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主 编 杨劲根 编写人员（按汉语拼音为序）
1. 序言 9 拓扑与微分几何
At some point, however, a prospective student in topology will have to learn homological algebra and this provides the most concrete approach I know to the subject. Algebraic topology is a lot of fun, but many of the previous textbooks had not given that impression. This one does. 书名： Lecture Notes on Elementary Topology and Geometry 作者： Singer & Thorpe 出版商：Springer Verlag (1967) 页数：232 适用范围：大学数学系本科高年级教材或参考读物 预备知识：微积分、线性代数、抽象代数 习题数量：少 习题难度：中等 推荐强度：9.2 书评： 本书是大数学家为本科生写教材的又一典范，在 60-70 年代曾用作麻省理工学院数学系本科高年级一学年的课程的教材。 时隔几十年，行家们仍然认为这是一本不可多得的拓扑和几何的优秀入门读物。 本书篇幅不大，包含的内容不少，深入浅出，引人入深。作者不追求完整性，比如前两章的点集拓扑的基本知识不拘泥于一些 公理的仔细探讨，而是简明实用地把几何中最常用的拓扑空间讲清楚。第三章只化 20 几页就把基本群和复叠空间及其关系写清楚的， 值得指出的是前三章看似枯燥的内容中时而出现非常有趣的例子。第四、五、六章是全书的第一重点，讲述拓扑空间的同调群及微分流形 的概念，高潮是 de Rham 定理的证明。这里充分体现了数学中各不同分支间的渗透。最后两章是黎曼几何的导引，讲述了曲面上的 Gauss-Bonnet 公式这一深刻的定理。即使从现在的角度去看，这本书的选材仍然是反映现代数学主流的。 从内容来看，本书是点集拓扑、微分拓扑、微分几何三合一，这正是作者开这门课的宗旨。我国综合性大学的数学系一般对本科生也设有 拓扑和微分几何的课程，一般各占一个学期，这些学校的学生在学完这些课程后常常不知道 de Rham 定理和 Gauss-Bonnet 公式。 Singer-Thorpe 的这本教材也许可以启发我们在拓扑和几何的本科生教学方面作些改革。 （杨劲根） 书名： Topology from the differentiable viewpoint 作者： John Milnor 出版商：The University Press of Virginia 页数：61 适用范围：大学数学系本科高年级参考读物 预备知识：微积分、线性代数、基础拓扑 习题数量：17 题 习题难度：从易到难 推荐强度：9.5 书评： 华罗庚前辈写过好几本题为《从 ... 谈起》的数学小册子，其中一本是 《从单位圆谈起》。我们可以把 Milnor 的这本小书起名为《从单位球面谈起》，事实上本书自始至终不离单位球面。 拓扑学家 John Milnor 是费尔滋奖得主，在 Princeton 执教多年，他在拓扑方面的很多系列讲座笔记被整理出版，成为脍炙人口的数学读物， 这本书就是其中一本。 一本可以让大学三年级学生能看懂的 60 页的小册子，却包含如此多的深刻的定理（从 Sard 定理直到 Hopf 定理）以及完整的证明，这是何等的不可思议！ 这是学拓扑或几何的学生的必读书。 （杨劲根） 国外评论选摘 i) Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book. ii) One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts. 书名： Algebraic topology 作者： Hatcher 出版商： Cambridge University Press (2002) 页数： 542 适用范围：大学数学系基础数学研究生教材 预备知识：点集拓扑、抽象代数 习题数量：中等 习题难度：中等 推荐强度： 8.5 国外评论选摘 i) No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's. ii) Allen Hatcher has gone to great length's in order to create a text which, albeit overly verbose, can be used as a gentle introduction to modern Algebraic Topology. Why 'modern'? Compare this text with the tried and tested texts of Spanier, Munkres as well as May and, almost immediately, you will see what I mean. The obvious example is Hatcher's use of CW-complexes as opposed to the more traditional build up beginning with simplices. For the die-hard mathematician who enjoys less fluff, this book is not for you and, in particular, if this is your first venture in Algebriac Topology, you enjoy the theorem-proof-theorem style with a light sprinkling of explaination, then I would recommend J.J. Rotman's text. Whereas, if you enjoy filler, background information, and lots of side-notes or examples, then Hatcher's text would be a perfect fit. Myself, I fall into the category of those who enjoy the more terse texts but, I purchased Hatcher's (the hardcover) because of the clarity and percision found in the proofs. The majority of other texts have a tendancy to obfuscate the underlying meaning that should be unerstood by the up-and-coming mathematician. Of course this approach has it's merits since, in particular, it forces the reader to fill in the blanks but, as a matter of insight, Hatcher's approach is also beneficial. Another positive strength of Hatcher's text lies in the fact that he effectively breaks the subject into it's prime sub-categories in such a way that the reader can begin with either of the four parts of the text without having to rely too much on previous sections. This novel feature allows someone interested in, say, Cohomology to pick up an begin learning about Cohomology without having to waste time making their way through material they are not interested in. Finally, yes you can get the book for free via Hatcher's website but I highly recommend purchasing the hardback text. It is well made, it will last for years, and it becomes truely mobile as compared to burning your eyes out while reading the text on your computer. Moreover, why waste the time printing it out. 