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What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

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It's no longer possible to add useful answers to this question (as there are too many!) and it's unclear whether this question would be "allowed" by modern standards -- far too broad. As it's been popping back to the front page fairly frequently, we've decided to close it. –  Scott Morrison Jul 11 '10 at 13:30
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See discussion on meta: tea.mathoverflow.net/discussion/499/… (and remember to vote this comment up, so it is visible to others) –  Victor Protsak Jul 14 '10 at 10:34
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closed as no longer relevant by Harry Gindi, Robin Chapman, Greg Stevenson, Harald Hanche-Olsen, Scott Morrison Jul 11 '10 at 13:29

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96 Answers

I liked Elements of Abstract Algebra by Allan Clark, which is mainly a problem book with a moderate amount of exposition, but the problems are so well-chosen that a diligent undergraduate student working through all of them will come out with a solid knowledge of group theory, classical ring theory, and Galois theory.

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Linear Algebra / Hoffman & Kunze - A book that truly develops linear algebra in a gradual manner. It starts with a basic discussion of systems of linear equations, matrices, Gaussian elimination, etc. and gradually progresses to the more abstract theory. Eventually it even touches upon subjects such as tensor products, the exterior algebra and the Grassmann ring. In short, it manages to cover a lot of linear algebra in a very leisurely and clear manner. I think that this is the quintessential example of a how an undergraduate level math book should be written. The only thing I don't like about it is the fact that quotient spaces aren't mentioned throughout the book (they're mentioned in the appendix, though).

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Atiyah and MacDonald, Introduction to Commutative Algebra

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I would also recommend the book entitled Analysis on Manifolds by James Munkres. I think that this is a good undergraduate textbook in mathematics for any student wishing to pursue multivariable calculus in greater depth. My only complaint is that Munkres often chooses to include details which can be seen easily after a little bit of thought. Perhaps this can be viewed as an effort to show the student how to "properly do analysis": doing analysis, just like doing any other branch of mathematics, requires you to carefully apply definitions and theorems, and it is important for the student to appreciate this early in his/her mathematical learning.

That said, the book is an excellent text overall for "advanced calculus". The student will need to be familiar with single variable analysis and perhaps some linear algebra. (Even a rudimentary knowledge of linear algebra will do since Munkres develops most of the necessary theory from scratch.)

Roughly speaking, the book splits into two parts. The first part covers most of the results students see when doing multivariable calculus that are stated "without proof" in their texts. For example, "the equality of the mixed partials", "double integrals can be done in any order", "a bounded function is Riemann integral if and only if it is continuous almost everywhere", "the change of variables theorem" etc., are (very) imprecise forms of some of the results Munkres establishes.

In the second part of the book, manifolds and their theory are introduced. Thus, for example, a rudimentary introduction to tensors is given, and this is supplemented by the basic theory of differential forms, the De Rham groups (of the punctured plane), Stokes' theorem etc.

I think that the exposition could be tightened: if you actually pick up the book and really make an effort to read it, it is quite possible to finish the first half of the book in the space of a week (that is, approximately 200 pages in a week) simply because certain topics are explained in more detail (at least in my opinion) than necessary. (One example is Munkres' proof of the linearity, monotonicity, additivity etc. of the Riemann integral. This is proved in three contexts separately: the case of the integral over a rectangle, that over a bounded set, and that of improper integrals, when essentially the proofs can be left as relatively easy exercises in some cases.)

As the above comments suggest, I think that this is an excellent book for undergraduate students, but perhaps less so for graduate students. (Spivak's Calculus on Manifolds is good for both undergraduate and graduate students, in my opinion, but some people may suggest that it is too hard for undergraduates.) And after reading this book, you should have more than enough preparation to read more advanced texts such as William Boothby's An Introduction to Differentiable Manifolds and Riemannian Geometry.

