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。
　　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，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。

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 California at Merced, Virginia Polytechnic Institute and State University, University of Massachusetts at Amherst, Florida State University, Georgia Institute of Technology, Oklahoma State University, Sonoma State University, St. Louis University, Winona State University, University of Rhode 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 having to do page upon page of number crunching. Use this book along with a graphing calculator and you too can learn to literally see what happens when equations are manipulated. A begining student conceptually gains an understanding of the subject with out getting bogged down in plugging and chugging and derivations. It's written in plain English.

　　ii) The authors of this text dislike the "plug and chug" methods of other texts, possibly necessitating an instructor more strongly than other books. The book stresses graphs and "real life" applications, making it more realistic and less abstract than other Calc books may seem. Contains 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 at Amherst, San Joze State University, Michigan State University, Tufts University, University of Michigan at Ann Arbor, University of Virginia'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 South Bend, Purdue University, University of Washington, Florida State University, California State University, Indiana University, Southeast Grinnell 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, University of Pennsylvania, California Polytechnic State University

　　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 of Charleston, Johns Hopkins University, Wake Forest University, Emory University, Florida State University, California State University at Stanislaus, Boise State University, University of Washington, The University of Western Ontario, Stony Brook State University of New York, 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 College

　　iii) 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, Boise State University, University of Colorado at Colorado, Springs University of Minnesota, Morris University of Rhode Island, Stony Brook University, Oberlin College, University of California at Irvine

　　iv) 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 State University, Stanford University, University of Minnesota, University of Tennessee, Northwestern University, University of Cincinnati, Utah State University, Oklahoma State University, University of Wyoming

3)
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. The author team’s extensive teaching experience and proven ability to write innovative and relevant problems has made this text a true bestseller. Exciting new real-world applications make this new edition even more meaningful to students in management, life and social sciences. This book will work well for those departments seeking a middle ground for their instructors. APPLIED CALCULUS, 3/E exhibits the same strengths from earlier editions including the “Rule of Four”, an emphasis on concepts and modeling, exposition that students can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativity and variety, continue to motivate and challenge students.

