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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.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
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，最高的是对物理系和力学系开设的数学，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, Harvard University, 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合力推出这本全新的微积分教材，简称为哈佛微积分，这是一套受美国国家自然科学基金会重金支持的教材。
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。这正是作者提到的“三步法则”(Rule of three): 图像、数值和解析式。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精心设计的文字、生动的图象来解释，然后再用一系列实例来展示其威力，最后在选学内容中写出精确的定义。
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。
本书还有后续本：多变量微积分，仍保持原来的风格，主编的次序改变了，编委名单也有一些改动，菲尔兹奖得主 David Mumford 的名字出现在编委名单中。（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
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) 书名：Introduction to Linear Algebra, 3rd edition
作者：Gilbert Strang
出版商： Wellesley Cambridge Press (2003)
页数：
适用范围：理工科大学本科二学年
预备知识：微积分
习题数量：大
习题难度：容易
推荐强度：8.3
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
国外评论选摘：
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) 书名：Linear Algebra and its Applications, 3rd ed.
作者：David C. Lay
出版商：Addison-Wesley
页数：445
适用范围：工科经济农医类本科二学级数学教材
预备知识：微积分
习题数量：大
习题难度：容易
推荐强度：9
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
书评：
作者是在序言中提到过的 1990 年美国线性代数第一教程研讨会的主要与会者之一，所以本书可以代表这个课程的主流。
这是一本标准的非数学专业的中等程度的线性代数教材，虽然有 445 页，但是浓度不大，可以在一个学期学完。
本教材的特点在于其应用性。每一章都已一个实际问题开始，该章结束时给出用本章的数学解决 那个实际问题的方法。这些问题及所在的章节罗列如下：
第一章 线性方程组 -----------经济学中的线性模型
第二章 向量和矩阵 -----------营养学中的问题
第三章 矩阵代数 -------------计算机图形和自动设计
第四章 行列式 ---------------平行六面体的体积
第五章 向量空间 -------------宇航和控制系统
第六章 特征值和特征向量 -----生态保护中的动力系统问题
第七章 正交性和最小二乘法----北美洲的统计数据之调整
第八章 对称矩阵和二次型 -----多通道图象处理
本书的另一个特色是图例丰富多样，有助于初学者比较直观地理解线性映射等抽象的概念。为了便于使用者学后查阅，书后还附有书中主要术语的小词典。（杨劲根）
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) 书名： Elementary Linear Algebra, 9th edition
作者：Howard Anton
出版商： John Wiley & Sons
页数：
适用范围：工科经济农医类本科二学级数学教材
预备知识：微积分
习题数量：大
习题难度：容易
推荐强度：8.5
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) 书名：Elementary Linear Algebra (application version), 9th edition
作者：Howard Anton
出版商：John Wiley & Sons
页数：
适用范围：农医人文科学类本科二学级数学教材
预备知识：微积分
习题数量；大
习题难度：容易
推荐强度：9.0
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
书评：
这是我用过的内容完整的最浅的线性代数教材，以计算和应用为主，但决不是一本“傻瓜书”。虽然绝大部分定理没有证明，但是诸如矩阵的列空间、秩、齐次线性方程组的解空间一类概念还是会让一部分学生头疼的。
不象很多教材在写完向量空间后就马上写线性变换，本书作者把线性变换放到最后，在它之前安排了内积空间和特征值，原因大概是线性变换这部分的内容比较抽象，放在后面较合适。由于深度的限制，通过这本教材不太可能在几何上对线性代数有深入的认识。对于时间宽余的学生，可以把这本书当作入门读物，学完后再念一本更深的教材。（杨劲根）
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) 书名：Linear Algebra with Applications, 3rd edition
作者：Otto Bretscher
出版商： 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.
7) 书名：Linear Algebra with Applications, 5th edition
作者：Steven J．Leon
出版商：Prentice-Hall
页数：491
适用范围：理工科本科二学级数学教材
预备知识：微积分
习题数量：大
习题难度：中等
推荐强度：8.5
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
书评：
本书目前已出版到第7版了，我这里只找到第5版。这本书的前六章的编排十分传统，内容也比较规范，从 Gauss 消元法、矩阵和行列式到向量空间和线性变换，再将正交性和特征值，没有讲线性变换的标准型。最后一章（第七章）讲数值线性代数，即线性代数的近似计算方法，一般的线性代数教材不含这方面内容。
本书几乎所有的定理都有证明，证明比较简洁，对读者理解有一定要求。如果用一个学期学本书的前一半还是比较轻松的，后半本显然要难一些。（杨劲根）
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.
