国外优秀数学教材选评

2023年12月10日发(作者：大人做小学数学试卷有用吗)

复旦大学外国教材中心国外优秀数学教材选评December 24, 2007本书内容为原作者版权所有，未经协议授权，禁止下载使用主 编 杨劲根副主编 楼红卫 李振钱 郝群编写人员（按汉语拼音为序）陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林1. 序言

2. 非数学专业的数学教材

3.数学分析和泛函分析4.单复变函数5.多复变函数6.代数7.数论8.代数几何9.拓扑与微分几何10.常微分方程11.偏微分方程12.概率论13.其他14.附录序言1.1 数学与数学教材数学是科学的一个重要工具，这已经是老生常谈的一个常识了。从中小学、大学直到研究生，数学课程始终占据显著的位置。 数学学科是庞大的，包含的分支很多，而且随着时间的推移，人类对数学的认识越来越深刻，数学的内容也越来越丰富，新的数学分支也 时常产生。然而，尽管数学学科在不断的发展，它的基本原理是相对稳定的。如果把现在的大学数学和50年前作比较，就会发现基础性的 内容是差不多的，那时的很多优秀数学书籍现在仍然奉为经典。这是数学和一些新兴学科的一个显著区别。数学大致分作两类：基础数学和应用数学。基础数学也叫做纯数学或理论数学，它是根据数学本身的需要而发展的。 应用数学是在纯数学的基础上产生的各种具有不同程度的应用性的各种学科，这是数学和其他学科如物理、化学、计算机科学、经济学等的桥梁。大学数学课程按学生的专业可以分成两大类：数学专业的和非数学专业的。按程度又分本科生课程和研究生课程两大类。数学专业本科生 有低年级的基础课程和高年级的专业课程选修课程。低年级的基础课程主要包括数学分析、线性代数、复分析、微分方程、抽象代数、实变函数、泛函分析等。非数学专业的本科生数学基础课程通常称‘高等数学’，内容以微积分、线性代数和微分方程为主， 只是比数学专业学生学习的内容要浅些。非数学专业的学生在数学课程中接受的训练主要是计算和应用的能力，而数学专业的学生主要 接受数学推理的能力训练。/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心学习数学的最主要的途径是看书。数学书籍大凡可以分教科书、学术专著和通俗读物三种。差不多所有数学分支都有一些不同深度的教科书。1.2 如何选择合适的教材对于在读的大学生或研究生，不需要化太多心思选择教材，只要用老师指定的教材就可以了。特别是如果在所修的课程中你感觉学到比较 扎实，习题基本会做，那也不一定去看太多其他同类的教材。对于学生来说，参考书是双刃剑，一方面它可以开拓视野，加深对所学知识 认识的深度，另一方面，由于不同的作者写书的构思不同，内容安排的次序也可能不同，甚至所用的术语也有区别，同时看几本同类的书 会造成混乱。所以建议在下面两种情形下去寻找合适的参考书：1）觉得课堂上用的教材太难，大部分习题不会做，这时可找一本浅一点的或者对基本概念解释得更仔细一点的书。2）能轻松对付课堂内容，又对该课程有浓厚兴趣，这时可请老师推荐更深一些的教材。对于自学数学的同志，选择合适的教材是十分关键的，千万不要随便抓起一本书就念。选错书是会走很多弯路的。对于初学者，光看书名、目录和序言是很难准确地判断这本书是不是适合于你，需要仔细看看里面的内容。可以到书店去浏览，有些书店有很多品种的数学书，但是有很多最好的书籍在书架上是没有的，因此图书馆是一个更好的选择。也可以在互联网上搜索，当然身边有高手指点就再好不过了。1.3 外国数学教材中国国内有不少好的数学教材，为什么还需要外国的教材呢？从中学数学教材到大学低年级的教材来看，光用国内的教材已经够了，但是 越到高的层次，对国外教材的倚靠就越明显了。不仅要使用翻译的教材，还要使用原版的。从语种来看，英语最为重要。世界上的数学大国是 美国、俄罗斯、德国、法国、英国，这五个国家堪称数学超级大国。 意大利、日本、印度和东欧诸国的数学也很强。中国虽然出些数学人才，但是和数学五大强国比差距仍不小，我们得摆正位置，老老实实学习人家先进的东西。日本和印度的数学家历来用英文写作。前苏联的数学教科书在60年代 对我国起很大影响，当时会俄文对学数学很有利。几十年前，非英语国家的数学教科书都用本国文字写。在当今的信息时代，英语几乎成了世界语，在数学中也不例外，连法国德国的数学家也经常用英文写作，在加上美国的数学界化了相当大的人力物力翻译数学名著。 对于数学工作者来说，只懂英语一种外语也够了。国外有几家著名的出版商如德国的 Springer Verlag, 美国的 Academic Press, 美国数学会(AMS) 和一些名校如英国的牛津、剑桥，美国的 Princeton 大学的出版社都是数学教材大户。非数学专业用的数学书的出版商不象基础数学那样集中，多数由一些综合性的出版社如 John Wiley, Prentice Hall, McGraw-Hill 等出版。数学类书籍的领头羊当数 Springer Verlag，它有很多系列丛书，主要给数学专业使用，最有名的几种是1）GTM, 即Graduate Texts in Mathematics, 至今以出版了200多种，覆盖面很广，但多数是基础数学方面研究生教材。2）UTM, 即Undergraduate Texts in Mathematics, 该系列比上面系列出现得晚一些，也没有列序号，因此品种也略少一些，似乎只有几十中，大部分是本科生数学教材。3）LNM, 即Lecture Notes in Mathematics, 这是规模最大的丛书，以专著和研究生课程的讲义为主，现已有几千种。Springer还有几个系列非常专门，这里就不介绍了。Springer的数学书差不多都是醒目的黄色封皮，印刷和装订都很考究。 书的数学质量也很高，很收读者欢迎。但是大部分书都适合于有较好的数学训练的人阅读的， 建议我国数学系研究生和高年级本科生使用。供大学生阅读的数学课外读物历来比较少。美国数学会在几年前推出的简装的系列丛书Student Mathematical Library 倒是针对大学生的。大部分书不到200页，选材比较有趣，非常有特色。美国数学会仿效Springer Verlag, 也出版一套黄封面的研究生数学丛书，里面不乏好书。此外，美国数学会有一个翻译书系列，以俄罗斯和日本的数学译著为主，多数是研究性的专著，但也有一些高质量的教材。 另一套值得推荐的系列丛书是伦敦数学会的Student texts,对象以数学专业高年级大学生和研究生为主，每本的篇幅为200页上下，内容覆盖的范围很广，基础数学方面的更多一些。下面谈谈供非数学专业使用的外国数学教材，其中最重要的是Calculus.美国的微积分教材品种很多，根据对象不同深浅也不一样，正象我国的高等数学课程分理工类、 医学类、经济类等等一样。美国的微积分教材篇幅很大，一本书一般都在600页上下，而且是大开本的。由于这是出版量最大的数学教材，印刷非常考究，校对也仔细，所以错误极少。我国的高等数学教材大部分比较简洁，/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心其优点是信息量大，缺点是不利于自学。美国的微积分教材一般浓度不大，非常注意由浅入深，描述和解释性的话比较多，特别注意讲解实例，习题也很丰富，一般的读者只要没有英语方面的问题读起来是很快的。1.4 外国数学教材的来源自改革开放以来，我国在引进外国教材方面作出了巨大的努力。教委和科学院每年都花费大量外汇购买 各种原版科技书籍。然而这些原版书价格非常昂贵，一般的读者很难承受，多数由图书馆采购，按我国现在的 条件，一般大专院校的原版书的数量是非常有限的。近十几年来，我国的一些出版单位如世界图书出版公司、高等教育出版社、机械工业出版社等购买了国外 一些大的出版公司的部分书刊的版权后在中国影印出版，其价格只及原版书的五分之一到十分之一，种类也 越来越多，这是喜欢外国教材的读者的一个重要书源。随着信息时代的到来，电子书籍成了一个最诱人的书源。虽然数学电子书不象文科书籍那样容易在互联网上找到， 但是它们的数量也是以惊人的速度增加。例如Springer在网上提供了它的全部电子出版物的收费网上资源，供集团订购。我国若干高校的图书馆（如清华、复旦） 已经订购，那些学校的师生可以在所在校园自由下载。象它的系列丛书GTM,UTM 自1997年来的教材几乎全部可以下载。1.5 本书的目标2007年复旦大学数学学院和校图书馆外国教材中心组织一批力量对国外大学的的数学教材进行调查研究。 选择一部分优秀的教材进行介绍，旨在帮助国内大专院校师生和自学数学的同志选择合适的外国教材，对于最常用的一些教材，我们对它们在国外的使用情况作了统计和调查。所有的书都由熟悉该书内容的教师书写介绍， 其中有不少书在教学中使用过，对于书的特色和难易程度都有较明确的评论。我们相信我们的选书标准是高的，所以数量相对来说不大，所覆盖的范围也并不是太广。对于我们选中的书籍，大部分都作了简评，结合中国高校的情况列出一些使用要点。为了使读者更加全面地了解所选的教材，我们还选载了一些国外读者的比较中肯的评论，不光是讲优点的评论，也有很多讲书中的不足之处的，评论者多数是使用过该书的教师和学生。在互联网上可以查到不少热心数学人士的网页上的一些读书指导，提供一些数学好书的清点，大部分都比较简略， 由于是个人行为，收集的面也有一定限制。我们尝试组织一批精通业务的专家合作也提供一些对国内师生 更有用的调查资料，起个抛砖引玉的作用。由于时间和人力物力的关系，这一次选的书的数量和范围有限， 我们希望这只是这个工作的一个开头，以后根据条件是可以大大扩充本书的内容的。本书分两个部分，第一部分是非数学专业（即公共基础课）的数学教材，第二部分是数学系的教材，它们又按不同 的数学分支进行编排。本科生和研究生教材就不分了，因为它们间也没有非常明确的界线。对于大部分书 除了一些基本资料外都有以下几项参考指标：适用范围，预备知识，习题数量，习题难度，推荐强度（最高是10）希望这些指标对读者选书提供帮助。2 非数学专业的数学教材在国内外高校中，高等数学是占课时最多的课程之一，因为几乎每个系每个专业都多少要学点微积分， 或许还要学线性代数、概率统计、微分方程等。这些数学和数学专业所学的数学有很大的不同，它们所强调 的是计算和应用，而数学专业的学生需要学系统的理论并且训练证明定理的能力，所以数学专业的 数学书籍有一定深度，不适合于工程类、医学和文科各专业的学生使用。理科有些专业（如物理、力学等）对数学的某些分支要求比较高，也可以使用数学系的教材。我国高校的高等数学按深浅一般分几类，有的学校分3类，有的分4类，最低的一般是文科数学，最高 的是对物理系开设的数学，国外大致上也是如此。我们对美国的微积分教材和线性代数教材分别进行了调查研究，各自精选了十本左右有影响力或 使用院校比较多的教材向读者介绍。我们列举的使用院校是根据非完全的统计，仅供读者选书时参考。/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心2.1 微积分微积分是大学数学最基本也是最重要的课程，可以毫不夸张地说高中的数学教育的目标就是为微积分 打基础。从历史上看，牛顿发明微积分是为了解决当时物理学不能解决的问题，这形成了数学的一个飞跃，随着数学的发展，为微积分建立严格的理论基础成为一个迫切的任务，经过数学家们不懈的努力， 在19 世纪就形成非常严格的微积分理论，被称为数学分析。现在国内大学数学系学的“微积分”大部分就 叫“数学分析”。而非数学系大学生学的“微积分”则含在一门叫高等数学的课程中。在英语国家中是没有Advanced mathematics 这门课的，他们的Calculus 课程对应我们的高等数学，他们的Mathematical analysis 或Advanced calculus 对应我们数学系学的数学分析。还有些书名含Real analysis 这词组，这就要看书的内容了，有可能是数学分析，也可能是比数学分析更深的实变函数论。如果一本书名是Vector analysis, 则它就是讲多变量的微积分，相当与我们高等数学后半部分的内容。1) Calculus, third edition作者：Hughes-Hallet,Gleason,McCallum et al.出版商：John Wiley & Sons, Inc. (2002) ISBN 0-471-40826-3页数：623适用范围：理工类大学本科生微积分教材预备知识：高中数学习题数量：大

习题难度：低推荐强度： 9.3使用学校：Duke University, University of California at San Diego, Northern Michigan University, University of Cincinnati, University of Californiaat Merced, Virginia Polytechnic Institute and State University, University of Massachusetts at Amherst, Florida State University, GeorgiaInstitute of Technology, Oklahoma State University, Sonoma State University, St. Louis University, Winona State University, University ofRhode Island, Berea College, The University of Arizona Jacksonville State University, Willamette University, Arizona State University,Western Oregon University, University of South Carolina, Marquette University, Western Washington University书评：在美国非数学专业的微积分教材中Thomas的Calculus统治了很多年，80年代我在美国任教时这是指定的标准教材。虽然该教材不断修改和再版，但这么多年由一本教材垄断并非正常。Hughes-Hallet,Gleason,McCallum 等一批有志于微积分教材 改革的人士在新世纪合力推出这本全新的微积分教材。本书的内容和传统的微积分没有任何不同，但是更突出重点。象交响乐的一个乐章里有陈述部、展开部和再现部一样，本书对每一个最 重要的概念从不同的角度反复讲解，这种一唱三叹的方法很容易让初学者抓住重点。另一个特点是降低微积分计算部分的要求而 重视对基本概念和方法的正确理解，作者认为用大白话 (plain English) 来理解数学比记住一些公式更重要。所以，象极限、导数、积分 这些概念的第一次出现都用大量的精心设计的文字、生动的实例和图象来解释，然后再用一系列实例来展示其威力，最后再在选学内容中再写精确的定义。本书的另一特色是习题的多样性，应用题的数学很简单，但涉及各科学，特别在生物、医学、经济和人文科学中的应用的习题数量很多，这是以前的微积分教材所没有的，在学习和做题过程中学生可以在早期就建立数学建模的思想。本人在2003年在美国使用此教材教过一学期，学生程度参差不齐，即使基础较差，凡用功的学生都能达到本教材的基本要求。经过实际使用，本人体会到作者在此教材上倾注的心血。错误极少，虽然是多人合作，但章节间的衔接非常自然。本书还配有习题详解Instructor\'s solutions manual, 760页和概念测验Concept tests 306页。目前已被包括哈佛大学、杜克大学在内的一批大学定为大一微积/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心分教材。 （杨劲根）国外评论摘选i) This is not the classic calculations approach to the subject. It is a totally new way of thinking and mastering the subject with out havingto do page upon page of number crunching. Use this book along with a graphing calculator and you too can learn to literally see whathappens when equations are manipulated. A begining student conceptually gains an understanding of the subject with out getting boggeddown in plugging and chugging and derivations. It\'s written in plain ) The authors of this text dislike the \"plug and chug\" methods of other texts, possibly necessitating an instructor more strongly than otherbooks. The book stresses graphs and \"real life\" applications, making it more realistic and less abstract than other Calc books may ns useful formulas and rules on inside covers and selected answers section at the back. Overall a great book to use in class.2)书名:Calculus 系列书作者:James Stewart出版商:Thomson/Brooks/Cole适用范围:非数学专业大学一年级预备知识:高中数学习题数量:大习题难度:从容易到中等都有