ii) This is a magnificent calculus book. It is aimed at students in business, the social sciences, and the life sciences. This is done by first the examples and problems. But perhaps even more important the wording of the text is such that these students will understand what they are trying to convey and to clearly show them how calculus can be used to solve problems in their particular field.
　　At the beginning of the book, three pages of the Preface, the applications discussed in the text are listed by: Business and Economics, Life Sciences 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 traditional problems 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 State University, San Diego State University, University of Rhode Island, University of California, University of Colorado, University of Central Arkansas, Fayeiteville State University, Brigham Young University, University of Calgary, Oregon State University, University of Illinois at 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 introduce you to the contemporary way to do multivariate calculus, including existence theorems connected to completeness. Very thorough treatment of 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 of exercises 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 they don'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 to know about analysis on manifolds but were afraid to ask". This book is a must-reading for the analyst. It covers everything from the most basic vector space concepts up to the fundamental theorems of classical mechanics, running through multivariate calculus, exterior calculus, 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 is an 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, The Behrend College, University of Colorado at Denver, Alamo Community Colleges, Johnson County Community College, The Community 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 years ago our university adopted the 6th Edition of Larson. As a pedagodical tool, this text is head and shoulders about all the others. The text uses abundand graphics, a clear design, concise writing, thoughtful examples, and carefully crafted exercises to make calculus accessible to students. 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, showing every 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 to spend on reading long discussions of theorems... and for someone who doesn't want to go too deep into material and wants to quickly get the 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 about this 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, Georgia Institute of Technology, Athabasca University in Canada, University of Washington, 台湾国立成功大学, New York University, The University of Texas at Austin, Georgia State University, University of Chicago, University of Illinois at Urbana-Champaign, New York University, 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 sound a little incongruous in relation to calculus) was the step-by-step build-up of knowledge with good, clear examples. Also, for the problems at the end of each section, all the odd problems have solutions, so one can get some practice (something that is unfortunately rare for many 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 key reason 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 book which made matters somewhat difficult. However, he was showing us the meaning of math which I found refreshing. This book serves its purpose as one which teaches the mechanics of solving problems but very little in developing an intuitive feeling for mathematics. I must admit that the multitude of exercises were very helpful in getting comfortable with difficult mechanical problems. For single variable calculus it is a standard book with good examples, excellent diagrams, and some applications. Getting into multivariables, the ideas are not connected 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 a text for multivariable calculus. If you are interested in theory I recommend Apostol's Calculus which covers a great range of material with rigorous 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: the State University of New Jersey, Whatcom Community College, University of Wisconsin at Green Bay, King's College, University of London, 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 90 percent 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 year my 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 other calculus textbooks. The book explains everything well and provides you with many examples. I am a math major and this book has been really 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
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 State University at Northridge, Syracuse University, Worcester Polytechnic Institute, Oregon State University, Saint Louis University, The Ohio State University, Southern Oklahoma Technology Center, Southern Illinois University at Edwardsville, Saint Louis University, Denison University, 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 teach Calculus. 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 in each 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. The problems are intended to be instructional. Indeed this book has a corresponding student solutions manual that helps students to check their work and see if they are "getting it". The problems in the book range from extremely elementary up to moderately challenging. If, instead of instructional problems, this book had given enough examples and text to explain all of the ideas, it would have to be over 2000 pages long. Students should think of the problems in each section as being part of the instruction instead of problems to test previously acquired skills.
    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 of time struggling with some of them, but for a talented and dedicated student, every problem in the book is accessible and most are extremely instructive. I should also mention that the book is very well written. Having never actually read a math text book from cover to cover 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 to read. 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 if money 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 lecture from the text and examples, but dig into the problems and carefully choose the most instructive ones for in-class presentations or homework assignments. If you only lecture from the text and examples, you'll only be teaching your class a small fraction of what this book 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 nothing but working problems. It might be best to work though the even numbered problems for your class, as the odd numbered ones all appear in the 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 arts colleges. It is not a meant to challenge exceptionally bright students. For this reason parts of chapter 2 seem inappropriate- specifically the sections 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 experience the 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 with them 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. For example, the chapter on integrating to find the volumes and surface areas of solids of revolution comes way too early while the chapters 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 exponential function and it's properties. I, and my students, find it more natural to define the Euler number, e, in terms of continuously compounded interest, and then derive the natural logarithm and its properties from the exponential function. It's a matter of taste, but the later approach seemed 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 Calculus sequence, but it also has chapters on analytical and numerical solutions to ordinary differential equations as well as an appendix containing more theoretical material for brighter students. If you find yourself teaching an unusually talented bunch of kids, the appendix on mathematical induction as well as the aforementioned sections on ODEs and proofs of continuity and existence of limits can make great supplements 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 value. It 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 an art: 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 relate the explanations to one another. The Purcell I read (the 1st edition - it was my dad's) is quite masterfull at that. Often, when my college standard 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 of Technology，SUNY Institute of Technology，Rivier College，New York University，Duke University，University of Colorado at Denver，Yale University，University of Houston，Loyola University，Drexel University，Tufts University，Stanford University，University of Regina，North Carolina State University，Brown University，Dartmouth College，University of Washington，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:\\ocw.mit.edu\ 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 a chance two years ago and adopted Gilbert Strang's linear algebra book for a large engineering course. We used the second edition of Introduction to Linear Algebra, Wellesley-Cambridge Press. Several colleagues said it couldn't be done, but students and the instructor survived nicely to see another day. Many students said they enjoyed the book. Gilbert Strang's enthusiasm for the subject matter comes through in the text and students find it a refreshing change. Another strong point is an extensive set of problems. Many problems probe the subject in a way that requires students to think about linear algebra. Routine problems are not forgotten. This is good. Students can work on problems that help them put the subject in their own voice. A third strength is the layout of topics. Matrix multiplication and elementary row operations from a matrix viewpoint are developed first, and this provides an opportunity to discuss row reduction, matrix inverse, and the decomposition with little extra effort. Other standard subjects follow in order and orthogonality arrives early. Computation is 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 for engineers has increased dramatically in the last two years. More than 250 students studied linear algebra and matrix theory at Drexel University 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 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 no wrong. 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 its theoretical beauty with a matrix based approach to vector spaces, and an ugly preoccupation with ${\mathbb R^n.$ It's clear that this book was not written to be either a lucid explanation of how to use linear algebra, nor was it intended to be an aesthetically pleasing exposition of theoretical linear algebra. It was written somewhere in between, and it is an unhappy medium. If you are interested in a theoretical treatment of linear algebra, there are sorrowfully few good texts available. The title of Axler's "Linear Algebra Done Right" is a result of this fact, and if you are seeking a mathematically pure treatment of the subject, that book is a much better choice. If you're not interested in 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 of Minnesota，University of Colorado at Denver，Cornell University，Massachusetts Institute of Technology，Loyola University，Drexel University，University of Maryland，Columbia University，Brown University，Rutgers, The State University of New Jersey，Michigan Interdisciplinary and Professional Engineering (InterPro)，University of Nevada, Reno，University of Alabama at Birmingham，College of 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 they need crystalline theorems. That's why it's so helpful to see applications like Markov models, Kirchoff's laws, and Google's analyses of the web. It's also helpful to see examples worked in Mathematica and MATLAB, the tools of choice for desktop exploration of numerical systems. 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 bar for the readership, but also opens up topics like change-of-basis in function space, including Fourier analysis. It also allows differential equations to be addressed as linear systems. Even without calculus, though, a reader is exposed to the singular value decompostion, QR and other matrix decompositions, and considerations in performing the computations. I found a few oddities, such as the description of a matrix's condition number. That has great physical meaning when it's taken as the ratio of the matrix's highest and lowest eigenvalues, but Strang gives a definition that I found less intuitive.
    Such oddities are rare, though. Even though this book covers many topics, its emphasis is on clear and applicable presentation. I recommend 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 algebra class. Math majors will not find the 'definition-proposition-lemma-theorem-proof-corollary' treatment here. Instead Strang, aware of the need to teach non-math majors the subject, explains linear algebra in a simple but effective way --examples, diagrams, motivations. This book 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 the classroom).

