芝加哥大学本科数学教材推荐目录Chicago undergraduate mathematics bibliograph

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INTERMEDIATERoughly, general rather than specialized texts in higher mathematics. I would not hesitate to recommend any book here to honors second-years, but they might not find easy going in some of them.Contents

• Foundations (5)
• General abstract algebra (7)
• Linear algebra (3)
• Number theory (5)
• Combinatorics and discrete mathematics (1)
• Real analysis (10)
• Multivariable calculus (2)
• Complex analysis (5)
• Differential equations (2)
• Point-set topology (5)
• Differential geometry (4)
• Classical geometry (3)
  

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FoundationsHalmos, Naive set theory

The best book for a first encounter with “real” set theory. Like everything Paul Halmos writes, it's stylistically beautiful. A very skinny book, broken into very short sections, each dealing with a narrow topic and with an exercise or three. It requires just a little sophistication, but no great experience with “real” math; we use this one for YSP kids sometimes too.

Fraenkel, Abstract set theory

Fraenkel was the F in ZFC, and he gives a suitably rigorous development of set theory from an axiomatic viewpoint. Unfortunately, for the philosophical foundations of the axioms he refers to another book (Fraenkel and Bar-Hillel, Foundations of set theory), which is missing from Eckhart Library. Good for culture.

Ebbinghaus/Flum/Thomas, Mathematical logic

The only logic book I can name off the top of my head, this is the 277 book. I found it readable but boringly syntactic (well, maybe that's elementary logic).

Enderton, A mathematical introduction to logic

Look, another logic book! This one might be preferable just because there's much more talking about what's going on and less unmotivated symbol-pushing than in E/F/T. The flip side of that is, the constructions may or may not be epsilon less precise. I'm not a logician; if you are, write some reviews so I can replace these lousy ones!

Landau, Foundations of analysis

This is the book that invented the infamous Landau “Satz-Beweis” (theorem-proof) style. There is nothing in this book except the inexorable progression of theorems and proofs, which is perhaps appropriate for a construction of the real numbers from nothing, but makes horrible bathroom reading. Read for culture.

General abstract algebra

The situation here is problematic, because there are many good books which are just a little hard to swallow for an average 257 student, but precious few good ones below that. But you learn by doing, so here we go:

(Difficulty: moderate)Dummit/Foote, Abstract algebra

[PC] I bought this for 257—I was at the age where I uncritically bought all assigned texts (actually, I may still be at that age; I don't recall passing on buying any course texts recently), but as Chris knows the joke was on me, since we used the instructor's lecture notes and not Dummit/Foote at all. So I didn't really read it that much at the time. I have read it since, since it is one of two general abstract algebra books in my collection. I think it's an excellent undergraduate reference in that it has something to say, and often a lot to say, about precisely everything that an undergraduate would ever run into in an algebra class—and I'm not even exaggerating. I would say this is a good book to have on your shelf if you're an undergraduate because you can look up anything; I used it this fall as a solid supplementary reference for character theory to Alperin and Bell's Groups and representations, and it had an amazing amount of material, all clearly explained. [Warning: there is an incorrect entry in one of the character tables; it's either A_5 or S_5, I can't remember which.] Look elsewhere, particularly below, for a good exposition of modules over a principal ideal domain; D/F's exposition is convoluted and overly lengthy. In fact, overall I would use this book as a reference instead of a primary text, because the idea of reading it through from start to finish scares me. It also has many, many good problems which develop even more topics (e.g., commutative algebra and algebraic geometry).

Herstein, Topics in algebra

This is a classic text by one of the masters. Herstein has beautiful and elementary treatments of groups and linear algebra (in the context of module theory). But there is no field theory, and he writes mappings on the right, which annoys many people. Sometimes he suffers from the same flaw of excessive elementarity as Spivak's calculus book, but overall the treatment is quite pretty. Many good exercises. (Not to be confused with Abstract algebra, which is a much-cut version for non-honors classes.)