推荐强度: 9书评:Stewart 的教材以前我不了解，这次调研外国高等数学教材的过程中发现了他的书 的使用率是在各同类教材中名列前茅的。仔细查查，他一个人大约写了八本不同的微积分教材， 应该是针对不同对象的，或者说分A,B,C,... 类的。我翻阅的一本是Calculus 第五版，一千一百多页，包含多重积分和二阶常系数线性微分方程。 我的印象是：这是一本朴实无华的相当标准的教材，包含了理工科一年级大学生应该学习的所有 内容，在很多关键章节的写法是很细致的。应用题很多，但以物理中的应用为主， 多少算是还微积分的本来面目，很多章节后还有一些供学生培养独立研究能力的课题，如 彩虹的原理，电影院里座位的视角分析等。在单变量微积分和多变量微积分之间插了几章关于空间 解析几何，其数量比较恰当。下面列举这个系列中的五本书的使用院校情况。 （杨劲根）i) Calculus : early transcendentals （2003 第五版）使用学校(30多所)：University of California at Berkeley, Columbia University, Saint Joseph\'s University, Louisiana University, Salisbury University,University of Minnesota, Rensselaer Polytechnic Institute, California State University at Channel Islands, University of Massachusetts atAmherst, San Joze State University, Michigan State University, Tufts University, University of Michigan at Ann Arbor, University ofVirginia\'s College at Wise, University of California at San Diego, Loyola University at Chicago, Tennessee Technological University,College of Charleston, Asheville Buncombe Technical Community College, University of West Georgia, Georgia University at SouthBend, Purdue University, University of Washington, Florida State University, California State University, Indiana University, SoutheastGrinnell College, Carnegie Mellon University, Vanderbilt University, Dartmouth College, California State University at Dominguze Hills,Idaho State University, Athabasca University in Canada, The University of Texas At Austin, University of Southern California, Universityof Pennsylvania, California Polytechnic State University/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心ii) Single variable calculus （2003 第五版）使用学校(20多所)：Hunter College of CUNY, Louisiana State University, Florida Atlantic University, University of Illinois at Urbana-Champaign, College ofCharleston, Johns Hopkins University, Wake Forest University, Emory University, Florida State University, California State University atStanislaus, Boise State University, University of Washington, The University of Western Ontario, Stony Brook State University of NewYork, College of the Holy Cross, San Diego State University, Oberlin College, University at Albany, State University of New York,Loyola College in Maryland, University of Missouri-Columbia, Saginaw Valley State University, Duquesne University, Rivier Collegeiii) Multivariable calculus （2003 第五版）使用学校(20多所)：Harvard University, Hobart and William Smith college, California state University at Dominguez Hills, University of Minnesota,University of Michigan, University of Connecticut, Rustgers the State University of New Jersey, University at Buffalo, Temple University,University of Minnesota at Duluth, Brown University, Kennesaw State University, Klarkson University, Binghamton University, BoiseState University, University of Colorado at Colorado, Springs University of Minnesota, Morris University of Rhode Island, Stony BrookUniversity, Oberlin College, University of California at Irvineiv) Calculus : concepts and contexts （2003 第三版）使用学校(近20所)：Mount Saint Mary College, Whittier College, University of Richmond, The University of Kansas, Kalamazoo College, Howard University,North Carolina State University, Northeastern University, Graceland University, Washington University in St. Louis, Wright StateUniversity, Stanford University, University of Minnesota, University of Tennessee, Northwestern University, University of Cincinnati,Utah State University, Oklahoma State University, University of Wyoming3)书名：Applied Calculus作者：Deborah, Hughes-Hallet et al.出版商：John Wiley & Sons, Inc. (2006) ISBN 0-471-68121-0适用范围：生命科学、管理和文科各类大学本科生微积分教材预备知识：高中数学习题数量：大习题难度：低推荐强度：9.2使用学校：Macalester College, Temple University, Indiana University,Purdue University, University of Rhode Island, Idaho State University,University of Sioux Falls,Loyola University Chicago国外评论摘选i) APPLIED CALCULUS, 3/E brings together the best of both new and traditional curricula to meet the needs of today’s students. Theauthor team’s extensive teaching experience and proven ability to write innovative and relevant problems has made this text a true/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心bestseller. Exciting new real-world applications make this new edition even more meaningful to students in management, life and socialsciences. This book will work well for those departments seeking a middle ground for their instructors. APPLIED CALCULUS, 3/Eexhibits the same strengths from earlier editions including the “Rule of Four”, an emphasis on concepts and modeling, exposition thatstudents can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativityand variety, continue to motivate and challenge ) This is a magnificent calculus book. It is aimed at students in business, the social sciences, and the life sciences. This is done by firstthe examples and problems. But perhaps even more important the wording of the text is such that these students will understand what theyare trying to convey and to clearly show them how calculus can be used to solve problems in their particular the beginning of the book, three pages of the Preface, the applications discussed in the text are listed by: Business and Economics, LifeSciences and Ecology, Social Sciences, Physical Sciences. Under these headings are subjects like: Value of a Car, AIDS, Cancer Rates,Abortion Rate and so on. These are subjects that will have some interest and applicability to students rather than the old traditionalproblems like water flowing into and out of a bucket that used to be the mainstream of teaching calculus.4)书名：Advanced Calculus, 2nd Edition作者： Patrick M. Fitzpatrick出版商： Brooks/Cole (2005),机械工业出版社影印

页数：590适用范围：数学系与理工科其他专业的本科生预备知识：高中数学习题数量：较大习题难度： 具有一定难度推荐强度：9.3

使用学校： University of Northern Iowa, University of Alberta, University of Colorado at Denver, University of Central Florida, Virginia StateUniversity, San Diego State University, University of Rhode Island, University of California, University of Colorado, University of CentralArkansas, Fayeiteville State University, Brigham Young University, University of Calgary, Oregon State University, University of Illinoisat Urbana-Champaign, University of Wisconsin at Whitewater

[作者简介] Patrick M. Fitzpatrick拥有格兰特大学博士学位，是纽约大学科朗研究所和芝加哥大学的博士后，1975年进入马里兰大学College Park分校任教，现在是数学系教授和系主任，同时它还是巴黎大学和佛罗伦萨大学的客座教授。他的研究方向是非线性泛函分析，在该方向著有50多篇论文。 书评: 本书以清晰、简洁的方式介绍了数学分析的基本概念：第一部分讲述单变量函数的微积分，包括实数理论、数列的收敛、函数的连续姓和极限、函数的导数和积分、多项式逼近等；第二部分把微积分的概念推广到多维欧几里得空间，讨论多变量函数的偏导数、反函数、隐函数及其应用、曲线积分和曲面积分等。 数学分析已经根植于自然科学和社会科学的各个学科分支之中，微积分作为数学分析的基础，不仅要为全部数学方法和算法工具提供方法论，同时还要为人们灌输逻辑思维的方法，本书在实现这一目标中取得了引人注目的成果。本书一方面按传统的和严格的演绎形式介绍微积分的所有主题，另一方面强调主题的相关性和统/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心一性，使读者受到数学科学思维的系统训练。 本书的一大特点是除了包含必不可少的论题，如实数、收敛序列、连续函数与极限、初等函数、微分、积分、多元函数微积分等以外，还包含其他一些重要的论题，如求积分的逼近方法、Weierstrass逼近定理、度量空间等。例如本书专门用一章讨论度量空间，从而把在欧几里得空间讨论微积分时使用的许多概念和导出的结果扩展到更抽象的空间中，引导读者作广泛深入的思考。 另外，与第一版相比，第二版增加了200多道难易不等的习题。全书贯穿了许多具有启发性的例题，并且本版还为教学考虑进行了许多实质性的改动，例如将选学材料与前后内容的关联度降到最低，单独放置，既不影响教学和读者自学的进度，又能让读者集中攻破一些难点，这样使得全书的叙述更简洁、更自然。本书曾于2003-2004年作为马里兰大学教材。 （高威） 国外评论摘选 i) A great book. Starts with two very good chapters on linear algebra, adapted to the needs of calculus, and then proceeds to introduceyou to the contemporary way to do multivariate calculus, including existence theorems connected to completeness. Very thorough treatmentof integration, including integration of forms on manifolds, up to the Stokes theorem, built upon a fine chapter on differential manifolds,exterior differential forms, riemannian metrics, etc. Good illustrations and beautiful typesetting add to the joy of reading it. Plenty ofexercises and chapters on applications to physics and differential geometry.

ii) This is the best book on mathematics I\'ve ever come across. The superbly written text succeeds in guiding the reader in an easy,clear-cut, graceful way through the realm of what he modestly calls \"Advanced Calculus\". Some minor misprints are to regret, but theydon\'t even come close to blurring the fact that this is - no doubt about that - an unsurpassable masterpiece.

iii) As Spivak\'s \"Calculus on Manifolds\", this book is labeled with a very modest title. It should be something as \"All you wanted toknow about analysis on manifolds but were afraid to ask\". This book is a must-reading for the analyst. It covers everything from the mostbasic vector space concepts up to the fundamental theorems of classical mechanics, running through multivariate calculus, exteriorcalculus, integration of forms, and many topics more, always keeping a very modern and rigorous style. The undergraduate may find it a little difficult, but the effort is worth it. For the graduate student and the working mathematician it isan almost-daily reference.

iv) This book is out of print, but is available from Sternberg\'s website. Search on his full name at Google.

5)书名：Calculus: early transcendental functions, 4th ed.作者： Ron Larson, Robert P Hostetler, Bruce H Edwards出版商： Houghton Mifflin, Boston (2007) ISBN 0-618-60624-6适用范围：对数学要求不高的专业的本科生微积分教材预备知识：高中数学习题数量：大习题难度：低推荐强度：9.2

使用学校 ： Houghton Mifflin College, Chandler-Gilbert Community College, South Carolina Technical College, Penn State University, TheBehrend College, University of Colorado at Denver, Alamo Community Colleges, Johnson County Community College, The Community/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心College of Baltimore County, Emory University, Jackson Community College, Michigan State University, Tri-country Technical College,Rivier College, Rutgers: the State University of New Jersey, Trident Technical College, Mississippi College, Jacksonville State University,Collin County Community College, District Hobart And William Smith Colleges, Oakland Community College 国外评论摘选 i) I have taught calculus for over 20 years, from about half a dozen books: Thomas, Swokowski, Anton, Stewart, and others. Two yearsago our university adopted the 6th Edition of Larson. As a pedagodical tool, this text is head and shoulders about all the others. The textuses abundand graphics, a clear design, concise writing, thoughtful examples, and carefully crafted exercises to make calculus accessible tostudents. I have never had so many students volunteer compliments about a text. This text is simply the \"best of the best.\"

ii) This textbook is much better than the one that is currently a bestseller (Stewart). It explains concepts and examples clearly, showingevery step so that we don\'t have to wonder how did something happened. It is best suited for someone who doesn\'t have a lot of time tospend on reading long discussions and for someone who doesn\'t want to go too deep into material and wants to quickly getthe concepts. But don\'t think it is some Dummies or Made Easy guide, it is still a textbook that takes time to read. What I like most aboutthis book is that the authors\' style of writing is very clear and friendly: Not many big words or abstract phrases.

6)书名：Calculus, 9th ed.作者： Saturnino L Salas, Einar Hille, Garret J Etgen出版商： John Wiley & Sons (2003) ISBN 0-471-23119-3适用范围：数学系、物理系或力学系本科生微积分教材预备知识：高中数学习题数量：中等习题难度：中等推荐强度：9.2

使用学校： Clark University, University of Houston, James Madison University, Johns Hopkins University, University of South Florida, GeorgiaInstitute of Technology, Athabasca University in Canada, University of Washington, 台湾国立成功大学, New York University, TheUniversity of Texas at Austin, Georgia State University, University of Chicago, University of Illinois at Urbana-Champaign, New YorkUniversity, National University of Ireland at Galway 国外评论摘选 i) This is a superb textbook and it\'s easy to see why the book is in its ninth edition. What I really enjoyed (yes, I know this may sounda little incongruous in relation to calculus) was the step-by-step build-up of knowledge with good, clear examples. Also, for the problemsat the end of each section, all the odd problems have solutions, so one can get some practice (something that is unfortunately rare formany textbooks). Before going through this book, I had minimal exposure to calculus and what I had seen wasn\'t very favorable. This book was a keyreason why I now really enjoy the subject and feel very comfortable in this area.

ii) I used this book in my first engineering calculus course. The professor was incredibly theoretical and did not teach from the bookwhich made matters somewhat difficult. However, he was showing us the meaning of math which I found refreshing. This book serves its/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心purpose as one which teaches the mechanics of solving problems but very little in developing an intuitive feeling for mathematics. I mustadmit that the multitude of exercises were very helpful in getting comfortable with difficult mechanical problems. For single variablecalculus it is a standard book with good examples, excellent diagrams, and some applications. Getting into multivariables, the ideas are notconnected well and seem segragated from the rest of material. I guess as a brief overview, it makes its point but should not be used as atext for multivariable calculus. If you are interested in theory I recommend Apostol\'s Calculus which covers a great range of material withrigorous foundation. As far as exercises go, Michael Spivak\'s Calculus is quite challenging and will keep you occupied for months. All-in-all, a great book for brush up and single variable material but not to be used for higher dimensional analysis.

7)书名：Calculus, 3rd ed.作者： Monty J Strauss, Gerald L Bradley, Karl J Smith出版商： Prentice-Hall (2002) ISBN 0-130-95005-X适用范围：对数学要求较高的专业的本科生微积分教材预备知识：高中数学习题数量：中等习题难度：中等推荐强度：9.2

使用学校： The University of Texas at Arlington, Texas Tech University, Devry University, Northwestern University, Utica College, Rutgers: theState University of New Jersey, Whatcom Community College, University of Wisconsin at Green Bay, King\'s College, University ofLondon, Dartmouth College 国外评论摘选 i) I learned calculus from this book, and i think that as a text it is excellent. I learned very little from my lecturer, and almost 90percent of my three good grades in calc 1,2 and 3 can be attributed to the pages of this book. On the other hand, by the end of the yearmy book had nearly fallen apart.

ii) Many people say that this book is bad. On the other hand, I think is very challenging. The exercises are not as simple as in othercalculus textbooks. The book explains everything well and provides you with many examples. I am a math major and this book has beenreally helpful.

8)书名：Calculus, 9th ed.作者： Dale E Varberg, Edwin J Purcell, Steven E Rigdon出版商：Prentice-Hall (2007) ISBN 0-131-42924-8适用范围：理工类本科生微积分教材预备知识：高中数学习题数量：中等习题难度：中等/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心推荐强度：9.2

使用学校： University of Wisconsin at Madison, The University of Chicago, Iowa State University, University of South Carolina, California StateUniversity at Northridge, Syracuse University, Worcester Polytechnic Institute, Oregon State University, Saint Louis University, The OhioState University, Southern Oklahoma Technology Center, Southern Illinois University at Edwardsville, Saint Louis University, DenisonUniversity, York University, The University of North Carolina at Chapel Hill, Virginia State University, 台湾国防管理学院

国外评论摘选 i) When I was 15, this was the book that I taught myself Calculus from. Now that I\'m a professor, this is the book that I use to teachCalculus. In this review I will give the pros and cons of using this book from both a student\'s and teacher\'s perspective. A Student\'s Perspective When learning Calculus, I read every page of this book and did every problem. Students will complain that examples and discussion ineach chapter seem inadequate to do all of the problems at the end of the section. I feel that this is part of the design of this book. Theproblems are intended to be instructional. Indeed this book has a corresponding student solutions manual that helps students to check theirwork and see if they are \"getting it\". The problems in the book range from extremely elementary up to moderately challenging. If, insteadof instructional problems, this book had given enough examples and text to explain all of the ideas, it would have to be over 2000 pageslong. Students should think of the problems in each section as being part of the instruction instead of problems to test previously acquiredskills. When teaching myself from this book, I was able to do all but a few of the problems. Granted I had to spend a considerable amount oftime struggling with some of them, but for a talented and dedicated student, every problem in the book is accessible and most areextremely instructive. I should also mention that the book is very well written. Having never actually read a math text book from cover tocover back then, I didn\'t have too much problem tackling this one. It\'s very rare that a math text be thorough, informative, and easy toread. This one manages to be all three. The main drawback of the book is that the students solutions manual is absolutely essential and will be an additional cost. Even ifmoney is tight, as it often is for students, make certain that you buy this manual. A Teacher\'s Perspective As I said above, the problems in this book are intended to be instructional. For this reason it is imperative that a teacher not just lecturefrom the text and examples, but dig into the problems and carefully choose the most instructive ones for in-class presentations orhomework assignments. If you only lecture from the text and examples, you\'ll only be teaching your class a small fraction of what thisbook has to offer. If you use this for a course, do as many examples as you have time for. I dedicate one lecture per week to doing nothingbut working problems. It might be best to work though the even numbered problems for your class, as the odd numbered ones all appear inthe student solutions manual. The layout of the book is a little bit flawed. This book is aimed at three semester Calculus sequences in state universities and liberal artscolleges. It is not a meant to challenge exceptionally bright students. For this reason parts of chapter 2 seem inappropriate- specifically thesections on the rigorous definition of limits and continuity. If you\'re teaching a calculus course to non-math majors at modest universities,why would you force students to wade through the muck of mathematical proofs of continuity and existence of limits? In my experiencethe students absolutely hate this part of the course and gain nothing from it. If you have a few bright kids in your class, you can work withthem on an independent study of the more theoretical areas such as this. Also, there are few chapters in the book that are out of place. Forexample, the chapter on integrating to find the volumes and surface areas of solids of revolution comes way too early while the chapters/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心on transcendental functions, inverse functions, and L\'Hopital\'s Rule come way to late. Overall the presentation of new ideas is very good in this book, with one notable exception. The book introduces the natural logarithm(ln x) through it\'s definition in terms of the antiderivative of 1/x. From there it uses the inverse function theorem to derive the exponentialfunction and it\'s properties. I, and my students, find it more natural to define the Euler number, e, in terms of continuously compoundedinterest, and then derive the natural logarithm and its properties from the exponential function. It\'s a matter of taste, but the later approachseemed more lucid to my students. You may want to supplement your lectures in this way. One of my favorite features of this book is that not only does it cover all the material from a traditional three semester Calculussequence, but it also has chapters on analytical and numerical solutions to ordinary differential equations as well as an appendix containingmore theoretical material for brighter students. If you find yourself teaching an unusually talented bunch of kids, the appendix onmathematical induction as well as the aforementioned sections on ODEs and proofs of continuity and existence of limits can make greatsupplements to challenge those eager to dive into mathematics.

ii) Ok, let me start by stating that because this is \"the shortest mainstream calculus\" text out there, it does not mean this has less would seem to be so, but this is the exception to the rule where shorter texts means dumber texts. Explaining mathematics is a bit of anart: you have to choose in what sequence things are to be layed out to the reader, so this means you have to choose how you will relatethe explanations to one another. The Purcell I read (the 1st edition - it was my dad\'s) is quite masterfull at that. Often, when my collegestandard text got the explanations too verbose and confused, I looked for my Purcell copy and there it was, crystal clear: short,mathematically rigorous, to the point.2.2 线性代数

在下面选的8本广泛使用的线性代数中，Hoffman-Kunze 的教材最深，适合于对线性代数要求高的专业使用，其次是 Strang 的Linear Algebra and its Applications. 其它教材一般都比较浅。

1)书名：Linear Algebra and its Applications作者： Gilbert Strang出版商： Thompson Learning, Inc. (1988) ISBN 0-15-551005-3

页数：505适用范围：理工科大学本科基础数学二学年的教材预备知识：微积分习题数量：大习题难度：容易到中等推荐强度：9

使用此书的部分院校 Massachusetts Institute of Technology , University of California，University of Delaware，Indian Institute of Technology,Bombay，University of Maryland，State University of New Jersey，Tulane University，State University of New York Institute ofTechnology，SUNY Institute of Technology，Rivier College，New York University，Duke University，University of Colorado at/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心Denver，Yale University，University of Houston，Loyola University，Drexel University，Tufts University，StanfordUniversity，University of Regina，North Carolina State University，Brown University，Dartmouth College，University ofWashington，Georgia Institute of Technology，Pennsylvania State University