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Algebraic Theory of Numbers, by Pierre Samuel

Assumes only elementary knowledge of group and ring theory (even less than a complete undergraduate course which covers Galois theory) and develops algebraic number theory, a beautiful subject which puts much elementary number theory into an interesting perspective.

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I would recommend Walter Rudin's classic text entitled Real and Complex Analysis for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: Principles of Mathematical Analysis.)

The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks.

While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)

The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's Real Analysis, Bartle's The Elements of Integration and Lebesgue Measure etc., very easy to follow.

The Amazon page for this book can be found here.

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Ireland and Rosen, A Classical Introduction to Modern Number Theory is a great second course in number theory. In spite of being part of "Graduate Texts in Mathematics" series and unlike Rudin's Real and Complex Analysis (see a comment above), this is a book at the undergraduate level. It only presupposes undergraduate algebra as in Herstein Topics in Algebra or M. Artin's Algebra, undergraduate analysis like in Rudin's Principles of Mathematical Analysis and basic number theory. In fact it recalls or proves many of the necessary results in each of those fields. A Classical Introduction to Modern Number Theory bridges the gap between basic number theory (that covers modular arithmetic, Fermat's little theorem and QR) and books like Lang's Algebraic Number Theory or Cassels and Fröhlich.

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GREAT book. The best number theory book for honor students. –  Andrew L Jul 7 '10 at 5:42
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Lebesgue Integration on Euclidean Space / Frank Jones - an extremely readable book on Lebesgue theory on $\mathbb{R}^n$ (lots of figures and geometric intuition). He constructs Lebesgue measure in a very down-to-earth manner, much more explicitly than other more abstract constructions (via Caratheodory's extension theorem or Riesz's representation theorem). In my experience, it's best to first study Lebesgue measure on $\mathbb{R}^n$ and only then point out that it's merely one instance of the general theory of measures, which is the way this book is written. It can't compare with the "tougher" books on measure theory (e.g. Big Rudin) since it doesn't discuss the Radon-Nikodym theorem and many other important theorems in measure theory, but then again the book is clearly intended for an undergraduate audience, and as for Lebesgue theory on Euclidean spaces, it provides a pretty complete picture.

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Algebra: Chapter 0, Paolo Aluffi

Best book on algebra I've had my hands on yet, and I love how it uses category theory. I wouldn't mind having a course taught from this one. Topics from group theory all the way through field theory, linear algebra, and homology. This book deserves more attention!

http://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813/ref=sr_1_1?ie=UTF8&s=books&qid=1278799249&sr=8-1

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Terrific book for first year graduate algebra or honors undergraduate. –  Andrew L Jul 10 '10 at 22:23
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Introduction to Analytic Number Theory - Tom M. Apostol.

When I bought this I really didn't want to put it down. It's a great book for exciting one's interest in the subject.

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A Concrete Approach to Abstract Algebra by W. W. Sawyer ($6 on Amazon!)

Though it goes a bit slow at times, it is by far the simplest, most intuitive book on Abstract Algebra in existence. Written for the non-mathematician, it does a great job of teaching the subject in simple, easy-to-understand prose. I couldn't put it down!

There are also two chapters on linear algebra, leading up to the final chapters, "vectors over fields" and "fields regarded as vector-spaces".

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Topics in Algebra by I. N. Herstein. A new edition will be coming out this year.

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I plan to post a complete reading list for undergraduates and graduate students at my blog this summer with my commentaries,but here's one I think that's available online and doesn't get nearly enough credit despite the fame of it's author: Gilbert Strang's Calculus. I wouldn't use anything else for a regular,non-honors calculus course. Carefully written,beautifully motivated with TONS of creative and SIGNIFICANT applications. I hope one day Strang finds the time to write a second edition-I have a list of improvements to suggest.

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Siegfried Bosch, Lineare Algebra

It's a very elegant, concise but beautifully written approach to Linear Algebra, and I love it.

Unfortunately for people who don't speak German, it has never been translated.