[PC] But this is the book I would use if I were a well-prepared undergraduate wanting to learn abstract algebra for the first time. Wonderful exposition—clean, chatty but not longwinded, informal—and a very efficient coverage of just the most important topics of undergraduate algebra. Think of it as a slimmed down D/F. “No field theory” is certainly an exaggeration; the exposition there is quite brief, and the restriction to fields of characteristic zero obscures the fact that much of the theory presented, including the Galois theory, is the theory of separablefield extensions, but even so, this is still the book I open first to remind myself about the Galois theory I'm supposed to know. The last main chapter of the book is quite lengthy and treats linear algebra and canonical forms in detail, which is one of the book's strongest features. Also, there are many supplementary topics—maybe Herstein really doesn't like field theory, since he inserts a section on the transcendence of e early on in his field theory chapter as something of a breather—but there's lots of good stuff to warm the heart of someone who likes to see his algebra applied to actual stuff, especially number-theoretic stuff; the famed Two and Four Squares Theorems are both proved in here!

Artin, Algebra

Artin's book is a nontraditional approach to undergraduate algebra, emphasizing concrete computational examples heavily throughout. Accordingly, linear algebra and matrix groups occupy the first part of the book, and the traditional group-ring-field troika comes later. This approach has the advantage of providing many nontrivial examples of the general theories, but you may not want to wait that long to get there. Supposed to be well written, though I haven't read it thoroughly.

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关键词：mathematics Mathematic Graduate Thematic Chicago 芝加哥大学 recommend abstract Complex general

(Difficulty: higher)Jacobson, Basic algebra I

Jacobson was my first real algebra book, and I retain an affection for it. The book is very densely written, and his prose has its own beauty but is difficult to get much from at first. The selection of topics is interesting: chapters 1–4 cover groups, rings, modules, fields (modules in the linear-algebra sense, that is, over principal ideal domains), while chapters 5–8 cover extension topics not usually found in general texts. He deliberately avoids modernist abstraction, preferring an explicit construction to a universal property and a commutative diagram (although the universal property is frequently given), and this complicates his notation and prose at times, especially in the module chapter. The field-theory chapter is fantastic. Some of the exercises are deliberately too hard.

Hungerford, Algebra

Many people like this book, but I don't. Hungerford covers the standard topics from group, ring, module, and field theory, with a little additional commutative ring theory and the Wedderburn theory of algebras. The field-theory chapter is horrible, and the rest of the book is okay but doesn't excite me. (And the typesetting is bad.)

Lang, Algebra

Well, do you like Serge Lang books, or not? Like every other Serge Lang book, this one is uncompromisingly modern, wonderfully comprehensive, and unpleasantly dry and tedious to read. Unlike most other Serge Lang books, this one has exercises, at least.

Mac Lane/Birkhoff, Algebra

I keep recommending this book to people because it's the only hard one whose contents correspond well to the 257-8-9 syllabus, and also because I like Mac Lane's treatment of linear and multilinear algebra. Mac Lane and Lang are the only books in this group which treat multilinear (tensor) algebra at all, and believe me, you'll need it eventually. Worth a look to see whether you find Mac Lane's style congenial. Not to be confused with Birkhoff/Mac Lane, A survey of modern algebra (a much shorter and easier book).

[BR] I used Mac Lane/Birkhoff's book pretty heavily in Math 257 and 258. Unlike most algebra books I've seen, they don't put all the group theory at the beginning and all of the field theory at the end, but prefer to develop each topic a little bit at a time and then develop it with more depth later. As a result, this book is hard to use as a reference. You can't get past rings without tackling categories and universal constructions which are used heavily throughout the remainder of the text. However, their treatment of categorical algebra is one of the more readable introductions to the theory I've come across.

Linear algebraHalmos, Finite dimensional vector spaces

This is a linear algebra book written by a functional analyst, and the crux of the book is a treatment of the spectral theorem for self-adjoint operators in the finite-dimensional case. It's a beautiful, wonderful book, but not a very good reference for traditional linear algebra topics or applications. You also have to read a fair distance before you even see a linear map, and the exercises are mostly too easy, with a few too hard. But this book was where I first learned about tensor products, and why the matrix elements go the way they do and not the other way (Halmos is very careful on this point).