8) 书名：Linear Algebra Done Right, 2nd edition
作者：S. Axler
出版商：Springer (UTM) 1997
页数：491
适用范围：理工科本科二学级线性代数教材
预备知识：高等数学
习题数量：中等
习题难度：一般
推荐强度：9
书评：
本教材比较适合已经学过一些基本的线性代数（如大学一年级的“高等数学”中的线性代数部分）的学生，内容和篇幅适合一个学期。
除了标准线性代数教材所需具备的条件外，本书最大的特点是线性变换的特征值的存在性的证明避开了行列式。按照常规的教程，先把线性变换的特征多项式定义为行列式 $|xI-A|,$其零点是特征值。利用复数域上的代数基本定理，特征值总是存在的。本书中的证明是这样的：对任意线性变换 $A,$ 任取非零向量 $u.$ 则 $u, Au, A^2u , /ldots, A^n u$ 线性相关，于是存在多项式 $f(x)$ 使$f(A)u=0.$ 将 $f(x)$ 分解成 $f(x)=c(a_1-x) /cdots (a_n-x).$ 则$(a_1 I - A) /cdots (a_n I -A) u=0.$ 因此某个 $a_i I -A$是不可逆的，这意味着 $a_i$ 是一个特征值。
作者认为行列式是一个非常不直观的概念，在线性代数教程中过早地使用行列式是违反学习规律的，这也成为本书起名的主要原因之一，虽然多少有点“狂妄”，但是线性代数似乎应该这样来学。
这本教材自出版来很快受到很热烈的欢迎，值得国内大学从事线性代数教学和改革的人员关注。由于本书的严密逻辑和行文的严谨，自学这本书也是比较容易的。（杨劲根）
国外评论选摘：
1) I have used this text for a beginning graduate course in linear algebra, mostly because I prefer its treatment of eigenvalues and eigenvectors over Hoffman and Kunze, and it sticks to the basics: complex scalars. It also has a good treatment of inner product spaces. The basic concepts and theorems are indeed presented cleanly and elegantly. Its use of linearly independent sequences (rather than sets) is a little nonstandard (what if the set of vectors is infinite?) but the adjustment is minor. Two things though I found treated in a less than desirable fashion: He pretends that we don't know about matrices, doesn't want to develop the machinery, and the treatment of coordinate vectors and matrix representations suffers. Students also get no sense of how to compute the solution of concrete vector space problems, which is easily done once the theory is established, and which is an essential skill to have after a second course in linear algebra. I have to give them supplementary notes. Second, the treatment of determinants suffers, apparently for ideological/political reasons. I think students deserve a straightforward development of determinants simply because that theory is widely used in applications, in engineering, and in discrete mathematics, and it has its own beauty. It is not hard to do, and I do it myself from notes, adapted from the treatment of Hoffman and Kunze. Now that undergraduate linear algebra courses have in many places dropped any substantial theorem-proving component, students need a serious course in linear algebra which can take them, e.g. all the way into Jordan form. There are not many good books for this, and this text does a good job with the basics without overkill on the abstraction, so I use it despite the drawbacks mentioned above.
2) I have no doubt that this is one of the most thought provoking math books that I have come across. I used this book for a linear algebra course last fall '08 and I learned a ton. Specifically about the structure of vector spaces and linear operators. However, the most important function that this book serves is to move students towards the methodology of mathematics, which means proof construction and counter examples. It also trains students to let go of their intuitions. But you can not self-study this book, there are no answers and more importantly the structure of the course begs for instruction. I would recommend before taking this course doing what i didn't do and have had to do since, make sure you have your first course of linear algebra solidly under your belt, and that doesn't mean having gotten an A in the prior class is sufficient. Go through the most difficult proof driven exercises in your first text, that should serve as practice for easiest homework problems in this book.
All that said, there are serious limitations to this book. It would be nice if the author worked out 1 comprehensive semi-difficult exercise in each chapter of the text. While struggling to solve the problems can be enlightening, there is only so many times I can read the same sections over and over again, looking for some insight from the kiddie exercises provided by the author. It would also help if some of the kiddie exercises were accompanied with graphs, especially when describing the sums of vector spaces. Sometimes a picture is worth a thousand words - sometimes!
Last but not least, the author has a copyright on the solutions to the book. Where he does not allow professors to post homework solutions to exercises. This had a devastating effect on the class I was in, because there were many students who were lost in the first couple of homework sets and basically were never given a chance to figure out what was going on. Pedagogically, this is unacceptable. Furthermore it sets a dangerous trend, math problems simply stated should not be copyrighted. For this reason I suggest that people not purchase this book, but I still strongly recommend that they get a hold of it.
2.3 其它
书名：Differential equations, 2nd ed.