书评： 本课程是麻省理工学院数学系为全校设置的王牌课程之一，至少已有30年的历史。 作者亲授的全套课程录象已经在 MIT的官方 网站上免费下载。本书从实用的角度包含了线性代数的全部内容，对基本概念的理解方面作者不惜用较多的文字 作解释，并且几乎手把手地教读者学会使用一些常规的线性代数方法。然而，本书决不是一本“傻瓜书”，它对 读者的预备知识虽然不高，对智商还是有一定的要求，比较适合我国重点大专院校使用。 我观看过该课程的部分录象，视频和音频质量不很高，有一定的英语听力的人可以听请每句话。 看录象比看书更有启发性。顺便提一下，Strang 教授在 MIT 开设的另外两门应用数学课程 18.085,18.086 也有录象，可在 http: 上找到。 在Amazon 网站上此书有67 篇读者评论，五颗星的31篇，一颗星的19篇（如下面所摘录的第5篇），中间的很少， 这在一定程度上说明了这本书的特点， 同时也提醒读者这本书是不是适合于你。 （杨劲根）

国外评论摘选

1) 就Linear Algebra 而言,我还没看到比 Gilbert Strang 的书更好的书。他的 Linear Algebra and Its Application 虽然旧，但经典，就像 Rudin 的书一样，难以被替代。他有一本比较新的书，Introduction to Linear Algebra，1993 年的。如果想深入，那么他的另一本巨著 Introduction to Applied Mathematics 则最适合不过了，这本书把 linear algebra 跟其他数学分支结合在一起，配上他启发性很强的描述，感觉好像在看小说，新奇，激动，期待。 现在 Gilbert Strang 的两门课 linear algebra 和 applied mathematics 都被 MIT 放到网上了，有全部的上课现场录像，还有很多相关的学习资料，上课的录像可以在线看或是下载下来看。 Gilbert Strang的讲课风格跟他的写作风格一样，充满睿智和启发性，还带点情节， 比起大部份的数学教育者沉闷的讲课模式和呆板的板书，Gilbert Strang的课很难让人睡着，当然前提是英语听力水平不能太差。建议去看看，感受一下大师的风采， 同时也感受一下 MIT 的气氛。

2) The Mathematics Department used linear algebra books by Howard Anton, Bernie Kolman, and David Lay for many years. I took achance two years ago and adopted Gilbert Strang\'s linear algebra book for a large engineering course. We used the second edition ofIntroduction to Linear Algebra, Wellesley-Cambridge Press. Several colleagues said it couldn\'t be done, but students and the instructorsurvived nicely to see another day. Many students said they enjoyed the book. Gilbert Strang\'s enthusiasm for the subject matter comesthrough in the text and students find it a refreshing change. Another strong point is an extensive set of problems. Many problems probe thesubject in a way that requires students to think about linear algebra. Routine problems are not forgotten. This is good. Students can workon problems that help them put the subject in their own voice. A third strength is the layout of topics. Matrix multiplication andelementary row operations from a matrix viewpoint are developed first, and this provides an opportunity to discuss row reduction, matrixinverse, and the decomposition with little extra effort. Other standard subjects follow in order and orthogonality arrives early. Computationis not ignored and the text is organized so that computation is optional. LU I worked to adapt my notes and style to the text. After a while,I discarded my old notes and discovered freshness in the subject that I had not known for some time. Enrollment in the course forengineers has increased dramatically in the last two years. More than 250 students studied linear algebra and matrix theory at DrexelUniversity in the spring of 2005. All day students taking linear algebra at Drexel used Gilbert Strang\'s book. I plan to use it again. Herman Gollwitzer,Mathematics Department,Drexel University

3) I had the opportunity to learn linear algebra from Prof. Strang\'s online video lectures at MIT. This book will be a good comapanion/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心to those lectures. All of you who hate Linear Algebra should take it from me : Watch the lectures along with the book, you will do nowrong. Strang\'s insights as he lectures, will make you fall in love with Linear Algebra. Rajesh Kumar Venugopal, Syracuse, New York

4) 這是本非常適合自修的書，書中的用字都是很基本的單字，讓英文不是很好的我也能輕鬆地閱讀； 内容由浅而深，观念清析，圖示更是一絕，封底有一個解釋 linear transformation 的圖，完全表達出 linear transformation 的精髓，令我嘆為觀止，解釋SVD 的圖也同樣令我印象深刻。另外，這本書在 2003 年出版了第三版 也已經在我必買的書單之中了。

5) Strang tells us in the preface that linear algebra is a beautiful subject, and he is correct. Yet he seems intent on strangling itstheoretical beauty with a matrix based approach to vector spaces, and an ugly preoccupation with ${mathbb R^n.$ It\'s clear that this bookwas not written to be either a lucid explanation of how to use linear algebra, nor was it intended to be an aesthetically pleasing expositionof theoretical linear algebra. It was written somewhere in between, and it is an unhappy medium. If you are interested in a theoreticaltreatment of linear algebra, there are sorrowfully few good texts available. The title of Axler\'s \"Linear Algebra Done Right\" is a result ofthis fact, and if you are seeking a mathematically pure treatment of the subject, that book is a much better choice. If you\'re not interestedin the theory, but only the applications, you should still be able to find a much better text than Strang\'s.

2) Introduction to Linear Algebra 作者：Gilbert Strang 出版信息： 2003, 3rd ed. Wellsley Cambridge Press

使用学校： Case Western Reserve University， College of the Redwoods，University of Houston，University of Miami，University ofMinnesota，University of Colorado at Denver，Cornell University，Massachusetts Institute of Technology，Loyola University，DrexelUniversity，University of Maryland，Columbia University，Brown University，Rutgers, The State University of New Jersey，MichiganInterdisciplinary and Professional Engineering (InterPro)，University of Nevada, Reno，University of Alabama at Birmingham，Collegeof the Redwoods，Wellesley College，Mount Holyoke College，University of Wyoming

i) People say that mathematical truths never change, and that\'s true enough. New concepts, applications, and techniques keep emerging,though, so math teaching needs to keep up with the times. Strang has done an outstanding job of keeping this book current and relevant. It\'s not a mathematician\'s math book - this is aimed at people who need results and needs computational techniques more than theyneed crystalline theorems. That\'s why it\'s so helpful to see applications like Markov models, Kirchoff\'s laws, and Google\'s analyses of theweb. It\'s also helpful to see examples worked in Mathematica and MATLAB, the tools of choice for desktop exploration of numericalsystems. It\'s startlingly easy to come up with a 100x100 system of equations, and just nuts to try to solve it by hand. Strang assumes some amount of calculus in this book, something that other books on linear algebra sometimes skip. That raises the barfor the readership, but also opens up topics like change-of-basis in function space, including Fourier analysis. It also allows differentialequations to be addressed as linear systems. Even without calculus, though, a reader is exposed to the singular value decompostion, QRand other matrix decompositions, and considerations in performing the computations. I found a few oddities, such as the description of amatrix\'s condition number. That has great physical meaning when it\'s taken as the ratio of the matrix\'s highest and lowest eigenvalues, butStrang gives a definition that I found less :///guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心 Such oddities are rare, though. Even though this book covers many topics, its emphasis is on clear and applicable presentation. Irecommend this to anyone studying linear algebra or who, like me, has to brush up on basics not used in many years. ii) Gilbert Strang is a very experienced teacher of Linear Algebra, and this book is written as a text based on his MIT linear algebraclass. Math majors will not find the \'definition-proposition-lemma-theorem-proof-corollary\' treatment here. Instead Strang, aware of theneed to teach non-math majors the subject, explains linear algebra in a simple but effective way --examples, diagrams, motivations. Thisbook is one of those with which you can skip class the whole semester and get good grades (but don\'t do it! get your education in theclassroom).

3) Linear Algebra and its Applications 作者：David C. Lay 出版信息： 3rd ed. Addison-Wesley

使用学校： Ohio Northern University，University of Kentucky，University of North Carolina at Charlotte，University of South Carolina,University of Memphis, Agnes Scott College，Alamo Community Colleges，Bates College，Boston University，Florida StateUniversity，Michigan Technological University，Salisbury University，Stony Brook University，University of Maryland，University ofConnecticut，University of Massachusetts Amherst，University of Missouri-Rolla，University of Oregon，University of Texas AtAustin，Boise State University，Brigham Young University，New Mexico State University，New York University，San Jose StateUniversity，Yale University，Westmont College，Rivier College，University of Delaware，University of London, University ofRichmond, University of Rochester, Eastern Mennonite University, Princeton University, University of Colorado at Denver, CityUniversity, Cornell University， University of Nebraska at Omaha

国外评论选摘 i) This text is a dream to read compared to many other mathematics texts. Lay\'s writing style is clear, and he rightly stays away fromusing wording that distracts the reader from the theory he presents. Mathematical notation is introduced before it is used, and proofs areplaced in an appendix. Overall, this is a very good book for undergraduate study. It won\'t carry you through graduate classes, but it mightbe useful as a support book if you have a weak background in the topic. Math majors who love concise formalism and extended proofsshould stay away from this book. Engineers, business, physical science, and social science majors will find the text very helpful. ii) Math texts are notoriously poorly written and difficult to follow for the typical undergrad without the guidance of a professor. Thisbook is an exception to the norm. Not everything, but most things, are presented in a way that most students will be able to absorb ontheir own.

4) Elementary Linear Algebra 作者：Howard Anton 出版信息： 9th ed. John Wiley & Sons

使用学校： The City College of New York，University of Texas at Dallas，Hartnell College，Rivier College，UC Santa Cruz，University ofColorado at Denver，McGill University，Athabasca University Canada\'s Open University，Victoria University of Wellington, NewZealand，Brandon University，Louisiana State University，Indiana University-Purdue University，State University of New YorkCollege at Brockport SUNY Brockport，University of Manitoba，The Richard Stockton College of New Jersey，Florida Atlantic/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心University，Saint Vincent Colllege，University of East Anglia，Norwich University，University college Dublin，CardiffUniversity，University of Essex，University of Calgary，Durham University，Queens College，wellesley College，LehmanCollege，Cayuga Community College

国外评论选摘 i) I used Anton in my linear algebra class a few years back and I have referred to it often since. Anton\'s approach is to introduce thenotation and basic tools, i.e. vector and matrix arithmetic, within the intuitive geometric settings of the Euclidean plane and space. Oncethe basic concepts of Euclidean vector spaces have been mastered, Anton moves into abstract vector spaces, linear transformations, andeigenvectors. One chapter is spent on complex matrices, and another chapter deals with numerical issues and least-squares only topic which is noticably missing is the singular value decomposition, but other than that, Anton is a remarkably complete definitions and theorems are clearly presented, along with the motivating intuitions. The exercises at the end of the chapter sectionsare a nice balance between computational and theoretical problems. Overall I highly recommend Anton as a first linear algebra text. ii) The Anton book appears to be the standard in teaching undergrad LA, but I personally didn\'t like it very much. Part of the problem isdue to several misprints in the early chapters. Some of the definitions of basic concepts are confusing at best, wrong at the worst. I foundmyself relying on the Hubbard-Hubbard \"Vector Calculus, Linear Algebra, and Differential Forms\" to get through the course. Theexplanations were more concise and easier to understand. If you\'r eteaching yourself, Hubbard-Hubbard is the way to go.

5) Elementary Linear Algebra: Applications Version 作者：Howard Anton, Chris Rorres 出版信息： 9th ed. John Wiley & Sons

使用学校： Murray State University，Stetson University，Athabasca University，The University of Tennessee at Martin，University ofToronto，City College of San Francisco，Drexel University，Eastern Michigan University，Towson University，University ofWales，University of Iowa，Stony Brook University，McMaster University，York University，University of SouthernIndiana，Binghamton University，University of Melbourne，University of Stirling，College of the Canyons，MiddleburyCollege，Elon University，Kennesaw State University，University of Manitoba，University of Colorado at ColoradoSprings，University of Guelph，University of West Georgia，University of Victoria，Chaffey College，Wayne StateUniversity，Rowan University

国外评论选摘 i) 這本書比較簡單，比較適合線性代數基礎比較差的學生，可當成入門的書籍，這本書的另一個重點在於它有三分之一的篇幅在談線性代數在各個領域的應用，可讓你看到線性代數抽象的數學背後廣大的應用。 ii) The book starts by describing matrix manipulations and determinants. These are very tangible things to most maths ingly, explaining how to take determinants or to invert a matrix lets you build confidence in your knowledge. Also, these topicslends themselves readily to many problems for you to do. After this, the book heads into more abstract territory. Null and range spaces and the rank nullity theorem, for example. You areexposed to the concept of an abstract vector space. Which invariably some students always trip over. So the grounding in the earlychapters can mitigate this :///guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心 The last chapter touches lightly on the interesting applications, like chaos and fractals. But mostly to pique your interest in proceedingfurther in the field. iii) This is the text I used this previous semester for my Linear Algebra class. I had no linear algebra background before taking thisclass. That being said, this was one of the roughest classes I\'ve ever got through only because the book kept going against the grain inevery way possible. I didn\'t even begin to understand the entire point of linear algebra until about chapter 7 and 8 when the chaptersstarted going into the general cases, and even now, I know how to \"solve\" all the problems without even knowing their meaning, whichseems totally pointless to me. The selected answers to the problems in the book are in no particular pattern. It\'s not \"all odds\" or \"allevens\"; it\'s just scattered and it made doing homework a nightmare. I felt like I was back in elementary school while reading this book,because back then all I did was learn \"methods\" of solving problems without understanding \"why\". The book almost never discussed thepurpose or main idea of the subjects it discussed. The \"explanations\" it gave would be based off of other vague topics. For example \"Whatis the Eigenvector Problem? Well, the eigenvector problem asks if there is a basis for $R^n$ in a $n times n$ matrix consisting ofeigenvectors of said matrix\", OK so What\'s a basis? \"A basis a set of vectors for a vector space S is linearly independant and/or set thatspans the space S\" and the cycle kept hitting me with one definition after another without giving me a big picture or anything. A bit of thebook is about \"applications\" of linear algebra, but doesn\'t help until you\'ve understood the meat of the book that came beforehand. Also,there were no teachers\' solutions manuals available when I took this class, because the distributers have been extremely lax about gettingthem out (why? who knows). I\'m not just saying this book is bad because I was lazy and didn\'t do well. I worked extremely hard to do\"well\" in this class. I must have read this book twice through and like I said before, I can solve all the problems but please don\'t ask me toexplain their significance or validate their existence, because I can\'t. STAY AWAY!

6) Linear Algebra 作者：K. Hoffmann and R. Kunze 出版信息：2nd ed. Prentice-Hall

使用学校： Central Michigan University，University of North Dakota，Indian Institute of Technology, Bombay，University ofPittsburgh，University of Texas，Johns Hopkins University，West Virginia University，University of Houston，Simon FraserUniversity，Washington University in St. Louis，University of Notre Dame，University of Wisconsin-Madison，CornellUniversity，University of South Carolina，University of Rhode Island，University of Missouri，University of Maryland，Stony BrookUniversity，University of Michigan，Purdue University，University of Kansas，United Arab Emirates University，RiceUniversity，Kenyon College，Temple University，Louisiana State University，Sonoma State University，North Carolina StateUniversity，University of Iowa

国外评论选摘 i) I got this book for my Linear Algebra class about four years ago. This is a great book if you are getting a degree in mathematics. Itwon\'t help if you are just trying to get by the class and don\'t like math. It is not very practical but if you are looking for a real math bookon Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn aboutLinear Algebra from a rigorous mathematical point of view this is it. This book taught me so much. ii) This was the textbook they used to use at MIT in the past few decades. Virtually, however, nobody uses this book in a regularundergraduate course anymore. Instead of developing the ideas in the familiar context of the real numbers, Hoffman and Kunze give amore abstract (and general) discussion. For example, the theorems about determinants work in all commutative rings. The rigorousness and/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心the wealth of information are overwhelming for most undergraduates to handle. You will not learn anything if you just glance through thepages. Every line requires deep thought. Down-to-earth applications are not included. So I do not recommend this book for engineers.

7) Linear Algebra with Applications 作者：Otto Bretscher 出版信息： 3rd ed. Prentice-Hall

使用学校： San Francisco State University，University of Utah，Pennsylvania State University，Agnes Scott College，Harvard University，JohnsHopkins University，University of Minnesota，McGill University，Colby College，Santa Clara University，University ofCalifornia，State University College at Buffalo, Queen\'s University, Georgia Institute of Technology, Northeastern University, PurdueUniversity, Loyola University, Iowa State University

国外评论选摘 i) The explanations and examples are generally very clear, and there isn\'t a lot of distracting nonsense. In many textbooks they try toohard to teach through \"Real World\" examples. i find such examples confusing because they obscure the math behind the example. I alsofelt this book had a nice mix of easy, medium and challenging problems. And it feels like the author really understands and strives toclarify many of the hurdles faced by Linear Algebra students. Make no mistake about it, Linear Algebra is a tough class that requires a lot of dilligence and abstract thinking. This book isn\'t going toguarantee you an A. But if you work through it, and if you have a helpful teacher, you\'ll be on the right track. By the way, I am a Computer Science major, and while I consider myself decent at math, I\'m by no means a math genius. :) ii) This text was developed by the author during his time on the mathematics faculty at Harvard for specific use in the second semesterof a two semester, undergraduate sequence on multivariable calculus and linear algebra. It is intended for physics, chemistry and stronglyquantitative economics majors. As such, in terms of complexity it is more par with a collegiate abstract algebra text, with a clear focushowever on linear algebra. The \"applications\" portion of the title is a bit of a misnomer, as examples only occur in the problems andalmost never in the examples (which are designed instead to show the theoretical precepts and continuity underlying the field). In general,this text is above the intellectual capabilities of but the most dedicated users of applied mathematics, and those especially is the fields ofeconomics and finance as generally taught at the undergraduate level would best look elsewhere. Most prominently, the text has almost noredundant examples, which makes it a enjoyably lucid read for those who grasp concepts quickly on the first go, but a dead end for thosewho come up short. I would not as professor think of assigning this book to non-Ivy caliber students outside of pure math; even Harvardstudents seemed to struggle with it at times. iii) I was required to purchase this book for a course called Linear Algebra with applications. This book seems to just cut out importanttheorems, proofs and other pieces of explanation commonly found in other text books I have looked through, and rather than making upfor it with a decent explanation or summary for what it omits, it leaves gaping holes in many topics. It gives partial proofs andexplanations at times and leaves other pieces \"for you to solve as exercises.\" It\'s like the [person] who made this book only wrote half amath book, and left the other half for you to figure out in problems at the end of the chapter.