[PC] I own this book and read through it often, but it's never taught me linear algebra per se. Let's agree that it's too abstract for a reasonable first introduction to linear algebra; it's really meant for students who already know (some) linear algebra to read through and appreciate one particular, and particularly elegant, presentation of the material. If you want to know about the linear algebra which surrounds functional analysis, then by all means read this book, but much of the material is nonstandard and a bit curious from the perspective of mainstream linear algebra; projections seem to be the most important linear map, and there are many sections lovingly devoted to commuting projections, decomposing projections, etc. I still am not sure why Halmos deifies the [,] as much as he does, and quite honestly, I would learn multilinear algebra anywhere but here.

Curtis, Abstract linear algebra

If you can stand terrible typesetting and an unexciting prose style, this tiny little book is a good rigorous reference for traditional linear algebra (i.e., it doesn't assume you're a tree). A nice bonus at the end is the Wedderburn theorem for division algebras over R, although the lack of sophistication makes for some unmotivated technical carpentry. I look in here whenever I can't remember what a positive-definite matrix is.

Greub, Linear algebra and Multilinear algebra

You may never need The Book on linear algebra. But one day, you may just have to know fifteen different ways to decompose a linear map into parts with different nice properties. On that day, your choices are Greub and Bourbaki. Greub is easier to carry. End of story.

Number theoryIreland/Rosen, A classical introduction to modern number theory

The first half is a coherent, systematic development of elementary number theory, assuming the basics of algebra. In the second half the authors explore more advanced topics of an algebraic/geometric flavor (zeta functions, L-functions, algebraic number fields, elliptic curves). Lots of exercises. This book helped make number theory make sense to me. You will find many introductory number theory texts pitched below I/R, but if you can read I/R, ignore the easy ones.

[PC] Yes, this is the standard and to my knowledge the best number theory text that is modern, broad, and reasonably elementary. It's a strange book in that it's really not written at any one level—if you've heard of something called unique factorization, you'll find the first few chapters easygoing material, but the algebraic sophistication rises slowly but surely throughout the book. Eventually you need to be comfortable with rings, fields and Galois theory at the undergraduate level, but they tell you at the beginning of the chapter when they require more background than before. There's an awful lot in here; this was my course text for Math 242 and I used it as one of the texts in a reading class on number theory, and I still haven't read through all the chapters. It's a great example of a book in which the authors have tried and succeeded in bringing advanced material down to the undergraduate level. Some good historical notes, as any self-respecting number theory text should contain. Recommended highly.

Burn, A pathway into number theory

[BB] The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. Can be tedious (you get to verify, say, Fermat's little theorem for maybe 5 different sets of numbers) but a good way to really work through the beginnings of the subject on one's own.

Hardy/Wright, Introduction to number theory

This is the classic, and Hardy is one of the great expository writers of mathematics. However, I remember that the last time I looked at this book it made no sense to me. If you like number theory you should probably at least look at it.

[PC] Oh, here I must fervently disagree (well, okay, maybe it didn't make sense to you at the time, but please go ahead and look again). I say that any student of mathematics should have this book on their shelf. Here's H/W's game: they explain number theory to people who can follow mathematical proofs but have no prior exposure to the subject or any advanced machinery whatsoever—hmm, maybe a little calculus at times, but not always. The one thing they do use is a little asymptotic growth notation, i.e., O, o, and the squiggly line, and for some reason they assume that people will know all about this without much comment. I seem to recall that one chapter towards the beginning is confusing because of this, and when I first bought the book it stymied me (I was sixteen at the time). But it's written so that you don't have to read it in order: they develop just enough theory about almost every branch of (elementary) number theory so that you can see interesting theorems proved. I have jumped around a lot, but over the years I think I've read almost every chapter. I really think it's the #1 “cultural enrichment” book for math students.

