An Introduction to Analysis, 4/e (GE-Paperback)

本書序言

Changes to the Exercises

• Computational exercises havebeen rewritten so that answers are simpler and easier to obtain.
• Calculus-style exercises atthe beginning of the book have been revised to be more conceptual, emphasizingthe same ideas, but at a higher level.
• Theoretical exercises ofmedium difficulty have been added throughout the book.
• New True/False questions inthe first six chapters confront common misconceptions that students sometimesacquire at this level. 

Content Updates

• A new section 1.1, Introduction,combines introductory material that was previously scattered over severalsections. This section includes two accessible examples about why proof isnecessary and why we cannot always trust what we see.
• The number of axioms hasbeen reduced from four to three by introducing the Completeness Axiom first,and using it to prove the Well Ordering Principle and the Principle ofMathematical Induction.
• The material on countable sets and inverse images of sets hasbeen postponed to Chapter 3, making it possible to begin discussing limits ofsequences even earlier than before.
• Coverage of Taylor's Formula has been moved fromChapter 7 to Chapter 4 to offer another example of the utility of the MeanValue Theorem.
• The Heine-Borel Theorem now has its own sectionand includes several exercises designed to give students practice in making alocal condition on a compact set into a global one.
• Section 12.1, Jordan regions,has been organized to simplify the presentation and make it easier to teach.

本書特色

Features

• Flexible presentation,with uniform writing style and notation, covers the material in small sections,allowing instructors to adapt this book to their syllabus.
• The practical focus explains assumptions so thatstudents learn the motivation behind the mathematics and are able to constructtheir own proofs.
• Early introduction of thefundamental goals of analysis Refers and examines how alimit operation interacts with algebraic operation.
• Optional appendices and enrichmentsections enables students to understand the material and allowsinstructors to tailor their courses.
• An alternate chapter on metricspaces allows instructors to cover either chapter independentlywithout mentioning the other.
• More than 200 worked examples and600 exercises encourage students to test comprehension of concepts, whileusing techniques in other contexts.
• Separate coverage of topology andanalysis presents purely computational material first, followed bytopological material in alternate chapters.
• Rigorous presentation of integers providesshorter presentations while focusing on analysis.
• Reorganized coverage of series separatesseries of constants and series of functions into separate chapters.
• Consecutive numbering of theorems,definitions and remarks allows students and instructorsto find citations easily. 

Pearson MyLab Math is not included. Students, if Pearson MyLab Math is arecommended/mandatory component of the course, please ask your instructor forthe correct ISBN. Pearson MyLab Math should only be purchased when required by aninstructor. Instructors, contact your Pearson representative for more information.

MyLab Math is not included. Students, if MyLab Math is arecommended/mandatory component of the course, please ask your instructor forthe correct ISBN. MyLab Math should only be purchased when required by aninstructor. Instructors, contact your Pearson representative for moreinformation.

本書序言

習題的變更：
- 重新撰寫了計算性習題，使答案更簡單且更容易獲得。
- 修訂了書籍開頭的微積分式習題，使其更具概念性，強調相同的觀念，但在更高的層次上。
- 在整本書中增加了中等難度的理論習題。
- 在前六章中新增了新的正確/錯誤問題，以解決學生在這個層次上可能產生的常見誤解。

內容更新：
- 新增了1.1節「引言」，將之前分散在多個章節中的入門材料結合在一起。該節包括兩個易於理解的例子，闡述了為什麼需要證明以及為什麼我們不能始終相信我們所看到的。
- 通過首先引入完備性公理，並使用它來證明良序原理和數學歸納法，將公理的數量從四個減少到三個。
- 將可數集和集合的逆像的內容推遲到第三章，使得可以更早地討論數列的極限。
- 將對泰勒公式的涵蓋範圍從第七章移至第四章，以提供另一個使用均值定理的實用例子。
- 海涅-博雷爾定理現在有了自己的章節，並包含了幾個設計用於將局部條件轉化為全局條件的練習題。
- 重新組織了第12.1節「喬登區域」，以簡化演示並使教學更容易。