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For (applied) ODEs: Nonlinear dynamics and chaos by Steven Strogatz.

A very inspiring book! The explanations are crystal clear with lots of pictures. And it's funny too – the "Romeo and Juliet" illustration of 2-dimensional linear systems (Section 5.3) is a classic.

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I found «A (terse) introduction to linear algebra» (Katznelson) to be a much better book than Axler. It's part of the Student Mathematical Library and published by the AMS.

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I am surprised this has not been mentioned before (is it too advanced?):

Bott and Tu, Differential forms in algebraic topology.

The best introduction to de Rham cohomology, spectral sequences, characteristic classes from the algebraic point of view, and countless other topics.

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... and seeing that Hatcher, Serre, Jacobson, Alperin, and Evans have been featured (some at the very top), I don't agree that it's "too tough for this list". –  Victor Protsak May 25 '10 at 4:33
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Complex Variables: Harmonic and Analytic Functions by Francis J. Flanigan

A nice little Dover paperback which turns the standard course on complex variables on its head. It begins by doing some multivariable calculus in the plane and harmonic functions, then proceeds to talk about complex numbers and to build analytic functions.

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Visual Complex Analysis by Tristan Needham is awesome!

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In fact, there are others that agree with you: the book is already in the list! –  Bjorn Poonen Mar 7 '10 at 7:46
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Hugo Steinhaus http://en.wikipedia.org/wiki/Hugo_Steinhaus book "Mathematical Kaleidoscope". It is kind of mathematical trivia sometimes very deep;-) It is not for learning math but for learning how to learn math in fun way.

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Here is an undergraduate level math book recommendation from an early undergrad's position:

I like "Linear Algebra Done Right". I've looked at a bunch of books on linear algebra, and the usual matrix approach is to me a big turn-off when what you're really interested in is the abstract machinery of transformations between vector spaces. I'm not a research mathematician. In fact, I don't even study linear algebra yet, but as a student of mathematics that like algebra, spaces, maps and all that good stuff, I find this to be a very readable account of linear algebra.

There are more abstract books on the subject, and my impression is that LADR prepares you for the next level way before you're usually "allowed to" by other accounts like Lax etc. The trade-off is that LADR is not a book for engineers, but this would be a sad world for a mathematician if that was something he had to worry about (in his spare time). Great for self-study. Reads like a novel. I'd probably prefer it if Axler used sets for span and bases instead of lists, but that's something you'll probably be able to shake off with the next book you read on the subject.

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See the other answer on this book for my comments. –  Gerald Edgar Dec 28 '09 at 15:24
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I am surprised no one has mentioned Halmos' Naive Set Theory or Finite-Dimensional Vector Spaces or Rudin's Principles of Mathematcal Analysis. There's also Sheldon Axler's Linear Algebra Done Right and Royden's Real Analysis.

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I believe we are supposed to put just one book per answer. It makes it easier to wote up. –  Grétar Amazeen Oct 17 '09 at 18:18
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Sheldon Axler (not Steven), for what it's worth. –  Tom Leinster Oct 24 '09 at 2:48
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I think that Linear Algebra by Friedberg, Insel, and Spence is a spectacular linear algebra book. It gets straight to the point, it provides a worked out example or two exactly when they're needed, and it has lots of interesting exercises.

There are way too many gigantic linear algebra books with colour pictures and contrived examples which often seem to obsfucate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.

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Karen Smith et al., An Invitation to Algebraic Geometry

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Since I haven't looked at this book but might be interested... This book is "undergraduate level" for whom? Presumably, many Harvard senior math majors would be able to tackle it. How many senior math majors at a mid-tier public research university? mid-tier liberal arts college? compass point state college? –  Alexander Woo Dec 28 '09 at 21:23
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A basic undergrad algebra book which I feel is not as well known as it should be is Michael Artin's Algebra. I have it in soft cover so I hope it's actually the one in this Amazon link. Anyway it's beautifully written, provides context and motivation and is just a pleasure to read or browse. How often do you find a basic text written by a world-class expert?