Chandrasekharan, Analytic number theory

[PC] Recommended to me by none other than Professor Narasimhan himself, it's actually a very elementary and readable introduction to the classic theorems of analytic number theory: Chebyshev's Theorem, Bertrand's Postulate, uniform distribution, Dirichlet's Theorem and the Prime Number Theorem. Requires epsilonics and just a little bit of complex function theory.

Apostol, Introduction to analytic number theory

[PC] If you've been reading this list, you know from Chris that Apostol writes terribly dry books. I've never read anything by him but this one, and it's fine, a bit more elementary than Chandrasekharan and easier to get your hands on (Apostol is a UTM; Chandrasekharan is an out of print Springer international edition). It starts out with a nice introduction to arithmetic functions, including the convolution product, and it covers much the same as the above, only a bit less briskly. A quick route to the proofs of the greatest theorems of 19th century mathematics.

Combinatorics and discrete mathematicsGraham/Knuth/Patashnik, Concrete mathematics

The first chapter of Knuth's immortal work The art of computer programming is an extensive study of combinatorics and asymptotics. G/K/P is an expanded and friendlier version, which emphasizes teaching the reader to solve things, rather than just showing how they are done. Contains many funny marginal notes from students in the Stanford class which gave birth to the book, as well as tons of great exercises. Not a reference work.

Real analysis(Elementary level: metric spaces, continuity, differentiation)Rudin, Principles of mathematical analysis

The first eight chapters of this little book form the best, cleanest exposition of elementary real analysis I know of, although few UC readers will have much use for the chapter on Riemann-Stieltjes integration. Like Rudin's other books, it is broken into bite-size pieces, so you can prove every statement in the book on your own if you're self-studying. If that isn't enough, there is a large collection of challenging exercises. Some people think Rudin is too skinny and streamlined, but I think it's beautiful. (Ignore chapters 9 and 10, which are a confusing and insufficiently motivated development of multivariable calculus. Chapter 11 is all right for Lebesgue integration, but there are better treatments elsewhere.)

[PC] I agree 100% with what Chris says, but I want to add my voice that this is (through chapter 8) the cleanest exposition I have ever seen. I still flip back to this to check things out.

[BR] I must insist that Chapters 9 and 10 are not THAT bad. They're worth revisiting if you are tired of Spivak and do Carmo.

Apostol, Mathematical analysis

Covers the same material as Rudin, plus a little complex analysis. Apostol assumes (hence, engenders) less maturity on the reader's part, writing most arguments out in “advanced calculus” detail rather than “real analysis” detail, if that makes sense. I find it terribly dry. Nevertheless the book is careful and comprehensive, with many exercises.

Gelbaum/Olmsted, Counterexamples in analysisThis little book contains a long list of examples, of strange objects which contradict the things that you think should be true but aren't. It starts off at a very elementary level and gradually builds up to include the Lebesgue theory and R^n. A good thing to have around on your first or second trip through analysis.

(Intermediate level: normed spaces, Lebesgue integration)Kolmogorov/Fomin, Introductory real analysis

When I started 207 I couldn't see why the material of this book was analysis: here was set theory, some linear algebra, some stuff about normed linear spaces, a little functional analysis... oh, here's that cool integral everyone talks about, but where are the derivatives? Now I know why it's analysis, of course, but the book as a whole is still a perplexing beast to the inexperienced. I think the primary reason it remains a text for 207 is that it costs $13, so why not? The style is distinctively Russian, which puts me off but turns other people on. Extended applications appear occasionally to lend context, but on the whole there is little motivation (and few exercises). The book is also difficult to use as a reference work, because the authors develop only the results they need to get where they're going.

[PC] Agreed. But it's cheap and though you may wonder why you're learning so much functional analysis before you see a Lebesgue integral, it's still clear and easy to read, so there's no reason why you shouldn't own it.