8) Linear Algebra with Applications 作者：Steven J．Leon 出版信息： 7th ed. Prentice-Hall/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心

使用学校： Rowan University, Arizona State University, Florida International University, Northern State University, University of Illinois atChicargo, University of Puerto Rico, Colorado State University, State University of New York Institute of Technology, SUNY Institute ofTechnology, University of Hawaii, Ohio State University, University of Minnesota, Texas A&M University, University of MassachusettsDartmouth, University of Texas at Dallas, University of New Mexico, Boise State University, Baruch College, University of Oslo,University of Missouri-Columbia, University of Mississippi, Utah State University, Kansas State University, University of California,Irvine, Brigham Young University, Cornell University

国外评论选摘 i) First of all, I would like to say this book is not for beginers. If you have no idea what a matrix is, don\'t use this book. However if youhave taken an introductory course in linear algebra or you already have a reasonably well foundation in this subject, then you should haveno problem in understanding following the text. Although the explaination in this book is not particularly outstanding, it does treat someadvanced topics like eigenvalues, numerical linear algebra elegantly. I would like to recommend this book to persons who would like toseek a more advanced linear algebra book for reference or self studying. ii) Leon\'s text on linear algebra isn\'t bad, but there is room for improvement. Chapters 1, 2, and 3 do a good job of introducing the basicconcepts of linear algebra, including matrix row operations, determinants, and linear independence. The book seems to lose claritybeginning in Chapter 4. The concepts become more abstract and Leon\'s notation interferes with the ability to clearly understand what he istalking about when it comes to linear transformations and issues regarding $R(A)$ and orthogonality. Very important results are frequentlyunderstated as well. In a few cases, there aren\'t enough examples to go around - especially in Chapters 4 and 5. It is ironic compared to therelative overexplanation found in Chapter 1, for example.

2.3 其它

书名：Concrete Mathematics, 2nd ed.作者： , , nik出版商： Addison Wesley (1994)

页数：624适用范围：大专院校计算机专业数学教材预备知识：基本微积分习题数量：大习题难度： 从容易的习题到研究性的题都有推荐强度：9.2

书评： 这是非常特别的一本教材。首先书名就与众不同，一不小心会误读为“离散数学”， 事实上从内容上看，它包含离散数学的很多内容，但作者在序言中声明书本书是“离散数学”和“连续数学” 的混合物。 三个作者排序是按姓氏的，本书的第一位作者 Ronald Graham 是组合数学的权威之一，曾任过美国数学会主席。第二作者Donald Knuth 是计算机科学界的传奇式人物，现任斯坦福大学教授，他的巨著《The Art of Computer Programming》是计算机程序/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心设计的圣经，本书包含了学习上述巨著的几乎全部数学知识。 上世纪末美国数学会曾在它的官方出版物上举行公开的辩论，探讨数学发展的方向，最后没有明确的结论。 现代数学是向抽象化的方向发展的，数学家更加注重数学问题定性的研究，其重要性是不容质疑的。 但有不少有识之士担心这样下去会有脱离实际的危险，所以他们提倡看得见的数学。这是这本书的初衷。 对此书有兴趣的读者不妨先看一下序言，以便更清楚地了解这本书的特点。 全书分9章，依次为：递归、求和、整值函数、数论、二项式系数、一些特殊的数、母函数、离散概率、渐近。 每章中包含丰富的内容，有很多问题和例子在其它同类书中很难找到， 一些比较难的问题的出处都一一写明。 本书的重点是讲述解决问题的方法，牵涉到很多数学的常用技巧，看上去比较初等， 但对读者的要求还是比较高的。 另外本书的趣味性很强。习题很全面，几乎所有习题有答案，这对自学非常便利。 数学系和计算机系的本科生阅读本书一定有不小收获。（杨劲根）

国外评论摘选

i) Unless you\'re very used to this type of mathematics, this book will, as other reviewers comment, prove hard work. However, evensomeone with little formal maths background like myself can get a lot out of it. It\'s beautifully written and well-presented, and on thewhole the pacing is OK, although sometimes it goes much too fast for casual reading. Once I\'ve made my way through it, I suspect it willmake a very useful reference book too; it\'s full of useful techniques for solving real-world problems, at least if you work in a field thatsometimes requires you to solve recurrences and work with tricky integer functions. Although often corny, the marginalia do give you something of the feeling of being on a course, rather than just reading a textbook. Aswell as daft jokes, there are hints as to the relative importance of some sections (including \"skip this bit on first reading\" as well as \"this isthe critical part\" -- both kinds very helpful).

ii) This book is not light reading, but it\'s worth it. It has most value as a reference tool, and covers well some areas of maths which areimportant to CS. Moreover, the information is presented in a light-hearted way, with lots of inline jokes (mainly very corny) and marginnotes from students who took the lecture course behind the book. The examples tend to help, and there are plenty of exercises withworked solutions. Also lots of references to the primary literature.

书名：Discrete Mathematics作者： Dossey,Otto,Spence,Vanden Eynden 原著，俞正光、陆玖改编出版商： Addison Wesley (2002) 高等教育社（2005），ISBN 7-04-016632-1

页数：562适用范围：大专院校计算机专业离散数学教材预备知识：基本微积分习题数量：大习题难度： 容易

推荐强度：8

书评： 离散数学并不是数学的一个分支，它是计算机和信息学专业的一门数学基础科，内容一般包括集合论、数理逻辑、 初等数论、抽象代数、组合数学等，但每部分内容都不是非常系统和完整。从某种意义上讲，这是一门大杂烩课程。由于 内容的繁多，要学完全部离散数学一个学期是不够的。对于一个学期的离散数学课，一般适合于选讲其中一部分。/guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心 本书从实用角度出发，以组合数学为主线安排了一个单学期的教程，最难的部分抽象代数完全没有，初等数论和数理逻辑也很少， 有一章讲述逻辑线路和有限自动机，涉及了最基本的布尔代数。叙述方面也以概念的直观解释和算法为主，不强调定理的证明， 所以比较适合于数学程度比较低的大学生使用。如果授课对象 是层次高的计算机专业学生，这本书就显的太浅，内容也不够丰富。 本书英文浅显易懂，例子非常多，作者们似乎花了工夫认真编写这本教材，错误非常少，习题虽然数量大， 但很有意思。 下面登载两篇国外的评论，代表两种观点。（杨劲根）

国外评论摘选 1) As a student at Illinois state, I\'m skeptical about all of the After all, these are the guys that consistently screwup addition in front of class. After having a chance to complete half of this book in my Discrete Math course (mind you, I\'m not a mathmajor) I have definitely gained respect for ISU\'s math department. I\'m not sure if most authors really teach classes, or if they write books to fulfill their publishing requirements. I can tell you that theauthors of Discrete math had the students in mind. I\'ve found this book to have exceptional examples, and well-explained, READABLE prose. If you wanted to pick up a copy for self study, this would be a Yes a professor would be nice, but these guys did a goodenough job that the book stands alone.

2) If you are looking for a book for a course in discrete mathematics where the emphasis is on graph theory, then this book willprobably satisfy your needs. However, for any other type of course, it will most certainly prove to be inadequate. Nearly half the book isdevoted to graph theory, and while many theorems are listed, very few are proven. The working computer scientist may find thatacceptable, but most mathematicians will find it inadequate. Logic and the basics of proof are relegated to an appendix. The first chaptercovers some combinatorics and the basics of algorithmic analysis, which is meant to be a primer. However, it requires the use of setterminology, set notation and basic counting techniques. Since set theory is covered in chapter 2 and counting techniques in chapter 7, Iconsider the order to be inappropriate. Recurrence relations, circuits and finite state machines are also covered in other chapters. There area large number of exercises and the solutions to the odd numbered ones are included. Sets of problems to be solved by programming acomputer are given at the end of each chapter, some of which are easy, but many of which are hard. Only students who have had aprogramming course could be expected to be able to do any of them without significant help. This is a book that does not satisfy myrequirements for a discrete mathematics textbook. I consider logic to be a critical topic that must be covered, so I will not consider usingany book where predicate and propositional logic are not covered in depth. While I do not expect my students to construct rigorous proofs,I do expect them to be able to construct simple proofs and follow some of the relevant more complicated :///guide/wjzx/[2008/7/13 13:58:40]复旦大学外国教材中心#3国外优秀数学教材选评December 24, 2007本书内容为原作者版权所有，未经协议授权，禁止下载使用主 编 杨劲根副主编 楼红卫 李振钱 郝群编写人员（按汉语拼音为序）陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林1. 序言

2. 非数学专业的数学教材

3. 数学分析和泛函分析4. 单复变函数5. 多复变函数6. 代数7.数论8.代数几何9.拓扑与微分几何10.常微分方程11.偏微分方程12.概率论13.其他14.附录3 数学分析和泛函分析书名：Introducton to Analysis作者： Arthur Mattuck出版商： Prentice Hall (1999) ISBN 0-13-081132-7页数：460适用范围：大学数学系本科基础数学学生教材预备知识：微积分初步知识习题数量：大习题难度：较大推荐强度：9.8

书评: 本书是麻省理工学院的 Arthur Mattuck 教授教授这门课程多年经验的基础编写而成的，是一本实分析的优秀入门教程，深受读者欢迎。 本书主要讲述单变量函数的分析理论，侧重于讲述实数理论的基本思想，特别是用分析的方法对函数进行估计。 本书从基本的实数理论讲起，内容主要包括数列与函数的极限和连续性，级数理论，微分理论，Taylor展开， Riemann积分理论，Lebesgue积分理论等等。本书的一个鲜明的特点是，对书中的定理不只是叙述，而是从来源讲起， 对读者以启发为主，侧重于揭示/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3数学思想。 例如，对微积分的两个基本定理，其证明较一般书中繁琐，但是其证明给出了微积分的重要思想， 即积分是微分的无穷积累， 微分是积分的局部化，并且，还分析了两个基本定理之间的关系。另外，书中还给出许多重要的应用。 本书比较适合作为我国综合性大学数学系实分析课程一学年的外文教材，也可以作为 程度较好的数学系本科生进一步深化实分析概念的课外读物。 （王泽军）

国外评论摘选

1) This is an unusual and beautifully written introduction to real analysis. The presentation is carefully crafted and extremely lucid, withwonderfully creative examples and proofs, and a generous sprinkle of subtle humor. The layout of the pages is exceptionally author has clearly put a great deal of thought and effort into producing an analysis text of the highest quality. Most of the bookconcentrates on real-valued functions of a single (real) variable. There is a gradual and careful development of the ideas, with helpfulexplanations of elementary matters that are often skipped in other books. For instance, prior to the chapter on limits of sequences, the bookhas a chapter on estimation and approximation, discussing algebraic laws governing inequalities, giving examples of how to use theselaws, and developing techniques for bounding sequences and for approximating numbers. Proofs involving \"epsilons\" and \"arbitrarily largen\" make their first appearance here. The overall presentation of the book is carefully thought out. Each chapter is broken up into small sections, and each section emphasizesone principle idea or theorem. The proofs of the main theorems are lovely, and give both intuitive explanations and rigorous ely interesting examples and problems illuminate the key ideas. Each chapter contains a mix of problems: \"questions\" that helpstudents test their grasp of the main points of each section, \"exercises\" that are intermediate in scope, and more difficult \"problems\". (Asolutions manual is available for instructors from the publisher.) The careful explanations, even of \"elementary\" matters, and two appendices on sets, numbers, logic, and methods of argumentation,make the book suitable for a first analysis course in which students have had no prior exposure to proofs. There is ample material for aone-semester, or in some cases a one-year, course. In summary, I believe that this is the best introductory real analysis book on the market. Students and instructors alike will find it a joyto read.

2) The book is slow to begin but it does a great job in explaining all the concepts. The author explains the proofs and theorems and itintroduces some intermediate ideas to understand the theorems and definitions. The book contains a lot of exercise of different nature anddifficulty. It covers a great range of subjects but not enough on the Rn. The book is basic in it contain, it is not difficult to read andfollow. It can serve as an introduction to analysis. I would recommend it if you want an introduction to analysis.

书名：Mathematical Analysis, Second Edition作者： Tom M. Apostol出版商： Addison Wesley (1974), 机械工业出版社影印 ISBN 7-111-14689-1页数：492适用范围：大学数学系本科生预备知识：高中数学习题数量：中习题难度： 中等推荐强度：9.8/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3

[作者简介] Tom M. Apostol, 美国数学家，生于犹他州。他于1946年在华盛顿大学西雅图分校获得数学硕士学位，于1948年在加州大学伯克利分校获得数学博士学位，1962年起任加州理工学院教授，美国数学会、美国科学发展协会（A.A.A.S）会员。对初等数论和解析数论有研究，他的著作很多，除本书外，还著有《Calculus, One-Variable Calculus with an Introduction to LinearAlgebra》、 《Calculus, Multi-Variable Calculus and Linear Algebra with Applications》等。

书评: 本书第一章以公理化的方式引入了实数系和复数系，接下来介绍了集合论和点集拓扑的一些基本概念和内容，为后面微积分理论的展开打好基础。从第四章开始，作者开始介绍极限、连续和导数等微积分的基本概念。在第六章作者引入了有界变差函数与可求长曲线的概念，接着就对Riemann-Stieltjes积分进行了介绍，而Riemann积分则是它的特例。第八第九章是对级数和函数序列知识的讲解。第十章介绍Lebesgue积分，第十一章介绍Fourier级数以及Fourier积分，第十二章介绍多元微分学，第十三章介绍隐函数与极值问题，接下来的两章是关于多重Riemann积分与Lebesgue积分的介绍，最后一章介绍了复变函数的Cauchy定理以及留数的计算。 本书是一部现代数学名著：自20世纪70年代面世以来，一直受到西方学术界、教育界的广泛推崇，被许多知名大学指定为教材。作为一本大学数学系的本科教材，本书仔细而又不累赘地向读者介绍了微积分的思想，涵盖了数学分析绝大部分的基本知识点，并配有覆盖各级难度的练习题，适用于初次接触数学分析的读者。无论对于教学还是自学，都不失为一本理想的教材。另一方面，本书对于实分析和复分析中的部分内容也有所介绍，这其实也是很多美国大学数学教材（Mathematical Analysis或者AdvancedCalculus）内容设置的共同点。例如作者在第十章有对Lebesgue积分的介绍。不过与一般实分析教材里的思路不同，作者采用了Riesz-Nagy的方法引入了Lebesgue积分，此方法直接着眼于函数及其积分，从而避免了对于测度论知识的要求；同时作者还进行了简化、延伸和调整，以适应大学本科水平的教学。 （徐晓津）

国外评论摘选

1) If you\'re the type of person who likes crisp and clear proofs but don\'t want to have the proofs be as skinny as Rudin\'s then this isthe perfect book. Apostol\'s writing style is not only accessible and clear but the organization of the text is excellent too. There are plentyof problems with a good mix of difficulty levels. He also throws in an example here and there to give you firm footing on some difficulttopics. If I had to recommend one analysis text this would be it.

2) I own analysis texts by Apostol, Rudin, Bear, Fulks, Protter, and Kosmala. This one by Apostol gets my vote as the best all-aroundtext on the subject. It\'s rigorous, elegant, readable, and has just the right amount of explanatory text. This would be my first choice as anundergraduate textbook, a self-study text, or as a supplemental reference to another text. I also recommend Bear for his elegance and wittystyle, and Kosmala for his thorough explanations. But if you are going to buy only one, make it this one.

3) I\'ve never been a big fan of Apostol. He tends to make things more difficult than they really are. Some of the reviewers commentedthat they are impressed with the elegance of the proofs, which makes me wonder if they are as confused as Apostol. As an example let\'sconsider his proof of the FTC. There is an easy and elegant proof which you find in most books, but Apostol tries to be cute and gives anobscure and ugly proof. Mathematics is an art, and Mr. Apostol is no Picasso.