书名： Differential forms in algebraic topology 作者： Bott & Tu 出版商： Springer GTM 82 (1982) 页数： 331 适用范围：大学数学系基础数学研究生教材 预备知识：基础数学本科生的大部分知识 习题数量：少 习题难度：中等 推荐强度： 9.3 国外评论摘选 1) The authors of this book, through clever examples and in-depth discussion, give the reader a rare accounting of some of the important concepts of algebraic topology. The introduction motivates the subject nicely, and the authors succeed in giving the reader an appreciation of where the concepts of algebraic topology come from, how they do their jobs, and their limitations. The de Rham cohomology, which is the main subject of the book, is explained in here in a way that gives the reader an intuitive and geometric understanding, which is sorely needed, especially for physicists who are interested in applications. As an example, they give a neat argument as to why de Rham cohomology cannot detect torsion. In chapter 1, the authors get down to the task of constructing de Rham cohomology, starting with the de Rham complex on R(n). The de Rham complex is then specialized to the case where only C-infinity functions with compact support are used, giving the de Rham complex with compact supports on R(n). The de Rham complex is then generalized to any differentiable manifold and the de Rham cohomology computed using the Mayer-Vietoris sequence. The discussion gets a little more involved when the authors characterize the cohomology of a fiber bundle. The all-important Thom isomorphism for vector bundles, is treated in detail. The authors give several good examples of the Poincare duals of submanifolds. The connection to ideas in differential topology is readily apparent in this chapter, namely transversality and the degree of a map. In addition, the first construction of a characteristic class, the Euler class, is done in this chapter. The Mayer-Vietoris sequence is generalized to the case of countably many open sets in chapter 2, and shown to be isomorphic to the Cech cohomology for a "good" cover of a manifold. Good examples are given for computing the de Rham cohomology from the combinatorics of a good cover. The authors then characterize Cech cohomology groups in more detail, introducing the important concept of a presheaf. Presheaves are usually introduced abstractly in most books, so it is a real treat to see them described here in such an understandable way. Computations of the case of a sphere bundle are given, and the role of orientability and the Euler class in giving the existence of a global form on the total space is detailed. The Thom isomorphism theorem and Poincare duality are generalized to the cases where the manifold does not have a finite good cover and the vector bundle is not orientable. A very concrete introduction to monodromy is given and nice examples of presheaves that are not constant are given. The authors treat spectral sequences in chapter 4, and as usual with this topic, the level of abstraction can be a stumbling block for the newcomer. The authors though explain that the spectral sequence is nothing other than a generalization of the double complex of differential forms that was considered in chapter 2. The crucial step in the chapter is the transition to cohomology with integer coefficients, which is necessary if one is to study torsion phenomena. The De Rham theory is then extended to singular cohomology and the Mayer-Vietoris sequence studied for singular cochains. The authors show that the singular cohomology of a triangularizable space is isomorphic to its Cech cohomology with the constant presheaf the integers. After a fairly detailed review of homotopy theory (including a discussion of Morse theory) the authors