Haaser/Sullivan, Real analysis

Covers the same material as K/F, with the addition of a chapter relating differentiation to Lebesgue integration (the fundamental theorems of calculus). H/S use the Daniell integral rather than K/F's concrete, bare-hands construction of Lebesgue measure; it's probably good to do it by hand once, but after that forget it. The sequence of topics makes a little more sense than K/F, although the chapter on inner product spaces is lonely at the end, where it lives because they want to do Fourier series. But the book is written in a ho-hum style, and the exercises are too easy. In this H/S shares the flaw of many books at this level, of making too big a deal of a little bit of abstraction which might be new to the reader. I went straight from little Rudin to big Rudin without much of a stop for either of these books.

Hewitt/Stromberg, Real and abstract analysis

This is an old, classic book which is worth a look. They develop many concrete classical topics (all those things like Legendre polynomials that you were always curious about) as exercises.

Dieudonné, Foundations of modern analysis

This book is a strange bird, the first volume of a nine(!)-volume treatise by one of the original Bourbakistes. I can't really describe it except to say that it's very formalistic, it has many good exercises, it's very hard to relate to other treatments of the subject, and it made a big impression on me.

(Graduate level: measure theory, basic functional analysis)Rudin, Real and complex analysis

The first half is the standard reference for real analysis (the second half is reviewed below). It's a very clean treatment of the topics it covers, again in bite-size pieces and with manychallenging exercises. Sometimes I get frustrated with the lack of motivation, or with Rudin's habit of proving exactly the lemmas he needs to do something, without any context for the results. Nevertheless it's a good reference or self-study book. Topics: Integration and L^p spaces, Banach and Hilbert spaces, Radon-Nikodym theorem and differentiation, Fubini's theorem, Fourier transforms.

[PC] Yes, how wonderful that there's one book whose first half contains all the analysis that you'll ever need to know! This book is advanced and the exposition is austere (“which gives (5). Applying (3) to (4), we get (6)”) but it is absolutely crystalline in its clarity (exception: is its proof of the L^2 inversion theorem for Fourier transforms valid? I'm not so sure.) Isn't this the one math book that every student must buy sooner or later (aside from Hardy and Wright, of course)? Some rainy day you'll discover that the book has a second half and find some very interesting theorems in there, but don't confuse it with a course on complex analysis, because it's a weird-ass treatment of complex analysis viewed through the eyes of a conventional analyst. Think of it as a bonus.

Lang, Real and functional analysis

Another Serge Lang book, but a Serge Lang book is about the only place you'll find the inverse function theorem systematically treated for Banach spaces (except Dieudonné, and Lang was a Bourbakiste too).

Royden, Real analysis

Royden is like Hungerford for me: a lot of people like it, but it annoys me for a number of semi-silly reasons. He denotes the empty set by 0 (zero) and the zero element of a vector space by lowercase theta. He proves many theorems three times in gradually increasing generality. He leaves whole proofs to the exercises, and then depends on them later in the text. And I don't like his construction of Lebesgue integration. (Nyaah, so there.)

[BR] This is such a terrible book! He leaves the hardest theorems to the reader and proves some really simple-minded things with too much machinery. For example, he assigns the Urysohn lemma for normal spaces as an exercise for the reader and then has to use the Baire category theorem to show that on Banach spaces, linear operators are continuous iff they commute with taking limits. If you have to take 208 or 272, find a supplementary text. You'll be happy you did.

Multivariable calculusSpivak, Calculus on manifolds

This is the book everybody gets in differentiation and integration in R^n, and it's a pretty good one, although the integration chapters are hard to read—maybe it was just my first encounter with exterior algebra that made it hard. As usual for Spivak books, clear exposition and lots of nice exercises. Unfortunately this one is old enough to be annoyingly typeset.

[PC] I don't really like this book, and I'm a big fan of Spivak in general. Does anybody else think that this rigorous multivariable Riemann integral theory is a dinosaur? And when Spivak starts talking about chains (in chapter four, I think), I don't know what the hell he's talking about. Presumably you could ignore that chapter and use the book as an introduction to differential forms. I can't suggest a substitute at the moment, other than Spivak's Comprehensive introduction volume 1, which is a wonderful book but which I still wouldn't want to read as a first introduction to forms. Come to think of it, I love forms to death, but maybe they're just plain confusing the first time around...

do Carmo, Differential forms and applications

This skinny yellow book has replaced Munkres's Analysis on manifolds as the text for 274, and I'm not sure it's an improvement. It's more like a modernized Calculus on manifolds. I haven't done more than glance through it, but the notation is reputedly horrible, and Spivak is definitely a superior expositor.