书名：Principle of Mathematical Analysis, 3rd edition作者： Walter Rudin出版商： McGraw-Hill (1976), 机械工业出版社影印页数：334/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3适用范围：数学系一、二年级学生与理工科高年级学生预备知识：高中数学，最好具备微积分的初步知识习题数量：287 道习题，较大习题难度： 较难，但是很多有难度的题目有提示推荐强度：9.5

[作者简介] Walter Rudin 1953年于杜克大学获得教学博士学位。曾先后执教于麻省学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究领域集中在调和分析和复变函数。除本书外，他还著有另外两本名著：《Functional Analysis》和《Real and Complex Analysis》，这些教材已被翻译成13种语言，在世界各地广泛使用，以本书作为教材的名校有加利福尼亚大学伯克利分校、哈佛大学、麻省理工学院等。

书评: 本书前二章介绍了从高中数学到大学数学过渡中的基本知识：实数与复数理论，基础拓扑理论。第三章介绍数列与级数。第四章介绍函数的连续性。第五章介绍微分的概念。第六章介绍Riemann-Stieltjes积分的概念。第七章介绍了数学分析中很重要的一个概念：函数序列与函数项级数的一致收敛性。在第八章作者列举了几个特殊的函数项级数，如幂级数、Fourier级数等作专门讨论。第九章介绍多变量函数。第十章介绍了微分形式的积分。在最后第十一章对勒贝格积分作了初步的介绍。 本书内容相当精练，结构简单明了，这是Rudin著作的一大特色。例如在第六章积分部分，作者直接介绍了Riemann-Stieltjes积分，而一般数学分析课程中的Riemann积分就是它的特例。书中的习题经过了精心挑选，有助于学生掌握数学分析的基本概念及提高逻辑推理的技巧。本书第3版经过了增删与修订，更加符合学生的阅读习惯与思考方式。 本书适合作数学系学生学习数学分析课程的参考书，也适合作为具有一定微积分知识的理工科高年级学生提高分析水平与能力的教材。 本书是一部现代数学名著，一直受到数学界的推崇。作为Rudin的分析学经典著作之一，本书在西方各国乃至我国均有着广泛而深远的影响，被许多高校用做数学分析课的必选教材。本书涵盖了高等微积分学的丰富内容，最精彩的部分集中在基础拓扑结构、函数序列与函数项级数、多变量函数以及微分形式的积分等章节。 [零星感悟] 作者从学生的角度出发来考察问题的接受难易程度，并在整本书的结构上做了精心的安排和调整。 比如说，从理论上讲，从有理数的概念出发引入实数的概念是非常正常和符合逻辑的，但是Rudin通过以往的教学经历发现学生对这样的做法不容易接受，因此Rudin从有序集与具有上（下）确界的性质入手来介绍实数，显得简洁而具有新意。 在第九章多变量函数中，一个关键的问题就是反函数存在定理的证明。记得以前看过的书上证明都比较复杂。在此书中，Rudin利用压缩映射的不动点理论，大大简化了证明过程。 (刘东弟)

国外评论摘选

1) OK... Deep It is not possible to overstate how good this book is. I tried to give it uncountably many stars but they only have five. Five is an insult.I\'m sorry Dr. This book is a good reference but let me tell you what its really good for. You have taken all the lower division courses. You havetaken that \"transition to proof writing\" class in number theory, or linear algebra, or logic, or discrete math, or whatever they do at yourinstitution of higher learning. You can tell a contrapositive from a proof by contradiction. You can explain to your grandma why there aremore real numbers than rationals. Now its time to get serious. Get this book. Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. Ifyou get stuck read a line or two until you see what to :///guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3 Thrust, repeat. If you make it through the first six or seven chaptors like this then there shall be no power in the verse that can stop you. Enjoygraduate school. You half way there. Now some people complain about this book being too hard. Don\'t listen to them. They are just trying to pull you down and keep youfrom your true destiny. They are the same people who try to sell you TV\'s and lobodemies. \"The material is not motivated.\" Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do will come. He\'s teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not needmotivation from the author as to why you need to know analysis. You should just feel a burning in you chest that can only be quenched byarguments involving an arbitrary sequence ${x_n$ that converges to $x in X.$ Finally, some people complain about the level of abstraction, which let me just say is not that high. If you want to see abstraction graba copy of Spanier\'s \'Algebraic Topology\' and stare at it for about an hour. Then open \'Baby Rudin\' up again. I promise you the feeling youget when you sit in a hottub for like twenty minutes and then jump back in the pool. Invigorating. No but really. Anyone who passes you an analysis book that does not say the words metric space, and have the chaptor on topologybefore the chaptor on limits is doing you no favors. You need to know what compactness is when you get out of an analysis course. Andit\'s lunacy to start talking about differentiation without it. It\'s possible, sure, but it\'s a waste of time and energy. To say a continuousfunction is one where the inverse image of open sets is open is way cooler than that epsilon delta stuff. Then you prove the epsilon deltathing as a theorem. Hows that for motivation? Anyway, if this review comes off a combative that\'s because it is. It\'s unethical to use another text for an undergraduate real analysisclass. It insults and short changes the students. Sure it was OK before Rudin wrote the thing, but now? Why spit on your luck? And ifyou\'r a student and find the book too hard? Try harder. That\'s the point. If you did not crave intellectual work why are you sitting in ananalysis course? Dig in. It will make you a better person. Trust me. Or you could just change your major back to engineering. It\'s more money and the books always have lots of nice pictures. In conclusion: Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me.

2) What has been said below is all true. Rudin really does have some excellent moments in this book, except perhaps the chapter onLebesque integration, which is one of the crappiest expositions of the topic I have found so far. Get the texts by E. Stein (Real analysis),or Bartle\'s small book on lebesgue integration. There is even a probability text by ev (\"Probability, 2nd edition\") which has a trulyamazing treatment of the Lebesgue integral. Anyhow, the rest of the book is excellent, concepts of single-variable analysis are very wellexplained, the proofs are short and enlightening. The multivariable calculus is also very well explained. I gave it 3 stars because it\'s not agood book for self-study. There are hardly any explanations, the beginning student will likely get really frustrated. In order to enjoy thisbook you either have to know analysis already, so this would be a second text, or you have to take a course that uses Rudin as you won\'t have any problems, since the teacher will probably end up explaining the stuff in class. I have heard complaints even bysome of the world\'s best mathematicians at Princeton against the fact that Rudin\'s text is so terse. To me that\'s not even impressive. It\'sjust arrogance or laziness on the behalf of the author. So don\'t feel too bad if you read it and find that things aren\'t explained very well;that\'s because they aren\'t! The false sense of reward comes from banging your head against the wall before finding the answer, and beingthankful you finally got somewhere rather than committing suicide (only thing is, if anything, you may have just reinvented )But by then, you\'ll have wasted a lot of time already. If you have nothing else to do, and are incredibly patient, this is no problem at all,otherwise, it\'s a real waste of your time. You could also be a genius, in which case, none of these points are even an issue; then you canprove all theorems in the world, so congratulations, I look forward to meeting you! Oh yeah, and there are no solutions to the problemsprovided to check if your answers are right or not, so good :///guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3

书名：Advanced Calculus, 2nd Edition作者： Patrick M. Fitzpatrick出版商： Brooks/Cole (2005), 机械工业出版社影印页数：590适用范围：数学系与理工科其他专业的本科生预备知识：高中数学习题数量：较大习题难度： 具有一定难度推荐强度：9.3

[作者简介] Patrick M. Fitzpatrick拥有格兰特大学博士学位，是纽约大学科朗研究所和芝加哥大学的博士后，1975年进入马里兰大学College Park分校任教，现在是数学系教授和系主任，同时它还是巴黎大学和佛罗伦萨大学的客座教授。他的研究方向是非线性泛函分析，在该方向著有50多篇论文。 书评: 本书以清晰、简洁的方式介绍了数学分析的基本概念：第一部分讲述单变量函数的微积分，包括实数理论、数列的收敛、函数的连续姓和极限、函数的导数和积分、多项式逼近等；第二部分把微积分的概念推广到多维欧几里得空间，讨论多变量函数的偏导数、反函数、隐函数及其应用、曲线积分和曲面积分等。 数学分析已经根植于自然科学和社会科学的各个学科分支之中，微积分作为数学分析的基础，不仅要为全部数学方法和算法工具提供方法论，同时还要为人们灌输逻辑思维的方法，本书在实现这一目标中取得了引人注目的成果。本书一方面按传统的和严格的演绎形式介绍微积分的所有主题，另一方面强调主题的相关性和统一性，使读者受到数学科学思维的系统训练。 本书的一大特点是除了包含必不可少的论题，如实数、收敛序列、连续函数与极限、初等函数、微分、积分、多元函数微积分等以外，还包含其他一些重要的论题，如求积分的逼近方法、Weierstrass逼近定理、度量空间等。例如本书专门用一章讨论度量空间，从而把在欧几里得空间讨论微积分时使用的许多概念和导出的结果扩展到更抽象的空间中，引导读者作广泛深入的思考。 另外，与第一版相比，第二版增加了200多道难易不等的习题。全书贯穿了许多具有启发性的例题，并且本版还为教学考虑进行了许多实质性的改动，例如将选学材料与前后内容的关联度降到最低，单独放置，既不影响教学和读者自学的进度，又能让读者集中攻破一些难点，这样使得全书的叙述更简洁、更自然。本书曾于2003-2004年作为马里兰大学教材。 （高威）

国外评论摘选

1) A great book. Starts with two very good chapters on linear algebra, adapted to the needs of calculus, and then proceeds to introduceyou to the contemporary way to do multivariate calculus, including existence theorems connected to completeness. Very thorough treatmentof integration, including integration of forms on manifolds, up to the Stokes theorem, built upon a fine chapter on differential manifolds,exterior differential forms, riemannian metrics, etc. Good illustrations and beautiful typesetting add to the joy of reading it. Plenty ofexercises and chapters on applications to physics and differential geometry.

2) This is the best book on mathematics I\'ve ever come across. The superbly written text succeeds in guiding the reader in an easy,clear-cut, graceful way through the realm of what he modestly calls \"Advanced Calculus\". Some minor misprints are to regret, but theydon\'t even come close to blurring the fact that this is - no doubt about that - an unsurpassable masterpiece.

3) As Spivak\'s \"Calculus on Manifolds\", this book is labeled with a very modest title. It should be something as \"All you wanted to/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3know about analysis on manifolds but were afraid to ask\". This book is a must-reading for the analyst. It covers everything from the mostbasic vector space concepts up to the fundamental theorems of classical mechanics, running through multivariate calculus, exteriorcalculus, integration of forms, and many topics more, always keeping a very modern and rigorous style. The undergraduate may find it a little difficult, but the effort is worth it. For the graduate student and the working mathematician it isan almost-daily reference.

4) This book is out of print, but is available from Sternberg\'s website. Search on his full name at Google.

书名：Advanced Calculus, 5th Edition作者： Wilfred Kaplan出版商： Addison Wesley (1991), 电子工业出版社影印页数：741适用范围：理工类本科高年级学生与研究生预备知识：高中数学和初步的微积分学基础习题数量：大习题难度： 有难度推荐强度：9.5

[作者简介] Wilfred Kaplan于1939年在Harward大学师从Hassler Whitney 获得博士学位，后任教于Michigan大学。 书评: 本书除了全面地介绍微积分的知识，还介绍了线性代数、矢量分析、复变函数、以及常微分方程、偏微分方程等方面的知识。全书共分为10章：前两章介绍了线性代数和偏微分；第三章介绍了散度、旋度和一些基本的恒等式，还介绍了n维空间中的张量；第四、五章介绍了积分理论，包括定积分、重积分、曲线积分、曲面积分、Stokes公式等；第六章介绍级数理论；第七章介绍Fourier级数理论；第八章介绍复变函数的解析理论；第九章介绍了常微分方程理论；第十章介绍了偏微分方程。本书内容丰富，编写深入简出，在每一章都有相当篇幅的内容打了\"*\"号，这些内容属于基础理论的深化与拓广，可供教师教学时选用，或供基础好的学生选读。 本书的前身是作者应他的一位工程学同事的建议所著，目的是让工科学生在掌握初等微积分的基础上进一步扩充数学知识，提高数学水平与能力。初稿写成后，曾用于工科大三学生的教学。付诸印刷后，被Michigan大学指定为理工科高年级学生的教材。 因为本书的写作初衷是提供给工科学生，并且作者认识到数值方法具有实用价值和帮助读者更深入的了解微积分理论，所以本书不仅介绍了理论知识，还涉及到相关数值方法，这也是本书的一个特点。 本书另一个特点是十分方便读者自学自测。比如说，本书中的定义都有明确标示，所有的重要结果都作为定理以公式的形式给出；书中不仅提供了大量难易不同的习题，更给出了习题答案；此外，还提供了大量的参考文献，并在每章的末尾给出了推荐阅读的书目。 本书为方便教师安排教学进度，注意在各章有机联系的同时，尽量减少每章节对前面章节知识的依赖程度。作者还在序言中为一学期每周四小时的课时提供了具体的教学安排建议。 (高威)

国外评论摘选

1) This book is simply the best that I have found for math texts. Kaplan does not expect much from the reader; he explains basicallyeverything besides Calculus I material. Kaplan\'s writing is lively and is (relatively) easy to read. He gets to the point and keeps everythingeasy to follow. I am still in awe about how much material (look below) he was able to fit into this relatively small book and still keep it/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3so clear. The examples are clear and concise. The problems in the book compliment the understanding of the material; they start out easyand guide the reader to do more difficult problems. This book is MORE THAN SUFFICIENT FOR SELF-STUDY.

2) Any student who is taking analysis/advanced calculus course should read chapter 2 of this book, especially if he is confused or isstruggling on the excellent but relatively abstract/concise texts of Rudin, Apostal, Bartle, Marsden et al. I\'ve never seen a book which canexplain the concept of Jacobian and Inplicit function theory in such a clear way!!

3) It is good and clear book. Excellent for undergrad students who want to dig into calculus a bit deeper. But it is too easy for anadvanced undergrad or a grad student in any technical field. I recommend the books published by Springer.

书名：Advanced Calculus作者： Lynn H. Loomis, Shlomo Sternberg出版商： Jones & Bartlett Pub (1989)页数：592适用范围：大学数学系本科生教材预备知识：高中数学习题数量：大习题难度： 中等推荐强度：9.6

[作者简介]Shlomo Sternberg，美国Harvard大学教授，他于1957年在约翰霍普金斯大学获得博士学位。Shlomo Sternberg是一位杰出的数学家，尤其因他在微分几何上的贡献而闻名。

书评: 本书第零章是关于集合、映射等基础知识，接下来对向量空间作了介绍；第三章引入了微分的概念，接下来作者又对紧性、完备性和点积空间进行了介绍。第六章是有关微分方程的简单讲解，第七章介绍了多重线性函数，第八章引入了积分，第九第十章介绍了可微流形以及流形上的微积分问题，第十一章介绍了外微分，第十二章介绍了位势理论，而最后一个章节对微积分在经典力学上的应用作了介绍，向读者展现了数学的威力。 本书是一部优秀的分析教材。与一般的微积分教材不同，它大体上可以分为两个部分：第一部分介绍了赋范向量空间上的微分知识，第二部分主要介绍了可微流形上的微积分知识。本书既有基础的章节，例如第一第二章对于向量空间的介绍，也有对于读者而言要求比较高的内容，比如第九章中关于切空间和李导数的概念。作者在用朴实的语言向读者介绍微积分的概念和思想的同时，也尽可能地展现了不同的观点：例如对于隐函数存在定理的证明，作者就给出了三种证明方法，揭示了数学的魅力。本书的另一大特色在于丰富的习题，练习题的题量大，并且难度不一，作者还把一些重要定理的证明放在了习题中，因此对于读者而言，尽可能多地完成书后习题可以更好的把握和巩固知识，提高分析能力。 本书可以根据教学的需要选取部分章节，程度较好的数学系本科生也可选用此书作为微积分的课外读物。 （徐晓津）

书名：Problems and Theorems in Analysis作者： George Polya and Gabor Szego出版商： Springer Verlag (1978)页数：第1卷389页；第2卷391页；共780页适用范围：数学专业高年级学生与研究生，数学教师与数学工作者/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3预备知识：数学分析，高等代数，复变函数习题数量：第1卷776道；第2卷884道。这是一套习题书习题难度： 难，有的习题甚至为研究者的最新成果，难度很大推荐强度：9.8

[作者简介] George Polya（1887-1985）匈牙利数学家，早年在苏黎世瑞士联邦理工学院任教，后入美国籍，1942年起在美国Stanford大学任教。Polya 在数学的广阔领域里都有深入的研究，特别在泛函分析、数理统计和组合分析等方面尤为突出。Polya不仅是数学家，也是一为优秀的教育家，他始终把高深的数学研究和数学的普及与教育结合起来。 Gabor Szego（1895-1985）匈牙利数学家，早年在柯尼斯堡大学任教，后入美国籍，也在美国Stanford大学任教。他主要的贡献是在数学分析与数理方程方面。 《分析中的问题与定理》一书是George Polya 与 Gabor Szego 最著名的著作。Polya曾经这样评论他与Szego的合作：这是一段美妙的时光；我们专心致志、充满热情地工作。我们有着同样的背景。我们象同时代其他匈牙利数学家一样，受到Leopold Fejér的影响。我们都是那个为中学生创办的强调解题的刊物Hungarian Mathematical Journal 的读者。我们又对同样的课题、同样的问题感兴趣，但往往是一个人对某一个课题知道得多，而另一个人对其他的课题知道得多。这是一次绝妙的合作。我们的合作成果-《分析中的问题与定理》，是我最好的工作，也是Szego最好的工作。

书评: 本书两卷，共分九个部分。第一部分主要收录无限序列与无限级数方面的问题。第二部分是有关积分的各种问题。第三、第四部分是关于单复变量函数的问题，内容包含了数学系本科生与研究生的复分析课程中的主要问题。第五部分主要涉及代数的零点确定问题。第六部分讲多项式与三角多项式。第七部分为行列式与二次型的问题。第八部分为数论方面的题目。第九部分为数学中与几何有关的一些问题。 本书与其说这是一部教科书，不如说这是一部字典，因为它收录了分析学中的各种问题和定理。这是一本有着突破传统意义的书。它对问题巧妙的系统性安排与归纳，给学生创造了自主性思考的可能，最大程度上启发学生的研究能力和创新能力，这也是它不同于其他一些平庸的习题参考书的地方。作者甚至试图用很多哲学的观点来阐释它所选出的题目的代表性，比如有关特殊和一般的问题，要知道早期著名的数学家迪卡尔曾经说过：\"我学数学是为了追求最终的哲学。\"正是这种理念的融入，使得这本书在学术界的地位尤为突出，不只是学生，很多教授和数学工作者都以此书为参考书，并对此书给予了高度的好评。 [零星感悟] 什么是好的教育？给学生一套完善的体系然后让学生在这样的体系下寻找机会自己去发现和解决问题，这样的完善的体系才是好的教育的关键。此习题书不同于其他习题参考书的特点也就在此。它给我们数学系高年级学生与研究生提供了在不同主题下精心安排的问题，启发我们独立思考和研究问题的能力，是一本不可多得的分析习题书籍。 第一部分的习题139让我们明白了很多问题就像两个点决定一条直线一样，是有两个极端的线性组合而得出的结论。 第六部分的习题92让我们明白了掌握一个领域的知识就像了解一个城市的所有交通路线。真正的掌握就是从任何一个出发点，你都可以找到最短的路线达到你想要达到的目的。(刘东弟)