作者：P. Blanchard; R. Devaney; G. Hall
出版商： Brooks/Cole Thomson Learning (2002)
页数：697
适用范围：理工科大学非数学专业数学教材
预备知识：初等微积分和线性代数
习题数量：大
习题难度：容易
推荐强度：9.4
使用院校：
Harvard University, Saint Joseph's University, Florida Atlantic University, Agnes Scott College, HaverFord College, Indiana University, University of Connecitut, University of Georgia, Boston University, Portland State University
书评： 这是一本应用性的常微分方程教材，在复旦大学外国教材中心将它列为哈佛教材的一种。它对学生的数学预备知识的要求不高，只要最基本的微积分和线性代数就够了。本书的内容
覆盖了常微分方程方面的所有重要内容。
这本教材最大的特点是在定性理论方面比同样深度的教材多而详细，作者并不把它集中在一章讲述，而是把它贯穿在全书中，用这样的方法由浅入深地把一些比较难说清楚的概念讲的非常清晰。
本书的英文非常通俗易懂，虽然理论性的证明很少，但是对主要定理的解说十分具有说服力。我曾经使用此教材教过一个学期，对象是相当于我国非重点工科大学的学生，学生反映此教材难度
比较合适，他们基本上能掌握其中的重要内容。
第一章是一阶常微分方程，在介绍了一些必要的基本概念和应用模型后作者立刻非常明确地用典型实例详细解释了三种方法：1. 解析方法-分离变量法 2. 定性理论 - 向量场方法3. 数值方法 - 欧拉折线法。每种方法都占十几页。接下来叙述并详细解释了解的存在性和唯一性定理。接下来讲述相直线和平衡解，又化整整一节讨论分歧理论。 这样的处理方式的一个显著的优点是用最短的时间是学生了解了常微分方程的概貌。
第二章是一阶常微分方程方程组，但以两个函数的方程组为主。其主题不变，继续讲述处理同一类问题的三种不同方法。由于维数的增高，向量场更有意思，书中举了各种形形色色的例子使读者知道各种变化。
第三章是一阶线性常微分方程方程组，这是任何一本常微分方程教科书必须包含的内容，本书也不例外，写的非常清楚，没有可挑剔之处。
第四章的标题是受迫振动和共鸣，其实就是二阶常系数线性方程。这也是标准内容，和第三章一起，这两章是以解析方法为主的。
第五章是非线性方程组，重点自然放在定性分析上，先详细讲述二维情形，在化较少的笔墨写三维情形。
第六、七、八章分别是 Laplace 变换，数值方法（特别是 Runge-Kutta 方法），离散动力系统。
前四章的内容是必学的，后四章的内容相对独立，可以根据不同的专业选将若干章。一般说来，一个学期讲完五章是绰绰有余的。（杨劲根）
国外评论摘选
i) As a differential equations instructor I used Boyce and DiPrima for many years. Its a good, solid presentation of differential equations and a great reference. However, I was always disappointed that my students ended up with no 'feel' for differential equations. Also I became convinced that more methods were needed for nonlinear differential equations. After using a couple of other books which seemed to be slanted toward more qualitative approaches I came across Blanchard's book. I used it as a textbook for my class for several years now and I have found it to be a near perfect match to my goals. Some consider it wordy but I appreciate the motivation and insight the authors try to bring to the concepts. As a result it is not a good reference but as a textbook it is great. There are plenty of graphical tools. Quite surprising to me is how much the book illuminates DE's by simply analyzing the components of the DE, even before any solution is attempted. These features, along with some integrated applications, gives students much more of the 'feel' for differential equations I have been looking for.
ii) This book is unique. Most differential equations textbooks simply provide formulae for different types of problems, but you don't really see the big picture. This book lets you see the big picture, but omits many of the most useful formulae that you may need in your career. This for that. It would be nice to see a book with the best of both worlds, but if you simply want to learn and understand the topic, this book is the way to go. Also, there is a good emphasis on qualitative and numerical techniques. Students often feel like they get less out of a mathematics class when qualitative and numerical techniques are emphasized over more analytic approaches. However, those of us who have worked in the 'real world' know that the qualitative and numerical techniques are probably even more important. I have worked as a research statistician and my research areas emphasize computing. When I'm presented with real problems and real data (which, in my career, usually comes in large, unmanageable quantities), do I usually pull out my notebook and tackle the problem in a very precise manner, working out an exact solution? No, quite often I cannot realistically do that. Now I'll admit that I don't use much from this particular field on the job, but it still applies. Moving on, I must also mention that the book does a very good job at explaining these qualitative and numerical techniques in addition to things that are more analytic, although it sometimes a little too verbose. Regarding applications, the book covers a lot of fields and does put a big emphasis on applications. Physics, biology (especially population growth models), and electrical/computer engineering receive the most treatment. Overall, I would say that the book does an excellent job at including plenty of applications and choosing meaningful ones.
I don't have much to say about the exercises. Most aren't too contrived and they mixed up the difficulty fairly well. However, I would have liked to see more 'hard' problems.
In summary, I'd recommend that you pick up a different book if you need a reference for work or research, but pick this one up if you actually want to learn and UNDERSTAND the basics of differential equations.
iii) I used this book in a 2003 summer course in DE, and found it to be a wonderful introduction to the subject. I am not sure what some of the other people meant by saying it wasn't for math majors- I am one and found it wonderful. Not everything needs to be concise, (I gave Rudin's book five stars too BTW, so I AM a fan of some concise books). It gave diverse examples of applications from all over--physics, EECS, ecology, biology, etc. The CD-Rom is a great learning tool. Ultimately analytic techniques are NOT what DE is about, and this book tries to show the student how to use qualitative and numerical methods early on. Anyone who wants to know DE must become familiar with numerics and the qualitative way of analyzing the equations. This book will show you how to THINK about DE, and not how to mindlessly attack an equation based on its form. This is the intro ODE book to which all others ought be compared.
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，特别是组合数学和数论，但作者在序言中声明本书是“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预备知识。
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
1) 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，俞正光等改编
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课，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.