Differential equationsArnold, Ordinary differential equations

Yes, Virginia, there is an interesting geometric theory of differential equations (of course!), not just the stuff you see in those engineering texts: stuff about stable and unstable points or manifolds, and other things with a dynamical-systems flavor. Nevertheless there is substantial material on how to reduce a differential equation to linear form and solve it, although no Laplace transform techniques or the like. Arnold explains it all coherently at an advanced-calculus level (manifolds appear at the end), complete with many beautiful diagrams. Another distinctively Russian book—read all the ones I describe that way, and you'll see what I mean. The third edition is substantially different from the second (which I have): the manifolds material is much expanded, and the typesetting is not so nice.

Hurewicz, Lectures on ordinary differential equations

A tiny book which covers material similar to Arnold, but more concisely. I haven't read it but it's frequently referenced, and worth a look if you need to know the basic theorems. (If all you need is the basic existence-uniqueness theorem for ODEs, it's also in Spivak volume 1 or Lang, Real and functional analysis.)

Point-set topologyMunkres, Topology: a first course

Munkres's book is a wonderful first encounter with topology; in fact it begins slowly enough to be a first encounter with abstract mathematics (after a traditional advanced calculus course). Every abstraction is carefully motivated, and there are tons of examples, pictures, and exercises. This is one of those books you could hand to a bright student of any age who knew some calculus (not a bad book to choose if you're coming back to mathematics at age 35). Most of the book is the traditional analysis-topology material, but there is a long last chapter on the fundamental group which covers enough to prove the Jordan curve theorem.

[PC] Yes, Munkres deserves to be the standard undergraduate point-set book. It doesn't have everything, but it has most of the standard topics and it's relentlessly clear.

Willard, General topology

But Willard is my topology book of choice. The level of abstraction is deliberately higher, and the book is better organized as a reference than Munkres. It's not nearly as friendly, but it's still clear and well-written (I think an unclear point-set topology book is probably no longer a point-set topology book). Willard is probably the best modern reference for analysis-topology, where “modern” means “excluding Kelley” (see below). You can learn from it too; it's organized bite-size like a Rudin book, so you can prove all but the hard theorems on your own (I did this with an initial segment, and learned a lot).

Kelley, General topology

[PS] Let me just say that Kelley's book on topology is horribly old-fashioned—I know because my advisor is forcing me to read it. Half the topics are things which I don't think are as important as they used to be. Nets, filters? I guess they're interesting in and of themselves. On the upside, it does have a nice appendix covering the rudiments of set theory.

[CJ] It is old-fashioned, but it's still the best book on topology for functional analysis, bar none. Nets are surprisingly necessary in infinite-dimensional topological vector spaces! The occasional proof is easier to read once recast in modern language, but doing so is a good learning exercise anyway. And Kelley has the nice habit (emulated less successfully by Willard) of treating substantial pieces of analysis as exercises; two of the exercises to Chapter 2 are titled “Integration theory, junior grade” and “Integration theory, utility grade”. It's really an analysis book disguised as a point-set topology book, but then much of functional analysis is really general topology on spaces that happen to be vector spaces too.

Steen/Seebach, Counterexamples in topology

This is a topology ‘anticourse’: a collection of all the screwed-up topological spaces which provide limiting counterexamples to all those point-set topology theorems with complicated hypotheses. It's a classic just for the content, but pretty well written too. This book and Gelbaum/Olmsted (above) are two parts of what should someday be the big book of counterexamples to everything. Read it and see just what you avoid by sticking to differentiable manifolds.

