分析（第1捲） [Analysis 1] 下載 mobi epub pdf 電子書 2025

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內容簡介

　　The present book strives for clarity and transparency. Right from the begin-ning， it requires from the reader a willingness to deal with abstract concepts， as well as a considerable measure of self-initiative. For these e&，rts， the reader will be richly rewarded in his or her mathematical thinking abilities， and will possess the foundation needed for a deeper penetration into mathematics and its applications.
　　This book is the first volume of a three volume introduction to analysis. It de- veloped from. courses that the authors have taught over the last twenty six years at the Universities of Bochum， Kiel， Zurich， Basel and Kassel. Since we hope that this book will be used also for self-study and supplementary reading， we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses， not only elegance and inner beauty， but also provides efficient methods for the solution of concrete problems.

內頁插圖

目錄

Preface
Chapter Ⅰ Foundations
1 Fundamentals of Logic
2 Sets
Elementary Facts
The Power Set
Complement, Intersection and Union
Products
Families of Sets
3 Functions，
Simple Examples
Composition of Functions
Commutative Diagrams
Injections, Surjections and Bijections
Inverse Functions
Set Valued Functions
4 Relations and Operations
Equivalence Relations
Order Relations
Operations
5 The Natural Numbers
The Peano Axioms
The Arithmetic of Natural Numbers
The Division Algorithm
The Induction Principle
Recursive Definitions
6 Countability
Permutations
Equinumerous Sets
Countable Sets
Infinite Products
7 Groups and Homomorphisms
Groups
Subgroups
Cosets
Homomorphisms
Isomorphisms
8 R.ings, Fields and Polynomials
Rings
The Binomial Theorem
The Multinomial Theorem
Fields
Ordered Fields
Formal Power Series
Polynomials
Polynomial Functions
Division of Polynomiajs
Linear Factors
Polynomials in Several Indeterminates
9 The Rational Numbers
The Integers
The Rational Numbers
Rational Zeros of Polynomials
Square Roots
10 The Real Numbers
Order Completeness
Dedekind's Construction of the Real Numbers
The Natural Order on R
The Extended Number Line
A Characterization of Supremum and Infimum
The Archimedean Property
The Density of the Rational Numbers in R
nth Roots
The Density of the Irrational Numbers in R
Intervals
Chapter Ⅱ Convergence
Chapter Ⅲ Continuous Functions
Chapter Ⅳ Differentiation in One Variable
Chapter Ⅴ Sequences of Functions
Appendix Introduction to Mathematical Logic
Bibliography
Index

前言/序言

　　Logical thinking， the analysis of complex relationships， the recognition of under- lying simple structures which are common to a multitude of problems - these are the skills which are needed to do mathematics， and their development is the main goal of mathematics education.
　　Of course， these skills cannot be learned 'in a vacuum'. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential， without being distracted by appearances and irrelevancies.
　　The present book strives for clarity and transparency. Right from the begin-ning， it requires from the reader a willingness to deal with abstract concepts， as well as a considerable measure of self-initiative. For these e&，rts， the reader will be richly rewarded in his or her mathematical thinking abilities， and will possess the foundation needed for a deeper penetration into mathematics and its applications.
　　This book is the first volume of a three volume introduction to analysis. It de- veloped from. courses that the authors have taught over the last twenty six years at the Universities of Bochum， Kiel， Zurich， Basel and Kassel. Since we hope that this book will be used also for self-study and supplementary reading， we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses， not only elegance and inner beauty， but also provides efficient methods for the solution of concrete problems.
　　Analysis itself begins in Chapter II. In the first chapter we discuss qLute thor- oughly the construction of number systems and present the fundamentals of linear algebra. This chapter is particularly suited for self-study and provides practice in the logical deduction of theorems from simple hypotheses. Here， the key is to focus on the essential in a given situation， and to avoid making unjustified assumptions.An experienced instructor can easily choose suitable material from this chapter to make up a course， or can use this foundational material as its need arises in the study of later sections.
　　……

分析（第1捲） [Analysis 1] 下載 mobi epub pdf txt 電子書 格式

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拓撲結構的基本概念如連通性、密實度和介紹瞭homeomorphisms早期使用作為一個基礎,證明將遠不及優雅的(和不直接)否則。例如,介值定理,證明瞭結果的連接的一個空間。一旦這是結果確定下來的普遍性,它討論瞭R。　　

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不錯的東西。。。。。。。。。。。。。

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這套書給人的感覺有點不上不下。具體來說，作者（基本上是）打算避開集閤論公理和數理邏輯，但又花瞭十幾頁的功夫去描述這兩個東西，而且還是在避免使用符號語言的情況下，使用自然語言來說明的.......嘛，因為原文是德文，說明上應該會比這英譯本的要嚴格一些，但是這英譯本就......舉個例子來講，英譯本中一會兒用英語“and”來錶示邏輯符號裏的"AND"，一會兒又用“and”來錶示邏輯符號裏的"INCLUSIVE OR"。都無語瞭......

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