書附有一本專門的習題冊，并且給出了習題的提示與解答。本書適合于多種學科界的讀者，如數(shù)學工作者、科學工作者、工程技術人員等。

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