compute the fourth and fifth homotopy groups of S(3). The last section of the chapter discusses the rational homotopy theory of Sullivan as applied to differentiable manifolds. The authors discussion is illuminating, and shows how eliminating any torsion information allows one to prove some interesting results on the homotopy groups of spheres. One such result is Serre's theorem, the other being the computation of some low-dimensional homotopy groups of the wedge product of S(2) with itself. The last chapter of the book considers the theory of characteristic classes, with Chern classes of complex vector bundles being treated first. The theory of characteristic classes is usually treated formally, and this book is no exception, wherein the authors formulate it using ideas of Grothendieck. They do however give one nice example of the computation of the first Chern class of a tautological bundle over a projective space. The Pontryagin class is defined in terms of a complexification of a real vector bundle and computed for spheres and complex manifolds. A superb discussion is given of the construction of the universal bundle and the representation of any bundle as the pullback map over this bundle. 2) This book is almost unique among mathematics books in that it strives to ensure that you have the clearest picture possible of the topics under discussion. For example almost every text that discusses spectral sequences introduces them as a completely abstract machine that pumps out theorems in a mysterious way. But it turns out that all those maps actually have a clear meaning and Bott and Tu get right in there with clear diagrams showing exactly what those maps mean and where the generators of the various groups get mapped. It's clear enough that you can almost reach out and touch the things :-) And the same is true of all of the other constructions in the book - you always have a concrete example in mind with which to test out your understanding. That makes this one of my all time favourite mathematics texts. 书名：Knot thoery 作者：Livingston 出版商：Mathematical Association of America (1996) 页数：258 适用范围：大学数学系本科生自学读物 预备知识：线性代数，群论 习题数量：多 习题难度：中等 推荐强度：8.8 书评：本书是美国数学协会出版的大学生系列丛书“Carus Mathematical Monographs" 中的一册，是拓扑学中纽结理论的优秀入门书。 本书的预备知识非常少，只要少量的线性代数知识。如果知道一些群论更好，但作者在用到群和二次型时都从头讲起。本书从纽结的历史和直观形象开始讲述纽结的分类，分别从组合、几何和代数三个方面引入各种重要不变量，如 Seifert 矩阵、 Alexander 多项式、 Conway 多项式、 Jones 多项式等。最后把 各种不变量的关系叙述得非常清楚。 本书图文并茂，习题非常丰富。内容安排从浅入深，章节的衔接紧凑。对初学者容易忽视的要点讲得很清楚。大部分定理有严格的证明，但又不拘泥于一些繁琐的证明细节，容易使读者掌握要点。 由于所用的准备知识少，所有的证明几乎都基于平面上的 Reidemeister 变换，纽结的基本群只是简单介绍一下，代数拓扑的工具没有使用。从这点来看对于具有较深数学基础的读者可以较快浏览本书后再选择更加高深的纽结理论的书籍阅读。 下面两段国外的评论中第一篇是一个数学教授写的，第二篇是自学过这本书的一个研究生写的，颇有代表性。（杨劲根） 国外评论摘选 1) This book is an excellent introduction to knot theory for the serious, motivated undergraduate students, beginning graduate students,mathematicains in other disciplines, or mathematically oriented scientists who want to learn some knot theory. Prequisites are a bare minimum: some linear algebra and a course in modern algebra should suffice, though a first geometrically oriented topology course (e. g., a course out of Armstrong, or Guillemin/Pollack) would be helpful. Many different aspects of knot theory are touched on, including some of the polynomial invariants, knot groups, Alexander polynomial and related abelian invariants, as well as some of the more geometric invariants. This book would serve as a nice complement to C. Adams "Knot Book" in that Livingston covers fewer topics, but goes into more mathematical detail. Livingston also includes many excellent exercises. Were an undergraduate to request that I do a reading course in knot theory with him/her, this would be one of the two books I'd use (Adam's book would be the other). This book is intentionally written at a more elementary level than, say Kaufmann (On Knots), Rolfsen (Knots and Links), Lickorish (Introduction to Knot Theory) or Burde-Zieshcang (Knots), and would be a good "stepping stone" to these classics. 