"Always study from the masters".

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As far as I know there is only one book by Michael Artin with that title. I have the hardcover and it looks like the one you link to. Apparently Artin is working on a new edition. –  Michael Lugo Nov 1 '09 at 17:15
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That book is so much better than Dummit and Foote for undergraduates. D&F is also useless at the graduate level, where much better texts like Lang blow it out of the water. –  Harry Gindi Nov 30 '09 at 12:16
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Artin is going to be rough going for undergraduates who are not well versed in basic geometry and linear algebra,fpqc.But you can't help but love the infectious passion with which Artin weaves his craft in front of the students.He loves algebra and he's trying to prosyletize his students to it. A book with a similar geometric bent,level and also by a master that students will probably find easier going is E.B.Vinberg's A COURSE IN ALGEBRA. But Artin's book is very good and it's good news for all of us that Artin is revising it. –  Andrew L Mar 27 '10 at 22:01
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From the quick look I've had, not much representation theory has been mentioned so here goes for undergrad level rep theory (perhaps suitable for 3rd/4th year in a standard sequence of undergraduate study), roughly in the order of difficulty (from easiest to hardest):

  • James & Liebeck - "Representations and Characters of Groups" (a very good introduction)

  • Sagan - "The symmetric group: representations, combinatorial algorithms, and symmetric functions"; (the first two chapters here at least are representation theory) OR James & Kerber - "Representation Theory of the Symmetric Group" (this one includes some modular representations of $S_n$)

  • Alperin - "Local Representation Theory" (basically, modular representation theory)

  • Hall - "Lie groups, Lie Algebras and Representation Theory" (a solid introduction to Lie theory); for a more advanced perspective Harris & Fulton - "Representation Theory: A first course" (but it could be slightly terse at points, but not necessarily)

For algebraic geometry, the one book I'd suggest is "Algebraic Geometry: A first course" by Joe Harris, very nice and full of examples. For algebraic number theory, a very good introduction is Janusz - "Algebraic Number Fields" (followed perhaps by Childress - "Class Field Theory", or Silverman - "The Arithmetic of Elliptic Curves" to go in a slightly different direction).

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I'm a big fan of John Hubbard's "Vector Calculus, Linear Algebra and Differential Forms" text. I was a TA for the course twice at Cornell and was amazed at how well it went. The text has an extremely pleasant "zest" to it. When Hubbard asked me to take a look at it my first response was the text is "overflowing with the spirit of calculus". I still believe that. I have a hard time containing my praise.

The main problem with the text is that it's so engrossing. It places more demands on the student than a traditional service course text would ever consider. But it's also far more rewarding. At Cornell it was taught as a branch of their traditional calculus sequence -- it was a course that was earmarked for keener students, mostly from other departments.

In short, if you want to have physics, engineering and economics students appreciating the derivative as a linear approximation, thinking Lipschitz bounds for functions are cool, being interested in the computation of norms of linear operators, etc, this is a great resource.

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For a long time, Kolmogorov-Fomin's Introductory Real Analysis was my standard for a great mahtematics textbook. I can't imagine a better introduction to serious analysis.

The translation I'm linking to is very good, and includes excercises (the original has many fewer), but it is incomplete (it's missing the chapter on Fourier Series). So if you can read Russian, I recommend you get the original.

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Miles Reid, Undergraduate Algebraic Geometry.

ps - anyone who thinks one can teach an undergraduate class out of Hatcher's Algebraic Topology (which is a great book) at more than 10 universities in the US is sadly deluded. Ditto for a few more things I've seen here.

pps - somewhere between a third and half of the math majors here could handle abstract algebra out of Artin. It would be great for those that could, but we're not going to ditch half our students.

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Very astute comments! Many answers to "best textbook" questions are way off the deep (macho) end. –  Victor Protsak May 24 '10 at 5:48
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