书名：Functional Analysis作者： Walter. Rudin出版商： McGraw-Hill Book Company ISBN: 0-07-054225-2页数：397适用范围：数学类专业本科高年级学生和研究生预备知识：数学分析 复分析 实分析 线性代数

习题数量：中等习题难度： 中等偏难推荐强度：9.5/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3

书评： 的《泛函分析》是一本分析数学方面的经典名著，多年来一直被国外一些高校用于研究生教学。 全书由三部分组成，第一部分是线性泛函分析基础，作者在线性拓扑空间的框架下建立了开映射定理、 闭图像定理、逆算子定理、共鸣定理和线性泛函延拓定理等基本定理，介绍了赋范线性空间的对偶性、 紧算子的概念与性质。作为这些理论的应用，作者还专辟一章介绍了Stone-Weierstrass定理、插值定理、 不动点定理、紧群上的Harr测度等知识。第二部分介绍了Fourier变换和广义函数理论， 并给出了这些理论在微分方程方面的应用。第三部分在Banach代数的基础上， 介绍Hilbert空间上有界正规算子的谱理论，并进一步建立了无界正规算子的谱定理，最后还介绍了 类算子半群。第一部分是全书的基础，第二部分和第三部分则是可供平行阅读的两个独立部分， 读者可根据需要选择使用。全书叙述严谨，条理清晰，理论的展开较为详尽。 该书既可用作泛函分析课程的教材，也可供数学工作者查阅参考。（童裕孙）

国外评论摘选 1) Hardly can I find words to highlight the goodness of this book. As mentioned by other readers ,it provides elegant, direct andpowerfool proofs of the three theorems which constitute the cornserstones of functional analysis (Hanh-Banach, Banach-Steinhaus andOpen mapping). These theorems are, in addition, studied in their most general context, namely topological vector spaces. Specially appealing is its treatment of distributions\' theory. It is, as far as I know, the only text which start by defining the riguroustopology on the set of test functions and then obtains the convergence and continuity of functionals (distributions) in terms of this topolgy,which is, indeed, the only way to present and gain insight into these concepts and to reach some results such as completness. In doingotherwise one risk definitions can emerge as artificial and rather arbitrary. It is, without any doubt, a must have book for those with interest in pure mathematics as well as for those who, eventually, realize thatthe only way to dominate their area is saling through mathematics.

2) No other book covers the elements of distributions and the fourier transform quite like Rudin\'s Functional Analysis. This is a mustfor every budding PDE-er!

3) I enjoy perusing Rudin\'s \"Functional Analysis\" at this stage in my life. It is fairly nice tome for functional analysis, and its generaltreatment of topological vector spaces (as opposed to the standard Banach space examples studied in a typical functional analysis class) isnow well-received. However, as a student, I was put off by this book. At times, I found it difficult to tie the theory present to the basic examples whichwere relevant at the time (such as $L^{p$ spaces). For a first time learner, I would suggest the book of Kolmogorov and Fomin (which isa Dover book, by the way), and would wait until later for this book.

4 单复变函数书名：Complex Analysis, 3rd edition作者：Lars V. Ahlfors出版商：McGraw-Hill (1979)

页数：331适用范围：大学数学系本科,数学专业研究生预备知识：数学分析和线性代数习题数量：小/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3习题难度： 中等推荐强度：10

书评： 出自数学大家之手的这本书已经公认为单复变函数论的经典著作， 既可以选取部分内容作为我国综合性大学数学系本科生的复变函数论教材， 又可以用来作为大学高年级学生和研究生的选修课内容，同时它又是从事复分析研究的标准参考书。 有关单复变函数论的教材、参考书不下几十种，但是除了干巴巴的概念、 定理的正确叙述与严格证明之外提供大量解释性文字的书本并不多见，而在这些叙述中既没有多余的话， 又能使读者开阔视野并感受到作者深厚功力的更为少见，本书恰恰为其中的佼佼者。这本书已经出了三版。 在这第三版中大部分内容未作更动，叙述依然简洁而流畅，但是作者彻底改写了第八章， 以层论的观点描述Riemann面上整体解析函数的存在性，使经典的内容现代化。（张锦豪）

国外评论摘选 1) This book has been, since its first edition in 1953, the standard textbook for rigorously learning complex analysis, and not without areason. The wonderful theory of this branch of mathematics is appropriately emphasized and thoroughly constructed, leading to moregeneral and precise results than most textbooks. While the constant appearance of new texts on the field can only help appreciate thesubject from a different perspective, few give you such a deep and serious treatment like this gem. Postscript: An earlier reviewer claimsthat Ahlfors never defines the set of complex numbers, while this is indeed done in the fourth through sixth pages in a much moreanalytical way than generally found elsewhere. It is quite possible to dislike this author\'s style or approach (or anybody\'s for that matter),but it would be difficult to charge Ahlfors with being sloppy with his writing.

2) How can anyone fail to read this book? The exposition is rigorous, coherent, precise without being either pedantic or overwhelming.A certain level of mathematical maturity is requisite, such as one might acquire in the course of digesting Rudin\'s \"Principles ofMathematical Analysis\" or Apostol\'s book. This is not a compendium of results and exercises for engineers or physicists, it is a conciseintroductory text in pure mathematics. In that sense it is too abstract and proof oriented for that aforementioned audience which would bebetter served by a text in mathematical methods. Even pure mathematics students would benefit from supplementing this book with moredetailed, computationally oriented books such as Conway or Boas. It\'s unrealistic to expect to find everything in one text and to furtherexpect it to remain cogent and approachable. Ahlfor\'s beautiful little book has justifiably remained a classic for four decades.

3) I\'m not sure why the other reviews are so positive. The book is very thorough and rigorous I\'m sure, but the explanations are ne I\'ve talked to in my class agrees that it\'s extremely difficult to learn from if you don\'t already know complex analysis, becausethe definitions and order of treatment are very un-intuitive. Example: residue at $a$ is defined as the number R that makes f-R/(z-a) thederivative of a single-valued analytic function in 0<|z-a|<δ; why didn\'t he even mention that it\'s the coefficient of 1/(z-a) in the Taylorexpansion? And he didn\'t even give any examples of specific residues. I ended up using a mathematical methods for physics book; it wasthe only way I could develop any kind of intuition for the subject.

书名：Introduction to complex analysis作者： Kunihiko Kodaira出版商： Cambridge University Press (1984)页数：256适用范围：大学数学系本科预备知识：数学分析和线性代数/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3习题数量：无推荐强度：9.8

书评： 日本岩波讲座的基础数学中由小平邦彦撰写过三本关于复分析的小册子，其中的I、II 分册被译成英文出版为本书。其内容与我国综合性大学的复变函数论课程基本相符。 本书体现了数学大家小平邦彦一贯的写书风格，起点低，过程详尽，深入浅出，流畅而易读。 本书以复可微（有连续导数）的条件引入全纯函数的概念，为后续的处理带来很大方便。 同时以Cauchy积分定理为主线，从简单到复杂，循序渐进地揭示了这个定理的成立与拓扑的关系。 以远较一般教科书为多的篇幅介绍了Riemann球，引入\"局部坐标\"\"齐次坐标\"等概念， 并顺理成章地接着用来导出分式线性变换的群伦性质。 本书处处体现了小平邦彦深厚的研究功力与广阔的视野。 对于希望将来在Riemann面、Teichmuller空间、多复变函数、复几何、 代数几何等方面进一步深造的有志者来说是一本不可多得的基础好书。（张锦豪）

书名：Functions of One Complex Variable作者： John B. Conway出版商： Springer-Verlag (1973)页数：313适用范围：大学数学系本科或数学专业研究生一年级预备知识：数学分析和线性代数习题数量：大习题难度： 从易到难都有，大部分中等推荐强度：9.6

书评： 本书虽然为大学生学习单复变函数论而写，但是内容十分丰富。 作者用整个一章介绍最大模原理，除了我国教材中通常出现的内容外， 还证明了Hadamard三圆定理与Phragmen-Lindelof定理。 对多复变函数的近代理论有深远影响的Runge定理、Mittag-Leffler定理、 Weierstrass定理等也给予详尽的介绍。本书还包含了大Picard定理等值分布理论的基础。 同时以解析函数芽层的现代观点描述了解析延拓这一重要现象，并引入Riemann面， 进一步再用复流形的现代概念进行提升，非常精彩。 本书内容远远超过我国综合大学复变函数论课程的需要， 所以同时可以用来作为大学高年级与研究生一年级的选课教材。 本书观点颇高，论述严谨，排版紧凑。 虽然作者声称只需基本微积分以及关于偏导数等少量预备知识即可阅读本书， 但由于介绍预备知识的叙述水平超过了一般的数学分析，因此初学者若没有一定的数学天赋则很难自学。 但毫无疑问，这是每一位学习或应用复变函数论者的极好参考书。（张锦豪）

国外评论摘选 1) We\'re using this book for my graduate level complex analysis course, and over all, I\'m pleased with it. Aside from some goofynotation (i.e., an empty box to represent the empty set?), it\'s pretty well written. The pace of the text isn\'t too fast or too slow, and thereare plenty of exercises of a varying degree of difficulty to help you learn the material.

2) An ideal text for a first-year graduate students in mathematics studying Compex Analysis. And this depend how the professorpresent the material. The exposition is complete and very clear, including a lot of optional material for the curious. which could be veryuseful to those preparing for a qualifying exam in analysis at the PhD level.

3) This book was the recommended textbook for a course in Complex Analysis I took at college. I had already done a 1st course on/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3analysis, but that didn\'t help me too much. This book, littered with loads of proofs and lemmas, is a little too terse, and the author expectsstudents to understand a lot on their own. Concepts in Complex Analysis need to be demonstrated using examples, and diagrams, ifpossible. Like for eg. the concept of branches in complex functions. The book starts of defining the complex logrithmic function. Theauthor never says what a branch exactly is. He writes down a hell lot of proofs and expects the student to figure out that the complexlogarithm is infact a multi-valued function, and that a branch is essentially a \"slice\" of this multivalued function. Similiar problems cropup when the author discusses fractional linear transforms. Instead of showing whats happening with simple diagrams, the author makesthings look extremely complicated with his equations and theorems. This book makes learning complex analysis a very mechanicalexercise, devoid of all fun.

书名：Complex Analysis, 3rd edition作者： Serge Lang出版商： Addison-Wesley (1993)页数：321适用范围：大学数学系本科或数学专业研究生一年级预备知识：数学分析和线性代数习题数量：中习题难度： 中推荐强度：9.7

书评： 本书第三版较之第一版增加了许多超出本科生学习的内容。全书分为三部分， 其第一部分与我国综合性大学的复变函数论教材大致相当，第二、第三部分为进一步学习的内容， 可供大学生高年级或研究生低年级的选修课之用。本书将Cauchy定理分为两部分介绍，从局部到整体， 从简单到复杂，使读者很容易接受。特别是在一般Cauchy定理的证明中借用了分析味更浓的Dixon证明， 避开了初学者理解拓扑内涵的困难。将对数函数的介绍与解析延拓的放在一起， 使读者从更一般的角度理解如何选取多值函数的单值支。 本书的另一亮点是介绍了Zeta函数并用来证明素数分布定理。作者是位著名的数学家， 学识广博，擅长撰写数学基础类教材。一些深刻的定理在他的处理下通俗易懂， 所以本书虽然述及到许多深入的复分析内容，读来也是毫无困难，值得向初学者推荐。（张锦豪）

国外评论摘选 1) A person with absolutely no knowledge of complex numbers could begin with page one of this book. However, I think that someexposure to analysis is helpful before finishing the first chapter, but not necessary. I found this book easier to read and understand thansome real analysis books, yet it helped me further understand real analysis in the process. I\'m sure this is due to mere repetition of someof those concepts over a different field. As the author mentions in his foreword, the first half of the book can be used as an undergraduatetext (Jr/Sn years) and the second half can also, but I would NOT have enjoyed it in undergraduate studies. I found it worthy of a firstcourse in complex numbers at the graduate level. I especially liked it after studying real numbers. The placement of the chapter subjectmatter can be altered (to some degree) to ones liking. I think Lang has provided good examples and problems. There\'s a solutions manual(by Rami Shakarchi) for this text somewhere.

2) if you want an introduction to complex analysis, I advise you to pass on this book, and read Churchill and Brown\'s introductorybook. Having said this, part I of Lang\'s book will seem mostly review if you follow my advice. Part II, on Geometric Function Theory, is/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3more advance material that is presented reasonably well.

5 多复变函数书名：An Introduction to Complex Analysis in Several Variables, 2nd edition作者：Lars H?rmander出版商： North-Holland (1973)

页数：254适用范围：数学专业研究生预备知识：实分析与泛函分析初步,复变函数,微分形式略知一二习题数量：无推荐强度：10

书评： 作者于1964年在Stanford大学介绍了多复变函数论，对其讲义稍作修改后即成本书第一版。 除了最后一章最后一节外，第二版基本保持原样。本书前后观点统一，读来似有一根红线贯穿始终。 作者处理单复变的预备知识的第一章会给习惯复变函数论方法的人以耳目一新的感觉。 反映作者将超定偏微分方程理论应用于多复变函数论所做巨大贡献的第四章，是本书的最精彩部分。 这一章系统地介绍了拟凸域上$bar{nabla-$算子的 理论及其应用，值得反复阅读，细细品味，否则会入宝山而空返，到此一游而已。本书叙述简洁而流畅。 阅读本书无需很多单复变函数论的知识，但若没有扎实的实分析功底，读时有如陷入泥泞之地。目前本书已被公认为多复变函数论的经典著作，是准备从事多复变函数论、复几何、 超越代数几何等方向研究的必读课本，也可以用来作为大学高年级学生的选修课内容。（张锦豪）

书名：Introduction to Complex Analysis Part II, Functions in Several Variables作者： B. V. Shabat出版商： American Mathematical Society

页数：371适用范围：数学专业研究生预备知识：复变函数习题数量：大习题难度： 中等推荐强度：9.5

书评： 本书为作者两卷书的第二部，第一部讲单复变函数。本书起点较低，通俗易懂。 对于深受前苏联教育体制影响的我国大学培养出来的学生来说， 它的内容与叙述方式以及对预备知识的要求都非常贴切我国学生的知识结构。 本书对概念的介绍非常清楚而到位，例子较其它书籍为多，还附有大量难度适中的习题 ，因此是值得推荐的多复变函数论入门书。但是本书没有提供系统的方法供研究者参考。 本书可作为大学生高年级与研究生的选修课教材。（张锦豪）

书名：Topics in Complex Function Theory I, II, III作者： Carl L. Siegel出版商： John Wiley & Sons, Inc. (1969,1971,1973)

页数：186,193,244适用范围：大学数学系本科或数学专业研究生/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#3预备知识：复变函数论，抽象代数初步，代数拓扑初步习题数量：无推荐强度：9.8

书评： 这是一本述及复分析高端论题的经典著作。本书根据作者于1964年在德国哥廷根大学 为时两学期的演讲基础上写成的。作者以椭圆积分与椭圆函数及单值化理论作为第一卷的内容开始， 继之以自守函数和阿贝尔积分成就第二卷，最后在第三卷将读者引入多变量阿贝尔积分与模函数。 本书写作风格独特：不追求概念的天衣无缝的表达，而侧重其内涵与相互联系的阐发， 同时包含了有关领域的几乎所有重要结果，以及在其它地方很难找到的一些熟知结果的证明。 因此，不仅初学者能沿着作者独辟的蹊径很快到达最前沿的研究领域， 许多专家也会从其叙述中感受到这位数学大家的深厚功力，得益良多。（张锦豪）/guide/wjzx/#3[2008/7/13 13:59:49]复旦大学外国教材中心#6国外优秀数学教材选评December 24, 2007本书内容为原作者版权所有，未经协议授权，禁止下载使用主 编 杨劲根副主编 楼红卫 李振钱 郝群编写人员（按汉语拼音为序）陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林1. 序言

2. 非数学专业的数学教材

3. 数学分析和泛函分析4. 单复变函数5. 多复变函数6. 代数7.数论8.代数几何9.拓扑与微分几何10.常微分方程11.偏微分方程12.概率论13.其他14.附录6 代数书名：Lectures on Linear Algebra作者： Gelfand出版商：INTERSCIENCE PUBLISHERS. INC.（60 年代）页数：185适用范围：大学数学系本科低年级教材预备知识：微积分习题数量：少习题难度： 比较大推荐强度：8.5

书评： 前苏联有一些数学大师写过一些给大学本科用的数学教材，就象我国华罗庚先生写过《高等数学引论》一样。 作为泛函分析祖师爷级别的人物 Gelfand 写的线性代数讲义也不失为一本极好的教材。 本书是学院式的，结构和叙述十分严格和简洁，具有 Bourbaki 的风格，但又不象 Bourbaki 那样地追求一般性， 所以不要求读者有很高的起点。不象一般的线性代数教材从解线性方程组或行列式开始，本书从 n 维内积空间 作为第一章，甚至包括复内积空/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6间，可谓开门见山。第二章讨论线性变换，特别是正交变换和酉变换。 第三章是 若当标准型，第四章是多重线性代数。 本书原文是俄文的，在 60 年代有多种文字的译本，包括中文本，由于是繁体字，现在已打入冷宫了。我在大学 刚毕业不久（1971年）阅读此书得益非浅，感觉对线性代数乃至泛函分析有更深的认识， 这是我把这本老书向读者介绍的主要原因。 本人认为此书层次较高，适合于已经有一些初等线性代数知识的人读，它不一定对提高解题能力有利， 但对提高数学观点是绝对有好处的。 （杨劲根）

国外评论摘选 1) The professor who recommended this book made the comment that every time you re-read it, you notice something else that youmissed the last time you read it. This is absolutely true. I must say, the first time I picked up this book, I did not like it. The notation was not what I was used to, and the book dives right in,assuming a lot of background (matrices, determinants, etc.) but covering material which many people find boring (bases, etc.). However,when you read deeper, there\'s a lot here. Once you get past the ugly notation, the proofs are extraordinarily clear. And in spite of thebooks small size, there is a remarkable amount of motivation and discussion. Like the other reviewer said, this is not a book to learn linear algebra from for the first time: this is an advanced book that is useful forgraduate students who have already had a linear algebra course and who want to learn more topics, or understand topics on a deeper level. This is an excellent book; the bottom line is that it\'s so cheap that there\'s no excuse NOT to buy it. 2) This is the best treatment of linear algebra that has been published. It starts with n-Dimenaional linear spaces and ends with anintroduction to tensors. An excellent description of dual spaces is consicely presented. NO INDEX! 3) Lucid and clear notation , complete explanations . This books was first published in 1937 but until now it remains best text book inthe field . 4) This is a good book if all you need is a condenced reference on theorems and proofs and it assumes that you go for practice (andinstruction) elsewhere. If you are trying to actually learn linear algebra (especially on your own and especially if you want to learn how tosolve practical problems) get one of Gilbert Strang\'s books and watch his videolectures at MIT web site. Another thing that I dislike aboutthe Gelfand\'s book is that it puts too much emphasis on index notation - instead of matrix notation wich is natural for linear algebra,almost all formulas and theorems are presented at very low level using expressions consisting of variables with multiple indices. Naturallyit gets very messy and hard to follow at times. This doesn\'t present any more information than equivalent matrix notation but introducesunnecceccary complexity and makes things that are really easy to understand very confusing.