[BR] Steen and Seebach have catalogued 143 of the most disgusting pathological topological creatures. They are invaluable for when you're first learning point set topology and need to understand why the definitions are necessary. They can also come in handy on tests: I used the one-point compactification of an uncountable discrete space three times on my Math 262 final. The text used for 262, Munkres, relies on three counterexamples to disprove everything: the Sorgenfrey line, S_Omega and I x I in the dictionary order. Steen and Seebach let you know that there are tons of other beastly topological spaces which violate the laws of common sense.

Dugundji, Topology

[YU]This is a point-set topology book. Less elementary than Munkres, but useful as a reference book for grad students.

Differential geometryGuillemin/Pollack, Differential topology

I didn't understand transversality at all until I saw this book. It's a very geometric (as opposed to formalistic), down-to-earth introduction to some of the most mystical areas of smooth manifold theory: transversality and intersection theory. Abstraction is avoided (manifolds are defined as embedded in Euclidean space, which annoys me just a bit), but without hand-waving important distinctions (they are careful to point out that for noncompact manifolds, an injective immersion need not be an embedding, that is, proper too). The last chapter treats integration and Stokes's theorem, but that's not what anyone reads the book for. Beautifully written, and fills an important hole in Spivak volume 1.

do Carmo, Differential geometry of curves and surfaces

We used this book for Corlette's differential geometry seminar two years ago (293). I didn't like it all that much because do Carmo is careful to keep the book to a post-advanced-calculus level: everything takes place in R^3, no vector bundles, lots of componentwise calculations. Nevertheless it's a nice treatment of the classical theory of curves and surfaces in space. Read it if you want to know about the Gauss map or the two fundamental forms, but don't want to work all the way through Spivak volume 2.

Spivak, A comprehensive introduction to differential geometry, 1

[PC] Volume 1 is the best introduction to smooth manifold theory and differential topology that I know of. Every chapter of this book has come in handy for me at one time or another. Ben and I like to describe the book as “locally readable”: his exposition is very careful, but sometimes he takes too damn long to explain a single concept. Luckily, despite Spivak's efforts to the contrary, you can flip around and read chapter by chapter, and I recommend this. There is so much good stuff in here.

[CJ] Buy it and read it over and over and over. Don't skip the exercises because that's where he puts all the freaky examples. It's true that sometimes he talks too much, but for the loving detail in which he lays out difficult concepts, he can be forgiven.

Spivak, A comprehensive introduction to differential geometry, 2

As Spivak puts it at the beginning, “Volume 1 dealt with the ‘differential’ part; in this volume we finally get down to some geometry.” Volume 2 treats the classical theory of curves and surfaces using the modern machinery developed in the first volume, which makes it (for me) a more comfortable read than do Carmo. Spivak is careful to motivate everything historically; surface theory is introduced by a long walk through Gauss's General investigations of curved surfaces (you should really have a copy of it to read this book), and the second half of the book goes through the (convoluted) stages of evolution of the definition of a connection. Not easy reading but every bit as rewarding as Volume 1. Unfortunately there are almost none of the wonderful exercises which characterize the first volume.

Classical geometryCoxeter, Introduction to geometry

This is an interesting book which I can't really describe. It contains a number of short treatments of undeniably geometric but nontraditional topics; one fascinating application is the relation between phyllotaxis (the arrangement of plants' leaves around the stem) and generalized Fibonacci-type numbers. Read for culture.

Hilbert, Foundations of geometryHilbert was very interested in finding coherent, minimal axiom systems for parts of mathematics; he was probably inspired by the long debate over Euclid's parallel postulate and the discovery in the late 19th century of consistent non-Euclidean geometries. (The Gödel incompleteness theorems solved negatively one of Hilbert's famous problems.) In this book Hilbert described a correct and complete axiom system for Euclidean geometry, with the dependence relations between axioms exhaustively determined, and then carefully derived most of Euclid from it. It's not a particularly fun read but its existence is philosophically interesting.Hartshorne, [Euclid revisited book]

The algebraic geometer of the famed book from hell (see below) recently finished another modern-Euclid book. I haven't seen it and don't even remember the title, but it might be interesting.

read it

谢谢谢谢谢谢

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