    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 State University，Michigan Technological University，Salisbury University，Stony Brook University，University of Maryland，University of Connecticut，University of Massachusetts Amherst，University of Missouri-Rolla，University of Oregon，University of Texas At Austin，Boise State University，Brigham Young University，New Mexico State University，New York University，San Jose State University，Yale University，Westmont College，Rivier College，University of Delaware，University of London, University of Richmond, University of Rochester, Eastern Mennonite University, Princeton University, University of Colorado at Denver, City University, 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 from using wording that distracts the reader from the theory he presents. Mathematical notation is introduced before it is used, and proofs are placed in an appendix. Overall, this is a very good book for undergraduate study. It won't carry you through graduate classes, but it might be useful as a support book if you have a weak background in the topic. Math majors who love concise formalism and extended proofs should 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. This book is an exception to the norm. Not everything, but most things, are presented in a way that most students will be able to absorb on their 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 of Colorado at Denver，McGill University，Athabasca University Canada's Open University，Victoria University of Wellington, New Zealand，Brandon University，Louisiana State University，Indiana University-Purdue University，State University of New York College at Brockport SUNY Brockport，University of Manitoba，The Richard Stockton College of New Jersey，Florida Atlantic University，Saint Vincent Colllege，University of East Anglia，Norwich University，University college Dublin，Cardiff University，University of Essex，University of Calgary，Durham University，Queens College，wellesley College，Lehman College，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 the notation and basic tools, i.e. vector and matrix arithmetic, within the intuitive geometric settings of the Euclidean plane and space. Once the basic concepts of Euclidean vector spaces have been mastered, Anton moves into abstract vector spaces, linear transformations, and eigenvectors. One chapter is spent on complex matrices, and another chapter deals with numerical issues and least-squares applications. The only topic which is noticably missing is the singular value decomposition, but other than that, Anton is a remarkably complete text. The definitions and theorems are clearly presented, along with the motivating intuitions. The exercises at the end of the chapter sections are 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 is due to several misprints in the early chapters. Some of the definitions of basic concepts are confusing at best, wrong at the worst. I found myself relying on the Hubbard-Hubbard "Vector Calculus, Linear Algebra, and Differential Forms" to get through the course. The explanations 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 of Toronto，City College of San Francisco，Drexel University，Eastern Michigan University，Towson University，University of Wales，University of Iowa，Stony Brook University，McMaster University，York University，University of Southern Indiana，Binghamton University，University of Melbourne，University of Stirling，College of the Canyons，Middlebury College，Elon University，Kennesaw State University，University of Manitoba，University of Colorado at Colorado Springs，University of Guelph，University of West Georgia，University of Victoria，Chaffey College，Wayne State University，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 students. Accordingly, explaining how to take determinants or to invert a matrix lets you build confidence in your knowledge. Also, these topics lends 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 are exposed to the concept of an abstract vector space. Which invariably some students always trip over. So the grounding in the early chapters can mitigate this awkwardness.
    The last chapter touches lightly on the interesting applications, like chaos and fractals. But mostly to pique your interest in proceeding further 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 this class. That being said, this was one of the roughest classes I've ever got through only because the book kept going against the grain in every way possible. I didn't even begin to understand the entire point of linear algebra until about chapter 7 and 8 when the chapters started going into the general cases, and even now, I know how to "solve" all the problems without even knowing their meaning, which seems totally pointless to me. The selected answers to the problems in the book are in no particular pattern. It's not "all odds" or "all evens"; 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 the purpose or main idea of the subjects it discussed. The "explanations" it gave would be based off of other vague topics. For example "What is the Eigenvector Problem? Well, the eigenvector problem asks if there is a basis for $R^n$ in a $n \times n$ matrix consisting of eigenvectors 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 that spans 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 the book 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 getting them 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 to explain 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 of Pittsburgh，University of Texas，Johns Hopkins University，West Virginia University，University of Houston，Simon Fraser University，Washington University in St. Louis，University of Notre Dame，University of Wisconsin-Madison，Cornell University，University of South Carolina，University of Rhode Island，University of Missouri，University of Maryland，Stony Brook University，University of Michigan，Purdue University，University of Kansas，United Arab Emirates University，Rice University，Kenyon College，Temple University，Louisiana State University，Sonoma State University，North Carolina State University，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. It won'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 book on Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn about Linear 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 regular undergraduate course anymore. Instead of developing the ideas in the familiar context of the real numbers, Hoffman and Kunze give a more abstract (and general) discussion. For example, the theorems about determinants work in all commutative rings. The rigorousness and the wealth of information are overwhelming for most undergraduates to handle. You will not learn anything if you just glance through the pages. 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，Johns Hopkins University，University of Minnesota，McGill University，Colby College，Santa Clara University，University of California，State University College at Buffalo, Queen's University, Georgia Institute of Technology, Northeastern University, Purdue University, 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 too hard to teach through "Real World" examples. i find such examples confusing because they obscure the math behind the example. I also felt this book had a nice mix of easy, medium and challenging problems. And it feels like the author really understands and strives to clarify 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 to guarantee 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 semester of a two semester, undergraduate sequence on multivariable calculus and linear algebra. It is intended for physics, chemistry and strongly quantitative economics majors. As such, in terms of complexity it is more par with a collegiate abstract algebra text, with a clear focus however on linear algebra. The "applications" portion of the title is a bit of a misnomer, as examples only occur in the problems and almost 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 of economics and finance as generally taught at the undergraduate level would best look elsewhere. Most prominently, the text has almost no redundant examples, which makes it a enjoyably lucid read for those who grasp concepts quickly on the first go, but a dead end for those who come up short. I would not as professor think of assigning this book to non-Ivy caliber students outside of pure math; even Harvard students 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 important theorems, proofs and other pieces of explanation commonly found in other text books I have looked through, and rather than making up for it with a decent explanation or summary for what it omits, it leaves gaping holes in many topics. It gives partial proofs and explanations at times and leaves other pieces "for you to solve as exercises." It's like the [person] who made this book only wrote half a math 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