2) I really do enjoy this book - but picked it up as a means of teaching myself Knot Theory... as was the case with many of my text books in college, brevity (for the sake of publishing costs) makes some concepts more of a challenge to grasp. Overall, the illustrations are great, and if you do the exercizes, the material tends to flow more easliy. It seemed to me the book worked backwards a bit - first covering a subject, than introducing it comprehensively later on - not what I'm used to. Keep in mind, I'm not a Mathematician, merely a graduate student of mathematics, who is interested in learning about this subject on my own. 书名： Riemannian Geometry, 3rd ed. 作者： M.P. Do Carmo 出版商：Springer Verlag (2004) ISBN-13: 978-3540204930 页数： 322 适用范围：数学专业研究生 预备知识：微积分，线性代数 习题数量：较大 习题难度： 适中 推荐强度: 10 书评：本书是一本标准的黎曼几何教材和参考书，其作者是巴西著名几何学家 Do Carmo 教授。作者写作风格清晰明了，全书共十三章，前四章介绍了黎曼几何的基本概念， 如黎曼度量、黎曼联络、测地线和曲率等；第五章介绍了 Jacobi 场这个重要的工具， 阐明了测地线与曲率的关系；第六章对等距浸入介绍了第二基本形式及相关的基本公式。 该书从第七章开始， 主要介绍了整体问题，涉及曲率与拓扑和比较几何中的一些基本结果。该书自成体系， 是目前黎曼几何最好的入门书之一。 对于想从事研究整体微分几何及相关领域的读者，该书也适合自学。 （东瑜昕） 国外评论摘选 i) "This book based on graduate course on Riemannian geometry covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results are treated in detail. contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004) ii) "This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris . Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." ( EMS Newsletter, December 2005) 书名： Foundations of Differential Geometry (in two volumes) 作者： Shoshichi Kobayashi & Katsumi Nomizu 出版商：John Wiley & Sons, Inc. (1996) 页数： Vol.I : 329 , Vol.II: 468 适用范围：数学专业研究生 预备知识：微积分，线性代数，微分流形、 Lie 群基础知识 习题数量：无 推荐强度：10 书评： 本书共两卷，旨在系统介绍微分几何的基础内容，其作者是著名的几何学家 S. Kobayashi 和 K. Nomizu 。第一卷首先概要地介绍了微分流形、李群和纤维丛的概念，然后主要介绍了主丛上的联络论、向量丛上的线性联络和仿射联络、黎曼流形上的黎曼联络，还涉及空间形式、仿射联络或黎曼度量的自同构群等。第二卷主要介绍了一些经典的专题 , 如子流形理论、Morse 指标理论 , 复流形、齐性空间和对称空间、示性类理论等。本书内容翔实、处理严谨，行文精练， 自二十世纪六十年代问世以来，一直被认为是经典的微分几何参考书。 1996 年John Wiley & Sons 出版社将其选入经典图书系列重印了其第三版，可见其影响。 对于想从事微分几何和相关领域研究的读者，这是一本很好的参考书。（东瑜昕） 国外评论摘选 1) The two-volume set by Kobayashi and Nomizu has remained the definitive reference for differential geometers since their appearance in 1963(volume 1) and 1969 (volume 2). Over the decades, many readers have developed a love/hate relationship with these difficult, challenging texts. For example, in a 2006 edition of a competing text, the author remarked that "every differential geometer must have a copy of these tomes," but followed this judgment by observing that "their effective usefulness had probably passed away," comparing them to the infamously difficult texts of Bourbaki. As a practicing differential geometer, I would argue that Kobayashi and Nomizu remains an essential reference even today, for a number of reasons. Volume 1 still remains unrivalled for its concise, mathematically rigorous presentation of the theory of connections on a principal fibre bundle---material that is absolutely essential to the reader who desires to understand gauge theories in modern physics. The essential core of Volume 1 is the development of connections on a principal fibre bundle, linear and affine connections, and the special case of Riemannian connections, where a connection must be "fitted" to the geometry that results from a pre-existing metric tensor on the underlying manifold, M. Volume 2 offers thorough introductions to a number of classical topics, including submanifold theory, Morse index theory, homogeneous and symmetric spaces, characteristic classes, and complex manifolds. The influence of the texts by Kobayashi and Nomizu can be seen in most of the subsequent differential geometry texts, both in organization and content, and especially in the adoption of notation. If there was a particularly