书名：Linear Algebra Gems

作者： David Carlson, Charles R. Johnson,David C. Lay, A. Duane Porter出版商： The Mathematical Association of America (2000左右)页数：328适用范围：大学数学系本科低年级参考读物预备知识：微积分、线性代数习题数量：123 题习题难度：大推荐强度：8.5

书评： 这本书不是线性代数的教材，而是兴趣浓厚的学生或教线性代数的老师的参考读物，同类 的书并不多见。美国最大的数学组织是美国数学会（AMS）, 其次就是美国数学协会（MAA）, 它 的主要目标是推动数学教学，尤其是大学本科的数学教学，和/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6AMS 一样，它也有不少出版物， 其中最主要的是美国数学月刊，简称 Monthly, 是一份历史悠久并且享有盛名的数学教育刊物，上面的文章质量高于我国的数学通报，另有一个刊物 College Mathematics Journal，我国数学界不太熟悉。 本书是从多年的Monthly 和 College Mathematics Journal 中选出几十篇与线性代数有关 的短文，又约稿请人写了若干文章，总共74篇按内容进行分类而构成的。大部分文章是教学心得和 若干有名的定理（比如若当标准型）和习题的进一步探讨。各篇文章互相独立，每篇文章一般在一个或半个小时内读完， 非常适合于充当大学生课外读物，特别是对大学生数学竞赛很有帮助。 内容共分十部分如下 PART 1 - PARTITIONED MATRIX MULTIPLICATION PART 2 - DETERMINANTS PART 3 - EIGENANALYSIS PART 4 - GEOMETRY PART 5 - MATRIX FORMS PART 6 - POLYNOMIALS AND MATRICES PART 7 - LINEAR SYSTEMS INVERSES AND RANK PART 8 - APPLICATIONS PART 9 - OTHER TOPICS PART 10- PROBLEMS 象第 1，2，5，6，7 部分一看就知道有不少有技巧性的内容。第十部分是习题，大部分是竞赛级别的题。 （杨劲根）

书名：Algebra

作者： Michael Artin出版商： Prentice Hall (1991) ISBN 0-13-004763-5, 机械工业出版社影印页数：618适用范围：大学数学系本科基础数学一学年的教材预备知识：微积分和线性代数习题数量：大习题难度： 各种难度都有推荐强度：9.8

书评： 本书是美国大数学家美国科学院院士 Michael Artin 的力作，从70年代早期开始就作为麻省理工学院数学系高年级 本科生教材，是一本极具特色的优秀教材，深受使用者欢迎。 与传统的抽象代数教材不同，本书以数学中的重要实例为主线索，引导出抽象的概念，对读者以启发为主，又不缺乏数学的 严格性。虽然教材的主要内容是基本的代数结构，但字里行间不乏现代数学的烙印。代数数论、代数几何、表示论中的一些基本思想 也时时涌现，如整二次型的原理和应用、二次域的理想类、不定方程、紧群表示等。 全书分14章，从矩阵运算引入群概念直到最后一章伽罗华理论一气呵成，不使人感觉600多页篇幅的冗长。 本书的习题是作者20多年积累而得，很多是作者独创的习题，例如有一道2x2魔方的问题是70年代3x3魔方游戏刚问世时作者 编制的群论习题。大约有四分之一的习题有一定难度。 本人80年代在 MIT 攻读研究生期间为此课程作过多次助教，主讲人为作者本人或其他资深教授，每次大约有三十人修课，主要 学生是基础数学各专业的学生，也有一些计算机专业的本科生及研究生选修的。学生反映此课程质量很高，但比较难。 本书比较适合我国综合性大学数学系抽象代数课程的外文教材，尤其适合一学年。对于半年的抽象代数课程，则可选用部分章节。 程度较好的数学系本科生可选用此书作为抽象代数的课外读物。 （杨劲根）

/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6 国外评论摘选

1) Pretty much any introductory abstract algebra book on the market does a perfectly competent job of introducing the basic definitionsand proving the basic theorems that any math student has to know. Artin\'s book is no exception, and I find his writing style to be veryappropriate for this purpose. What sets this book apart is its treatment of topics beyond the basics--things like matrix groups and grouprepresentations. I suppose many introductory books shy away from much of the material on matrix groups in Artin\'s book because itinvolves a little analysis (and likewise for the section on Riemann surfaces in the chapter on field theory). However, Artin correctlyrealizes that a reasonably mathematically mature student--even one who doesn\'t know much analysis--will be able to profit from andenjoy the relatively informal treatments he gives these slightly more advanced topics. Of course these topics can also be found in graduate-level texts, but I for one would much rather be introduced to them via an example-based approach such as that in Artin than through thediagram-chasing obscurantism in more advanced books. I happened upon this book a little late--in fact, only after I\'d taken a semester ofgraduate-level algebra and already felt like analysis was the path I wanted to take--but I\'m beginning to think I would have been morekeen on going into algebra if I\'d first learned it from a book like this one. 2) I bought this book for a class that I ended up dropping. In the beginning, I hated this book. I found Herstein\'s \"topics in algebra\"much better, and more to the point. It was only when I was getting bored with Herstein that I bothered to pick this up again. I waspleasantly surprised. A lot of the material flowed very smoothly - exactly as if Artin was teaching the material to you. It must however benoted that people tend to love or hate this book. This is predominantly due to the author\'s writing style. Given how expensive this book is,you might perhaps want to peruse it somewhere before deciding to buy it. But if you do, you\'ll get a solid exposition on most of theintroductory topics in algebra as well as some insight on groups and symmetry, lie groups, representation theory, galois theory andquadratic number fields. And a whole lot of intuition as well, for the more regular topics. Give this book a chance - it\'s worth the effortand money. 3) As an undergraduate I learned, or tried to learn, algebra from this book. Artin\'s pedagogical methods just didn\'t work for gh his idea of teaching through concrete, geometric examples sounds great in principle, in practice it\'s not so successful. It is veryhard to see the forest for the trees, since Artin is so chatty and discursive. When he is discussing examples, he sometimes puts specializedresults on par with more general theorems, which may be misleading. Many proofs are only sketched, and occasionally theorems are statedafter their proofs, necessitating a rereading of the preceding paragraphs in order to grasp the points of the proof. The chapters onrepresentation theory (Ch. 9) and arithmetic of quadratic number fields (Ch. 11) are nonstandard topics and interesting in themselves, butagain, the level of detail tends to obscure, rather than enlighten. The one saving grace of the book is the excellent problem sets at the end of each chapter. In doing them you will learn the algebra thatthe main body of the text attempts to impart.

书名：Codes and Curves作者： Judy Walker出版商： American Mathematical Society (2000) ISBN 0-8218-2628-X 系列：AMS Student Mathematic Library, Volume 7

页数：66适用范围：大学数学系本科高年级参考书预备知识：抽象代数习题数量：小习题难度： 容易

/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6推荐强度：8.5

书评： 这本小册子是1999年美国数学会在 Princeton 组织的暑期学校的一门课程的讲稿，是代数几何码的入门读物。代数几何码是新发现的 一种纠错码，目前仍有大量问题在研究。本书前一半对纠错码的基本知识和若干经典的纠错码作了扼要的介绍，重点是 Reed-Solomon 码，因为代数几何码是它的推广。然后，作者不加证明地清楚地叙述了有限域上平面代数曲线的基本知识， 最后介绍了代数几何码以及好的代数几何码的构造方法。 本书的一个显著特点是提供了六个供本科生研究的课题。本人曾指导复旦大学数学系毕业班的六名学生报告这本书，并围绕 六个课题查阅文献资料，写作毕业论文，取得很好的效果。 （杨劲根）

国外评论摘选

1) The book gives an overview of algebraic coding theory. The first chapter introduces error correcting codes, the Hamming distance,Reed-Solomon codes, and concludes with a brief exposition of cyclic codes. The second chapter discusses some upper bounds on theminimum distance of a code such as the Singleton and Plotkin bounds. The second theme of this book are algebraic curves. Chapter 3 contains the basic definitions and some examples of algebraic concept of a nonsingular curve is explained in Chapter 4. This chapter also contains a half page explanation of the genus of a Riemann-Roch theorem is finally covered in Chapter 5. The two themes come together in Chapters 6 and 7. These chapters discuss the basic principles of algebraic geometry codes. This little book gives the reader a first taste of an intriguing field. The most surprising part is how much is covered in so few pages [themain text without appendices has 44 pages]. The explanations are always accessible for undergraduate students of mathematics, computerscience, or electrical engineering. The prerequisites are some knowledge of abstract algebra, but most material is reviewed in theappendices. It is a lovely little book that is written in a lively style. The book nicely complements the typical college courses on coding theory. Ifyou want to get an idea what algebraic geometric codes are and you want a quick answer, then this is the book for you. 2) There is a free version of the book available on the website of the University of Nebraska-Lincoln. 目录： Chapter 1. Introduction to Coding Theory 1.1. Overview 1.2. Cyclic Codes Chapter 2. Bounds on Codes 2.1. Bounds 2.2. Asymptotic Bounds Chapter 3. Algebraic Curves 3.1. Algebraically Closed Fields 3.2. Curves and the Projective Plane Chapter 4. Nonsingularity and the Genus 4.1. Nonsingularity 4.2. Genus Chapter 5. Points, Functions, and Divisors on Curves Chapter 6. Algebraic Geometry Codes Chapter/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6 7. Good Codes from Algebraic Geometry Appendix A. Abstract Algebra Review A.1. Groups A.2. Rings, Fields, Ideals, and Factor Rings A.3. Vector Spaces A.4. Homomorphisms and Isomorphisms Appendix B. Finite Fields B.l. Background and Terminology B.2. Classification of Finite Fields B.3. Optional Exercises Appendix C. Projects C.1. Dual Codes and Parity Check Matrices C.2. BCH Codes C.3. Hamming Codes C.4. Golay Codes C.5. MDS Codes C.6. Nonlinear Codes

书名：Introduction to Commutative Algebra作者： Michael Atiyah & ald

出版商： Addison-Wesley Publishing Company (1991) ISBN 0-13-004763-5页数：126适用范围：大学数学系本科基础数学高年级或研究生低年级教材预备知识：抽象代数和点集拓扑习题数量：大习题难度： 较大推荐强度：9

书评： 英国皇家科学院院士 Michael Atiyah 是当代大数学家，曾或菲尔滋奖。本书是交换代数的入门书籍，是一本优秀教材，特别适合于代数几何、代数数论和其他代数专业的研究生使用。本书的篇幅虽小，内容却很丰富，包含了交换代数的核心内容。学过一学期抽象代数的人可以顺利学习本书前九章，学习第十和第十一章需要点集拓扑的基本知识。正文中的定理的证明简明易懂，有很多重要的定理安排在习题中，所以要掌握此书内容必须化工夫做每一章后的大部分习题。 本书以诺特交换环和有限生成模作为重点，这正是代数几何和代数数论中出现最多的代数结构。 作者在序言中说到域论没有 涉及，这可以从别的优秀教材（如 Nagata 的“域论”）中得到补充。 国外很多名校的数学教授将此书作为交换代数教材的首选。我国引进此书也很早，它很受师生的欢迎。 （杨劲根）

国外评论摘选

1) Some people believe that, for getting into algebraic geometry (by this I mean Grothendieck-like AG, with schemes and all that), oneneeds a monolithic training in commutative algebra (something like both volumes of Zariski-Samuel, for example). I disagree. This little/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6book seems to be specially suited to those who want to learn AG. It\'s a bit too brisk, specially at the beginning - if you don\'t already havean acquaintance with the basics of groups, rings and ideals, you may run into trouble - but very illuminating. Masterful choice of topics,great exercises (as a matter of fact, about half the topics of the book, and more specifically the ones that are directly related to AG, aretreated in the exercises, some of them quite challenging) - like one said before, it looks like a \"chapter 0\" of Hartshorne\'s book on AG. Theauthors consciously estabilish relations between the commutative algebra and the modern foundations of AG over and over along the way,illuminating both topics. For the algebra itself, it also gets on well with Rotman\'s \"Galois Theory\" and MacDonald\'s out-of-printintroduction to AG, \"Algebraic Geometry - Introduction to Schemes\", besides being the perfect preamble in commutative algebra to thebooks of Mumford and Hartshorne. A gem. 2)The strongest aspects of Atiyah & MacDonald\'s book are its brevity, accessibility to undergraduates, and subtle introduction of moreadvanced material. Audience: I think an undergraduate with a solid understanding of material from a first course in abstract algebra (i.e., the chapter onrings--the modules chapter would help, but isn\'t necessary--from M. Artin\'s book \'Algebra\' is more than sufficient) and some basic point-set topology from an intro real analysis course (or ch1-4 of Munkres) would be sufficient for fully appreciating the material. I thinkhaving experience in PS Topology is important for understanding parts of this book well; doing the exercises is possible if you learn it \"onthe fly,\" but I hadn\'t seen Urysohn\'s Lemma before, and even that caused me some \"intuition\" hangups; to fully appreciate the material, Iwould recommend doing a healthy number of problems in topology first. Material: The material uses concepts from homological algebra, though in a disguised form; students with experience in category theorywill find offhanded comments that recast some of the material in that language, but CT is absolutely not essential to understand thematerial well. It also provides exercises that lead naturally into topics from Algebraic Geometry and Algebraic Number Theory quitereadily; a nice set of problems in CH1 walk a student through construction of the Zariski topology, prime spectrum, etc., and somefunctional properties of morphisms between spectra. Algebraic Number Theory starts showing up after chapter 4 in greater detail, andwould lead comfortably into Lang\'s GTM on ALNT by CH9 (though I only read a bit of Lang, the first chapter felt natural). The \"details left to the reader\" are usually reasonably tackled with the tools made available so far, and the book is short enough thatone can cover a lot of ideas in a reasonable amount of time; the commentary made by the authors is brief, to the point, and neverredundant as far as I can recall, so I consider this a highly efficient book (but not too efficient, it\'s self contained enough and notuncompromisingly terse). Exercises: They are quite good, I think. Very few of them follow from \"symbol-pushing\" or \"robotic theorem proving,\" and usuallyrequire some constructive argument. The exercises are mostly chosen to introduce more advanced material, and do a good job in thatregard. The longer chapters have 25-30 exercises, and shorter chapters (a few pages) have maybe 10, so there are plenty of problems to do. Hazards: The material on modules is brisk, the propositions in the first three sections on modules are mostly left without proof;however, the proofs follow from their analogues for rings, and aren\'t that hard, just be sure to actually do them because they arementioned only briefly. Also, the book is not typo-free, but this only caused me one major hangup during the semester. After Chapter 3,the proofs are mostly complete, with a spattering of \"left to the reader\" exercises, which I usually found helpful. Companion Material: I think Lang\'s \'Algebra\' GTM would make a nice reference for the material on Homological Algebra and othermiscellaneous things that come up in the proofs; I remember once a proof in the book required the notion of the adjoint of a matrix over aring, and so I had to look it up in Lang, and also the basic category theory covered in CH1 of Lang would at least introduce (though in avery rapid way) the \"abstract nonsense\" mentioned offhandedly here and there. If you have a lot of money, or access to a good library,\'Categories for the Working Mathematician\' is a slower and more thorough introduction to that language, and I would recommend at leasthaving a look, though this isn\'t really central to the material from Commutative :///guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6 3) This is how mathematics texts SHOULD be written. As in technical writing, the smaller text is the better written text. Everything isclean and direct, with clairity obviously a prime consideration. One never gets mired down. The proofs are always as close to a \"THEBOOK\" proof as possible, with illuminating examples, and plenty of excercises, many with outlines for solution, which makes the bookideal for self study. This book is a revelation. If I had to take only one math text with me to a desert island, this would be the one. 4) This is a difficult book for undergraduates, even ones who have already had some abstract algebra. Many refer to the book\'s style as\"terse\", meaning that there is little explanation, few examples, and proofs are very condensed.