本書特色：

- 靈活的呈現方式，統一的寫作風格和符號，以小節的形式涵蓋材料，使教師能夠根據自己的教學大綱調整本書。
- 實用的焦點解釋了假設，使學生了解數學背後的動機，並能夠構建自己的證明。
- 早期引入分析的基本目標，引用並探討極限運算與代數運算的交互作用。
- 可選的附錄和豐富的章節使學生能夠理解材料，並使教師能夠量身定制課程。
- 關於度量空間的另一個章節使教師能夠獨立地涵蓋其中一個章節，而不提及其他章節。
- 超過200個解題示例和600個習題，鼓勵學生測試對概念的理解，並在其他情境中應用技巧。
- 分開涵蓋拓撲和分析，先呈現純計算材料，然後在交替章節中呈現拓撲材料。
- 嚴謹地介紹整數，提供更短的介紹，同時專注於分析。
- 重新組織了對數列的涵蓋範圍，將常數數列和函數數列分為不同的章節。
- 定理、定義和備註的連續編號，使學生和教師能夠輕鬆找到引用。

本書不包含Pearson MyLab Math。學生如果課程中推薦/強制使用Pearson MyLab Math，請向您的教師索取正確的ISBN。只有在教師要求時才應購買Pearson MyLab Math。教師請聯繫Pearson代表獲取更多信息。

本書不包含MyLab Math。學生如果課程中推薦/強制使用MyLab Math，請向您的教師索取正確的ISBN。只有在教師要求時才應購買MyLab Math。教師請聯繫Pearson代表獲取更多信息。

William Wade received his PhD in harmonic analysis from the University of CaliforniaRiverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than forty years. During that time, he has received multiple awards including two Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.
Wades research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis,is now in its fourth edition.
In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.

William Wade在加州大學河濱分校獲得了他的調和分析博士學位。他在田納西大學數學系擔任教授已經超過四十年。在此期間，他獲得了多個獎項，包括兩個富布萊特獎學金、校長研究和創意成就獎、院長特殊服務獎和全國校友協會傑出教學獎。

Wade的研究興趣包括唯一性、增長和二進制調和分析等問題，他在這些領域發表了許多論文，出版了兩本書，並在三大洲上做過多次演講。他目前的著作《分析導論》已經出版第四版。

在閒暇時間，Wade喜歡旅行並拍攝照片來記錄他的旅程。他也對音樂有興趣，喜歡演奏古典音樂，主要是巴洛克風格的小號、直笛和鋼琴。

Part I. ONE-DIMENSIONALTHEORY
1. The Real Number System
2. Sequences in R
3. Functions on R
4. Differentiability on R
5. Integrability on R
6. Infinite Series of Real Numbers
7. Infinite Series of Functions
Part II. MULTIDIMENSIONAL THEORY
8. Euclidean Spaces
9. Convergence in Rn
10. Metric Spaces
11. Differentiability on Rn
12. Integration on Rn
13. Fundamental Theorems of Vector Calculus
14. Fourier Series
Appendices
A. Algebraic laws
B. Trigonometry
C. Matrices and determinants
D. Quadric surfaces
E. Vector calculus and physics
F. Equivalence relations
Answers and Hints to Selected Exercises

第一部分. 一維理論
1. 實數系統
2. 實數列
3. 實數函數
4. 實數的可微性
5. 實數的可積性
6. 實數的無窮級數
7. 函數的無窮級數

第二部分. 多維理論
8. 歐幾里得空間
9. 在 Rn 中的收斂
10. 度量空間
11. Rn 上的可微性
12. Rn 上的積分
13. 向量微積分的基本定理
14. 傅立葉級數

附錄
A. 代數法則
B. 三角學
C. 矩陣和行列式
D. 二次曲面
E. 向量微積分和物理學
F. 等價關係
選擇性練習的答案和提示