fine point in your favorite introductory differential geometry text that you never completely understood, the odds are good that you will find the answer, fully developed and presented at an entirely different mathematical level, in Kobayashi and Nomizu. It is not an unreasonable analogy to say that learning differential geometry without having your own copy of Kobayashi/Nomizu is like studying literature in the complete ignorance of Shakespeare. Let there be no mistake about the advanced level of these texts. The Preface to Volume 1 clearly states that the authors presume the reader to be familiar with differentiable manifolds, Lie groups, and fibre bundles, as developed in the (now classical) texts by Chevalley, Montgomery-Zippin, Pontrjagin, and Steenrod. Today's reader is far more likely to have studied these subject from more recent books like those by Boothby, Hall, and Husemoller, but whatever the source, a familiarity IS presumed. The "lightning review" provided in Chapter I of Volume 1 will be extremely tough going for the reader who is new to these topics. It should also be noted that in 329 pages of Volume 1 and 470 pages of Volume 2, not a single diagram or picture is to be found! Those drawn to geometry for its visual aspects will find Kobayashi/Nomizu totally lacking in visual aids. As with so many classic references in mathematics, the hardbound edition of Kobayashi and Nomizu is no longer in print. Copies appear sporadically on the used book market at absolutely obscene prices. The Classics Library paperback edition is still available, but the serious student will find that the paperbacks simply do not fare well under serious, sustained use. 书名： Introduction to Lie groups and Lie algebras 作者： A.A.Sagle & R.E.Walde 出版商： Academic Press (1973) 页数： 361 适用范围：大学数学系研究生低年级教材 预备知识：抽象代数 习题数量：较小 习题难度： 中等 推荐强度： 9.5 书评： 本书详细介绍了李群和李代数的基本知识以及半单李代数的结构。本书的特点是起点很低 , 对欧氏空间中的微分、张量积、模及其表示以及微分流形和 Riemann 流形的基本内容都作了一定的介绍， 学生只需要有最基本的群论知识就可学习李群和李代数，而不需要事先掌握较多的几何基础 , 同时本书对于李群和李代数的介绍又是相当完全的。 （周子翔） 10 偏微分方程 书名： Hyperbolic Partial Differential Equations 作者： Peter D. Lax 出版商： American Mathematical Society, Providence, Rhode Island 页数： 217 适用范围：数学专业研究生 预备知识：泛函分析，常微分方程 习题数量：无 推荐强度： 10 书评： 本书是柯朗研究所系列讲义丛书之一。其作者是美国著名数学家 Peter D. Lax 教授。他在双曲型方程尤其是拟线性双曲型守恒律方程组及其计算，孤立子理论， 拟微分算子理论等诸多方面作出了开创性及里程碑式的工作。 本书恰是作者对双曲型方程理论知识方面的讲解。 全书包含十章及五个附录，介绍了线性双曲型方程及方程组的一些基本概念，能量估计， 解的存在性以及解的其他性质，还介绍了线性双曲型方程及方程组的差分格式及其差分格式的稳定性， 以及散射理论和拟线性双曲型守恒律方程组一些基本理论。 附录中最后一章由美国著名数学家 Morawetz 教授执笔。对于想从事双曲型方程研究的读者来说， 这是一本很好的入门书。（张永前） 书名： Partial Differential Equations, An Introduction 作者： Walter A. Strauss 出版商： John Wiley ＆ Son Inc. 页数： 425 适用范围：大学数学系本科高年级学生或低年级研究生 预备知识：微积分，线性代数，常微分方程 习题数量：较大 习题难度： 较难 推荐强度： 9 书评： Walter A. Strauss 是美国 Brown 大学数学系教授，著名的偏微分方程专家。本书自出版以来， 被美国多所著名大学作为本科生的偏微分方程课程的教科书。 全书包含十四章及一个附录，介绍了几类重要的偏微分方程的来源和基本性质以及基本的研究方法， 并在最后两章中介绍了一些来自物理学的非线性偏微分方程方面的进一步课题。 书中习题类型甚广，而且还配有部分习题的答案供读者参考。 本书适合作为偏微分方程的教材及参考书。（张永前） 国外评论摘选 1 ） This 1992 title by Strauss (professor at MIT) has become a standard for teaching PDE theory to junior and senior applied maths and engineering students in many American universities. Last year, being an informal teaching assistant for the class, I found many of the students struggling with the concepts and exercises in the book. Admittedly the style of writing here is very dense and if the reader does not have a very strong background in the topic, chances are high he or she will face a grand level of frustration with the exposition and the subject as a whole. One would need perseverance and dedication working numerous hours with this text before things start to settle in. After about the second or third chapter onward, those who were still taking the class had an easier time understanding the material and doing the excercises. Contentwise, after a brief and important introductory chapter (which should not be