书名：HOPF ALGEBRAS作者： MOSS E. SWEEDLER出版商： W. A. BENJAMIN, INC. (1969)页数：336适用范围：大学数学系本科、数学专业研究生预备知识：代数、环模基础理论习题数量：小习题难度： 容易推荐强度：9

书评： 1941年，德国数学家H. Hopf在研究代数拓扑时引入了Hopf代数的概念。真正引 起人们对这类代数结构普遍关注的是1965年J.W. Milnor 和J.C. Moore的有关分 次Hopf代数的文章；到上世纪80年代末，量子群概念的出现及其在Knot不变量理 论中的应用将Hopf代数的研究推向一个新的高潮。如今，Hopf代数理论正在诸如代 数群、李代数、表示论、组合论以及量子力学等学科的研究中发挥着重要的作用。M.E. Sweedler所著的textquotedblleft Hopf Algebratextquotedblright是历史上第 一本系统介绍这方面Hopf代数知识的书籍。 这本书是从Sweedler给研究生的系列讲座内容中整理出来的，介绍的对象主要是非分 次的Hopf代数。域$k$上一个增广（augmented）代数$H$，若带有一个余结合的 （coassociative）和余单位的（counitary）代数映射$Delta colon HrightarrowHotimes H$，则称$H$是一个双代数（bialgebra），Hopf代数是指带 有antipode的双代数。本书开始先引入sigma-符号；而后一步步将上世纪70年代 以来有关Hopf代数的最新结果，其中大部分是作者与其合作者当时取得的进展，呈现 给读者；最后一章以证明域上由所有有限维交换、余交换的Hopf代数组成的范畴是交 换的（abelian）范畴作为结束。整本书的内容简洁，易懂，且自我包含，是一部很好 的关于Hopf代数知识的入门教材。它的不尽完美之处是没有列出任何参考文献 。 本书共有16章，有些章节非常短。前4章，给出了余代数、余模、及Hopf代数的 初步介绍，其中包括从模中构建余模的有理模构造方法及余代数的基本定理---该定 理阐明了在一个余代数中，任意有限个元素均包含于一个有限维子余代数中，故而任一 余代数都是有限维余代数的直接极限。第5章讨论了积分（integral）， textquotedblleft积分textquotedblright这一名称是因它很像紧群上关于 Haar测度的积分运算而得名。这一章的重要结果是证明任一有限维Hopf代数必存在 一维的积分空间；作为一个推论，本章中导出了群环的著名Maschke定理。 第6章对一个代数引入并讨论了它的对偶余代数。第7章到第9章主要介绍了度量 、smash积和外积等概念，为第13章余交换点（pointed）Hopf代数的结构定理的 证明作了前期的准备工作。 第10章的主要内容是Hopf代数作用的Galois理论；第11章对本原元进行了重点 讨论并引入了分次双代数的概念，在这一章以及后面的第14章中余代数的基本定理起 到了很大的作用。第12章考虑了shuffle代数和相关既约的点余代数万有映射性质 ，此外还讨论了divided power。 第13章中证明了著名的结论：任一既约的点余交换Hopf代数一定同构于Lie代数 或限制Lie代数的包络代数；据此导出了余交换Hopf代数的结构定理。剩下的两章 讨论了仿射群和由交换、余交换Hopf代数构成的Abelian范畴。 （朱胜林）/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6

7 数论书名：Elementary Methods in Number Theory作者： Melvyn son出版商： Springer-Verlag页数：509适用范围：大学数学系本科基础数学学生、数学专业研究生预备知识：微积分习题数量：大习题难度： 中等，多数习题很容易推荐强度：9.5

书评： 本书是Springer-Verlag出版的研究生系列教材中的一本，编号第195，2000年出版。全书分 为三个部分，第一部分介绍了初等数论的基本内容，整除性，同余，原根，Gauss二次互反律，有限 交换群上的Fourier分析，以及abc猜想的一个简单介绍。第二部分讨论了一些算术函数的性质， 给出了素数定理的初等证明。第三部分介绍了加法数论中的三个问题，即Waring问题， 正整数表为整数的平方和的问题，以及分拆函数的渐进估计的问题。 本书的一个特点是给出了许多深刻的数论定理的初等证明， 比如，Selberg的素数定理的初等证明，Linnik关于Waring问题初等证明， 一个整数表示为偶数个整数的平方和的个数的Liouville方法，以及Erdos关于分拆函数的渐进估计的结果。 事实上，本书的所有的证明都只使用了初等的方法，不涉及解析方法以及其他的高等方法， 因此本书也是一本很好的大学生数论教材。本书第一部分和第二部分作为大学生一个学期的课程是合适的。 （王巨平）

国外评论摘选

1) Every serious student of number theory should have this classic book on their shelf. Even though only \"elementary\" calculus andabstract algebra are used, a certain mathematical maturity is required. I feel the book is strongest in the area of elementary --notnecessarily easy though -- analytic number theory (Hardy was a world class expert in analytic number theory). An elementary, but difficultproof of the Prime number Theorem using Selberg\'s Theorem is thoroughly covered in chapter 22. While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theorygiven (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique. It covers more disparate topics in numbertheory than any book I know of. The fundamental results in classical, algebraic, additive, geometric, and analytic number theoryare all covered. A beautifully written book. Other recommended books on number theory in increasing order of difficulty: 1) Elementary Number Theory, By David Burton, Third Edition. Covers classical number theory. Suitable for an upper levelundergraduate course. Primarily intended as a textbook for a one semester number theory course. No abstract algebra required for thisbook. Not a gem of a book like Davenport\'s The Higher Arithmetic, but a great book to seriously start learning number theory. 2) The Queen of Mathematics, by Jay Goldman. A historically motivated guide to number theory. A very clearly written book thatcovers number theory at a graduate or advanced undergraduate level. Covers much of the material in Gauss\'s Disquisitiones, but withoutall the detail. The book covers elementary number theory, binary quadratic forms, cyclotomy, Gaussian integers, quadratic fields, ideals,algebraic curves, rational points on elliptic curves, geometry of numbers, and introduces p-adic numbers. Only a slight bit of analyticnumber theory is covered. The best book in my opinion to start learning algebraic number theory. Wonderfully fills the otherwise/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6troublesome gap between undergraduate and graduate level number theory. Full of historical information hard to find elsewhere, very well researched. To cover all the material in this book would likely take twosemesters, though most of the important material could be covered in one semester. Requires a background in abstract algebra(undergraduate level), and a little advanced calculus. Some complex analysis for sections 19.7 and 19.8 would be helpful, but not at all arequirement. The author recommends Harold Davenport\'s The Higher Arithmetic, as a companion volume for the first 12 chapters;according to Goldman it is a gem of a book. 3) Additive Number Theory, by Melvyn Nathanson. Graduate level text in additive number theory, covers the classical bases. This bookis the first comprehensive treatment of the subject in 40 years. Some highlights: 1) Chen\'s theorem that every sufficiently large eveninteger is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun\'s sieve for upper bound on thenumber of twin primes. 3) Vinogradov\'s simplification of the Hardy, Littlewood, and Ramanujan\'s circle method. 2) My initial reaction through the first chapters was one of embarrassment at my lack of understanding. I could not believe a book,hailed by so many as a standard and essential resource, could be so much out of my reach. Then, amid the last page or so of chapter 1 Ihad an epiphany. The book, from that point on, was completely clear and logical while retaining an extraordinary amount of breadth incoverage. Add my staunch support and recommendation to the long list of kudos that this book has accrued. There are, to my knowledge,no better books for the beginning student of number theory. If you have any interest whatsoever in the theory of numbers, this book isessential.

书名：A course in arithmetic

作者： J.-P. Serre出版商： Springer Verlag (1973) ISBN 0-387-90041-1页数：113适用范围：大学数学系本科基础数学高年级或研究生低年级教材预备知识：抽象代数,复分析习题数量：很少习题难度： 较大推荐强度：10

书评： 法国大数学家，菲尔滋奖和阿贝尔奖获得者 Serre 写过不少短小精悍的小册子，大部分从他亲自所讲授的课程的讲稿整理而成。 本书是他非常有代表性的本科生高年级的数论教材，曾在西方评为某年度世界最佳数学教材。 本书并不是数论的系统教程，作者选择数论中三个重要专题扼要叙述了它们的内容和方法，这三个专题是：二次型、素数的Dirichlet 定理 和模形式。读者可以化较少的时间学到一些近代数论的知识。最令读者欣赏的是定理的证明将大数学家的技巧展现得淋漓尽致，阅读中不禁 拍案叫绝。非定型幺模偶整格的分类定理非常漂亮，但其完整的证明在很多代数教科书中很难找到，本人所知道的就是本书以及 Milnor 和 Husemoeller 写的 Symmetric bilinear form 一书中的证明。这两本书都是70年代出版的，经过这两位菲尔滋奖得主之手的证明已经很难再 作改进，因此后人写的书大多只是引用而不再重写了。 具备抽象代数的知识就可以读懂前半本书，后一半需要复分析的准备知识。由于叙述简洁，习题数量少，作为教材使用会有一定困难。 作为自学的参考书对读者的数学素养也有较高的要求。 （杨劲根）

国外评论摘选

/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6 1) The book is divided into two parts -- algebraic and analytic. I\'ve only worked through the analytic part. Anything by Serre is worthits weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre\'s style is extremelycondensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can\'t digest more than half a pageof Serre a day; however if one wants to understand the structure of a theory, Serre is ideal. I worked through \"A Course in Arithmetic\"over a decade back. As I recall I covered Riemann\'s zeta function and the Prime Number Theorem, the proof of Dirichlet\'s theorem onprimes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modularfunctions. All of this material is also covered in Apostol\'s two books on analytic number theory (\"Introduction to Analytic NumberTheory\", and \"Dirichlet Series and Modular Functions in Number Theory\"); Apostol goes further than Serre in the analytic part -- which isonly to be expected since he is devoting two whole texts to the subject. 2) Serre\'s work could best be summarized in one word - Elegance. The book comprises of two distinct parts. The first one is the\'algebraic\' part. Serre\'s goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries toreaching this goal, he introduces the reader to quadratic reciprocity, $p-$adic fields and the Hilbert Symbol. After these three, he spendsthe next chapter detailing the properties of quadratic forms over ${mathbb Q$ and ${mathbb Q_p$ (the $p-$adic field). The reason towork over ${mathbb Q_p$ is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if andonly if it has solutions in ${mathbb Q_p$). Using Hensels Lemma, checking for solutions in ${mathbb Q_p$ is (almost) as easy aschecking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note:the classification goal is already achieved in previous chapter). The second half of the book is the \'analytic\' one. The first chapter in thissection gives a complete proof of Dirichlet\'s theorem while the second one studies the properties of modular forms (these are good!) Dueto the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it\'s not and I\'ll explain why. The booktakes some effort to read because it\'s terse and it often takes a while to figure out why something is \'obvious\'. However, once you see itall, you\'ll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that theauthor sets out for himself. Also, I\'d rather think for myself and read a smaller book than be given a huge fat tome where the authordetails his own thought process. This book was my first foray into number theory and I absolutely enjoyed it. If you\'re considering readingit, I wish you joy in your pursuits.

书名：Introduction to Analytic Number Theory作者： Tom Apostol出版商： Springer Verlag (1976) ISBN 0-387-90163-9

页数：328适用范围：大学数学系本科数论教材预备知识：微积分,复分析习题数量：大习题难度：一般推荐强度：9

书评： 这是一本非常受欢迎的数论入门教材，写得极其清楚仔细而又不烦琐。虽然书名是解析数论，事实上也包括了初等数论。 由于书的自封性能好，习题又经过精心挑选，适合于大学低年级的数论教材。 本书由于其良好声誉而多次再版，被选入 Springer 的 UTM 系列。同一作者的微积分教材（见本书的另一篇书评）也有好口碑。（杨劲根）

/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6 国外评论摘选

1) I think that there will be little harm if the title of the book is changed to \'Introduction to elementary number theory\' instead. Theauthor presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots,and quadratic reciprocity are included. Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chaptersare on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This laydown the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the authorexplain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is toprepare for the proof of Dirichlet\'s theorem and introduction of Gauss sums, the character theory is developed just to the point which is allthat\'s needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet\'s theorem on primes in arithmeticprogression. The proof still uses $L-$function of course, but the estimates, like the non-vanishing of $L(1)$ , are completely elementaryand is based only on the first 2 chapters. The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sumscan then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contourintegration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained bycrossing into the complex domains. The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zetafunction, L function ,are introduced. It\'s shown that the $L-$functions have meremorphic continuation to the whole complex plane byestablishing the functional equation $L(s)=$ elementary factor $* L(1-s).$ The reader should be familiar with residue calculus to read thispart. Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it\'s the Prime NumberTheorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the PrimeNumber Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, thenestimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quitesmoothly. As is well known , a key point is that one can move the contour to the line $Re(s)=1,$ and to do this one have to verify that$zeta(s)$ does not vanish on $Re(s)=1.$ The proof , due to de la vale-Poussin, is a clever application of a trigonometric unately, the method does not allow one penetrate into the region $0<1,$ where distribution contain information about flunctuation$Pi(x)$ around $x log x.$ famous Hypothesis states zeroes region lis line $Re(s)=\"1/2.$\" After than 100 years, Riemann Hpothesisnatural generalisation fields, neither indicates difficulties Recently some new directions, related quantum statistical mechanics, has beenconnected old problem. If RH proven, then set prime numbers , although looks completely random locally like occurences twin primes),governed by clear-cut laws large after all. last Chapter quite differnt flavour, so-called additive theory. Here author only focusses on ---theunrestricted partition. interesting phenomeon occur already at this level. first result theorem, simple recursion formula function p(n). proofsare given. beautiful no doubt a combinatorial due Franklin. third establishing Jacobi triple product identity, leads besides Euler?spentagonal number theorem. Jacobi?s original uses his theta but it turns out that power series manipulaion all that?s needed. The endsindication deeper aspect partition theory--- Ramanujan?s remarkable congrence and identities simplest being $p(5m+4)=\"0\" pmod{5$ ).To prove mysterious identites, ?natural?way plow through theory modular functions, which Ramanujan had left lots more theorem (unfortunately most without proof). However elementary proof of one these identites outlined in exercises. This book is well written, withenough exercises to balance the main text. Not bad for just an ?introduction?. 2) This book has excellent exercises at the end of each chapter. The exercises are interesting and challenging and supplement the maintext by showing additional consequences and alternate approaches. The book covers a mixture of elementary and analytic number theory,and assumes no prior knowledge of number theory. Analytic ideas are introduced early, wherever they are appropriate. The exposition isvery clear and complete. Some novel features include: three chapters on arithmetic functions and their averages (including a simpleTauberian theorem due to Shapiro); Polya\'s inequality for character sums; and an evaluation of Gaussian sums (by contour integration),/guide/wjzx/#6[2008/7/13 14:00:29]复旦大学外国教材中心#6used in one proof of quadratic reciprocity.

8 代数几何书名：Introduction to Commutative Algebra and Algebraic Geometry作者： Ernst Kunz出版商： Birkhauser Boston, (1985) ISBN 3-7643-3065-1页数：238适用范围：基础数学本科高年级或研究生低年级预备知识：线性代数，群，环，域扩张，Galois 理论的基础知识，最基本的点集拓扑习题数量：大习题难度： 大部分中等，少量难题推荐强度：9

书评： 作为交换代数的入门书，它不如 Atiyah-McDonald 的有名，也不如 Eisenbud 的大， 但是我认为对于有志学习代数几何的大学生来讲，这是最好的入门书。此书与大学基础课程 的衔接非常紧密，不管是自学还是用此书当教材都比较轻松，如果再认真做习题则效果更好。 交换代数的内容甚广，作者完全按经典代数几何的需要选择交换代数的内容，重点是多变量 多项式环的商环及它上的有限生成模。除了基础性的材料外，也有少量研究性的题材。现在 代数几何最流行的研究生教材是 Hartshorne 的Algebraic Geometry, 本书可以认为是 Hartshorne 的教程的前续课程。本书作者是著名的交换代数专家，原书用德文写作，后翻译成英文，美国 代数几何大师 David Mumford 写了序言，称此书是美国学代数几何学生久等的一本书，它填补 了一个空白。（杨劲根）

书名：Basic Algebraic Geometry (Second, Revised and Expanded Edition)作者： Shafarevich出版商： Springer-Verlag (1988) ISBN 3-540-54812-2

页数：上册 303 下册 269适用范围：基础数学研究生预备知识：近世代数、复分析、点集拓扑习题数量：大习题难度：较难推荐强度：9

书评： 本书是俄罗斯的数学大师 Shafarevich 的力作，由英国著名代数几何学家 Miles Reid 翻译成英文。本书内容非常丰富且不枯燥，叙述和证明清晰，比较容易读，是非常收欢迎的一本 代数几何书，国内外不少院校开设代数几何课曾将此书选为研究生教材。全书分三大部分， 第一部分是射影簇，内容包含经典代数几何，一直讲到代数曲面的分类和奇点，其中不乏其它 教科书中不多见的内容。这一部分占了整个上册，作为一个学期的课程内容够多的。 第二部分是概形理论，用现代的语言来刻画代数簇，最后讲到 Hilbert 概形。本部分内容比较 简要，基本上讲清概形和层论的威力。第三部分是复代数流形的拓扑和几何，很多内容如代数簇的拓扑 分类和 uniformization 在其它代数几何教科书很难找到。总之，本书基本上讲述了代数几何的所有方法。 习题非常丰富，大部分的习题很有意思，可以看出是作者和他的助手们多年积累而编成的。 还有一个显著的特点是本书不需要交换代数的预备知识，当然学过交换代数在看此书更加轻松。 （杨劲根）/guide/wjzx/#6[2008/7/13 14:00:29]