    使用学校：
    Rowan University, Arizona State University, Florida International University, Northern State University, University of Illinois at Chicargo, University of Puerto Rico, Colorado State University, State University of New York Institute of Technology, SUNY Institute of Technology, University of Hawaii, Ohio State University, University of Minnesota, Texas A\&M University, University of Massachusetts Dartmouth, 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 you have taken an introductory course in linear algebra or you already have a reasonably well foundation in this subject, then you should have no problem in understanding following the text. Although the explaination in this book is not particularly outstanding, it does treat some advanced topics like eigenvalues, numerical linear algebra elegantly. I would like to recommend this book to persons who would like to seek 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 basic concepts of linear algebra, including matrix row operations, determinants, and linear independence. The book seems to lose clarity beginning in Chapter 4. The concepts become more abstract and Leon's notation interferes with the ability to clearly understand what he is talking about when it comes to linear transformations and issues regarding $R(A)$ and orthogonality. Very important results are frequently understated 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 the relative overexplanation found in Chapter 1, for example.


2.3 其它

书名：Concrete Mathematics, 2nd ed.
作者： R.Graham, D.Knuth, O.Patashnik
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, even someone with little formal maths background like myself can get a lot out of it. It's beautifully written and well-presented, and on the whole 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 will make 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 that sometimes 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. As well 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 is the 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 are important to CS. Moreover, the information is presented in a light-hearted way, with lots of inline jokes (mainly very corny) and margin notes from students who took the lecture course behind the book. The examples tend to help, and there are plenty of exercises with worked 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。
    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 professors abilities... After all, these are the guys that consistently screw up 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 math major) 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 the authors 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 good book.... Yes a professor would be nice, but these guys did a good enough 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 will probably satisfy your needs. However, for any other type of course, it will most certainly prove to be inadequate. Nearly half the book is devoted to graph theory, and while many theorems are listed, very few are proven. The working computer scientist may find that acceptable, but most mathematicians will find it inadequate. Logic and the basics of proof are relegated to an appendix. The first chapter covers some combinatorics and the basics of algorithmic analysis, which is meant to be a primer. However, it requires the use of set terminology, set notation and basic counting techniques. Since set theory is covered in chapter 2 and counting techniques in chapter 7, I consider the order to be inappropriate. Recurrence relations, circuits and finite state machines are also covered in other chapters. There are a large number of exercises and the solutions to the odd numbered ones are included. Sets of problems to be solved by programming a computer are given at the end of each chapter, some of which are easy, but many of which are hard. Only students who have had a programming course could be expected to be able to do any of them without significant help. This is a book that does not satisfy my requirements for a discrete mathematics textbook. I consider logic to be a critical topic that must be covered, so I will not consider using any 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 ones.