skipped by any reader!) the book first focuses on the properties and methods of solutions of the one-dimensional linear PDEs of hyperbolic and parabolic types. Then after two separate chapters, one on the trio of Dirichlet, Neumann, and Robin conditions and the other on the Fourier series, the author embarks upon the discussion of elliptic PDEs via the methods of harmonic analysis and Green's functions. Subsequently there is a brief introduction to the numerical techniques for finding approximate solutions to the three types of PDEs, mostly centered on the finite differences methods. The beginning of roughly the second half of the text is devoted to the higher-dimensional wave equations and boundary conditions in plane and space, utilizing the machinery of Bessel and Legendre functions, and ending up with a section on angular momentum in quantum mechanics. In the following, Dr. Strauss brings up the discussion of the general eigenvalue problems, and then proceeds with a treatment of the advanced subject of weak solutions and distribution theory. (This topic is normally skipped in an undergraduate course.) The last two chapters are a pure delight to read, dealing with the PDEs from physics as well as a survey of the nonlinear phenomena (shocks, solitons, bifurcation theory). A few appendixes at the end, summarize the analysis background needed for the course and must be consulted before and during the first reading. All in all this is a very splendid source for all the applied maths and engineering students, that can be used in conjuction with other references to help break through the conceptual barriers. In fact, I recommended the book by Stanley Farlow to our students and many found the presentation there very modular and accessible. For example, some of the Strauss' homework problems, such as solving the Poisson equation on an annulus, were subjects of a single chapter in Farlow. In any event, I am very much hoping to see a new and more student-friendly edition of the Strauss' text be prepared and issued in the near future. 2 ） I used this book in a tough applied math course, and the quality of this book did not help matters much. There are a couple of good things about this book. The material chosen is appropriate and reasonably comprehensive for an intro PDE text. In other words, the table of contents is a nice read. The notation is very clean and concise throughout, as is the typesetting. The bibliography was also useful, pointing me to some great supplementary texts. Now for the bad parts. An intro PDE book should explain clearly the basic concepts behind PDEs, including how certain famous equations (wave, heat, Laplace , etc.) arise in physical modeling. It should explain in detail the various computational techniques for finding analytical solutions to these equations. It should explain relevant elementary theorems needed for these computational techniques. This book attempts to do all of these things, but does so poorly. The basic problem is that the book's explanations and examples are too terse and incomplete for an introductory text. Analytically solving a PDE is a relatively difficult task, involving several computational steps and techniques. Examples of these techniques should be worked in detail, but in this book, they frequently omit steps or fail to explain where or how a particular technique is being applied. Theorems are often not stated, or if they are, proofs are either omitted or partially sketched. This makes the book difficult for beginners, but it is not a terrible reference if you have already been exposed to the material. My advice: given the price of this book and its mediocre quality, you would do better by looking elsewhere for an intro PDE text. 书名： Partial Differential Equations 作者： Lawrence C. Evans 出版商： American Mathematical Society, Providence, Rhode Island 页数： xviii+662 适用范围：数学专业研究生 预备知识：微积分，线性代数，常微分方程 习题数量：较大 习题难度：适中 推荐强度： 10 书评： Lawrence C. Evans 是著名的偏微分方程专家，美国加州大学伯克利分校的教授。本书 被美国多所著名大学采用作为研究生偏微分方程课程的教科书或者参考书。 作者在这本教材中介绍了偏微分方程的许多重要课题，重点介绍了偏微分方程各种现代处理方法。 内容主要分为三部分。第一部分介绍了一些偏微分方程解的表示， 其中包含了一阶非线性偏微分方程的特征线方法， Hamilton-Jacobi 方程和双曲守恒律方程组解的表示方法 以及一些特殊非线性方程的行波解等。 第二部分主要介绍了处理二阶线性椭圆型方程及抛物型方程的现代方法。 第三部分介绍处理非线性方程的变分和非变分方法以及处理 Hamilton-Jacobi 方程和双曲守恒律方程组 的一些现代方法。每章之后都给出了相关内容的出处的说明以及与正文内容紧密配合的习题。 本书极适合作为研究生的偏微分方程教科书。（张永前） 国外评论摘选 1) This is a textbook for a first-year graduate course in PDE (for mathematics students). You should take courses in analysis (on the level of Rudin) and measure theory before you expect to understand everything in this book. This is by far the best book on PDE. The text is extremely clear, and most of the rather technical proofs are prefaced with "heuristic" calculations to help the reader understand what is going on. The chapter on the calculus of variations is the best exposition I have found of the subject, and Evans completely dispenses with the awful "delta" notation which never made any sense. The text doesn't make much use of the Fourier transform and doesn't even mention distributions, and this gives his book a definite nonlinear flavor (which is a good thing). This should become the standard introduction to PDE on the graduate level. 2) I have taught a one-year course in PDE based on Evans' book and found it extremely cogent and stimulating both for myself and for the students. The treatment is up-to-date, with a definite nonlinear flavor. Beyond that, the exercises are very good, and the treatment is sufficiently detailed to make class preparation fairly fast. It does demand mathematical dexterity and maturity of the students right from the start, though. 3 ） I've seen a lot of positive reviews of this text, and I feel the need to explain some cons of this book. Before that, I will say this is probably the best introduction to PDE theory out there. This is NOT a book for people looking for a dissertation on undergraduate methods of solution (separation of variables, fourier series, etc.). If that is what you are looking for, go to Haberman or perhaps Strauss. Ok, so here are the problems I see with this text. First, there is no mention of distributions in this book. Evans addresses this in the intro., saying it's not necessary. I find that hard to swallow, given that fundamental solutions play a big part in the text. Despite this, Evans devotes parts of the book to going into very esoteric subjects like mean value theorems for the heat equation. The other glaring gap in this text is the absence of Schauder estimates; a corner-stone for linear elliptic theory. On a note of personal preference, I would have like to have seen more of the book dedicated to a functional analytic foundation; the appendicies that are present are simply not enough. Overall, the book gives a decent introduction; but is far from being self-contained and is not enough of a foundation for people wishing to pursue research in PDE. Evans does acknowledge this in his introduction, but I think its something that is frequently overlooked in reviews of this text. 书名： Partial Differential Equations 作者： Fritz John 出版商： Springer-Verlag, New York-Berlin 页数： x+249 适用范围：大学数学系本科 预备知识：微积分，线性代数，常微分方程 习题数量：中等 习题难度：适中 推荐强度： 10 书评： 本书是系列丛书《 Applied Mathematical Sciences 》的第一卷。其作者是已故著名数学家﹑美国纽约大学 Courant 研究所 Fritz John 教授。其内容包括：一阶偏微分方程， Cauchy-Kovalevskaya 定理 , Holmgren 定理 , Lewy 的著名反例，波动方程及 Poisson 方程﹑热传导方程和相应高阶方程及对称双曲组的性质及 相应定解问题的基本求解方法。既介绍了特征线方法等经典方法， 也介绍了差分方法及 Hilbert 空间理论等近代方法。本书已被四次再版， 作者在每个新版中加入了一些新的内容，如在第四版中加入了关于 Cauchy-Kovalevskaya 定理解的存在区域大小的讨论等。 1991 年 Springer-Verlag 出版社又重印了其第四版， 可见其影响。我国科学出版社也在 1986 年翻译出版了其第四版。该书由朱汝金教授翻译。 书后习题与正文内容紧密配合，有助于对所介绍方法的理解。 适合偏微分方程的初学者作为入门及参考书。（张永前） 11 概率论 12 计算数学 国外评论摘选 13 其他
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