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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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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

96 Answers 96

Algebraic Topology by Hatcher (available online here).

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If you like print books (easier to carry around, scribble in, etc.), Hatcher's book sells for $37, which seems pretty reasonable. –  Michael Lugo Oct 16 '09 at 20:12

Generatingfunctionology by Wilf is fun, free, requires very little in the way of prerequisites, and is as good an introduction to the methods of analytic combinatorics as could be asked for. It's long been one of my favorite textbooks.

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The free edition is the second (early 1990s?); there's a third edition (2005), which as of now is not free. The third edition doesn't differ that much from the second one, though. For a more advanced book that massively expands upon Wilf, I recommend Flajolet and Sedgewick's <i>Analytic Combinatorics</i>, published 2009 (also available <a href="algo.inria.fr/flajolet/Publications/AnaCombi/… online!</a>) -- but this is really a graduate-level text. –  Michael Lugo Oct 17 '09 at 5:11

Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.

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I finally gave in and bought this book last week, after realizing that at any given moment over the last few years I was more likely to have it checked out of the library than not. –  Michael Lugo Oct 17 '09 at 5:04

For undergraduate level topology (mostly point set topology) I recommend "Topology" by Munkres. I learned topology from this book as an undergrad and I remember this being one of my favorite books at the time.

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Spivak, Calculus

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I agree, with the caveat that the subject of the book is usually called "Elementary Real Analysis" these days. That said, when I first read this book I loved it so much that it made me want to be a mathematician. It's both rigorous AND intuitive, in a way that both qualities complement one another. –  John Goodrick Oct 16 '09 at 21:46

Milnor, Topology from the differentiable viewpoint.

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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

Serre, A Course in Arithmetic.

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Not exactly one of the topics in the question, but I particularly liked Silverman and Tate's Rational Points on Elliptic Curves.

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Ordinary Differential Equations by Vladimir I. Arnold

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Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo is an excellent introductory book.

http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

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Needham, Visual Complex Analysis. I read this while in high school, and it's simply beautiful. I recommend this book as a supplement to any first course in complex analysis (a different book should probably be used for the main textbook since Needham's is very pretty, very engaging, but not very rigorous).

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Real and Complex Analysis by W. Rudin is a beatiful and extremely well written book which presents the fundamentals of real and complex analysis highlighting the interactions between different results and ideas.

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I'm all for "adult rudin"... but undergraduate? I don't see it. Have you seen it used at the undergraduate level? –  Michael Hoffman Oct 30 '09 at 13:28
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Actually, Rudin has an undergraduate-level book also, the "small Rudin". I learned from it, and it was fine. I loathe the big Rudin, though, even for graduate level. I never managed to learn anything from it; I especially hated the way every proof refers to a million previous results as "Lemma 12.1.8" without mentioning what they actually are. As a result, reading anything required flipping through the whole book after every line just to know what he is talking about. –  Ilya Grigoriev Jan 23 '10 at 7:16
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I don't like either Rudin particularly,to be honest.Adult Rudin tries to put too much into one book.Folland is the same level and is just so much more pleasant to read. –  Andrew L Mar 18 '10 at 20:44

Spivak, A Comprehensive Introduction to Differential Geometry. There is a nice geometrical philosophy and plenty of motivation.

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Especially the newer version which is typeset in LaTeX! The typesetting of the older version is still charming in its own ugly way, though. Also, I'm not sure if Spivak should count as an undergraduate level book... –  Kevin H. Lin Dec 28 '09 at 15:24

Complex Analysis, by Lars Ahlfors

http://en.wikipedia.org/wiki/Lars_Ahlfors

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It is a complete mystery to me why people are still using this monstrosity. Actually,it's not-it's because Ahlfors was a God at Harvard and they're afraid of being struck down by lightening using anything else.I can think of at least a half a dozen texts now that are better then this one. –  Andrew L Mar 18 '10 at 20:42
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I'm not a huge fan of this book, either... –  Qiaochu Yuan May 25 '10 at 6:32
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'Elementary Theory of Analytic Functions of One or Several Complex Variables' by Cartan is a far superior text. –  Saikat Biswas May 10 '11 at 23:51

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

As an undergraduate I loved Shafarevich's book Basic notions of algebra. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields.

I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.

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The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).

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Dummit and Foote's Abstract Algebra is an excellent book for learning group theory, ring theory, and module theory. There's also a section on basic algebraic geometry and homological algebra.

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For discrete mathematics, I would recommend Van Lint-Wilson's "A Course in Combinatorics" as a good introductory text. It consists of 38 (in my edition) chapters that give (often largely self-contained) introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper.

My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book.

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Guillemin and Pollack, "Differential Topology"

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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

Basic Algebra by Jacobson.

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Volume I is a great guide for the advanced undergraduate, but I think that volume II is beyond what all but the most sophisticated undergraduates can deal with. –  David Speyer Oct 30 '09 at 13:05

I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:

  • Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov and S. V. Fomin
  • Theory of Functions of a Real Variable by I. P. Natanson (this I think, it's hard to find)
  • Theory of Functions of a Complex Variable, Second Edition (3 vol. set)
  • Elements of Functional Analysis by L.A. Lusternik and V.J. Sobolev
  • Problems in Mathematical Analysis by B. Demidovich
  • Calcul intégral et differentiel (2 vol. set) by N. Piskounov. In English should be something like Differential and Integral Calculus
  • A Course of Mathematical Analysis (2 vol. set ) S. Nikols'skii
  • Differentialrechnung und Integralrechnung (3 vol. set), by Gregor M. Fichtenholz. Unfortunately, there is no English translation of this book, only the German translation that it's mentioned. I think this was THE calculus book on Russia. THIS BOOK SHOULD BE TRANSLATED INTO ENGLISH, and I suspect that there is no copyright, it appears around 1959 I think.
  • Mathematical Analysis (2 vol. set) by Vladimir A. Zorich. This is a very recent book from a great mathematician which is in Moscow university. The book is based upon is lecture.

I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus". It is very good as an undergraduate book. The spanish translation (for those who want to search) is Calculo diferencial e integral

Books that undergraduate should not touch (in my humble opinion) are books written in bourbaki's style.

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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

Ok, this is not a single book, but I have often found books from the Springer Undergraduate Mathematics Series (SUMS) to be excellent. Here is a list of titles.

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Another one I like is "An introduction to Lie algebras." by Erdmann and Wildon.

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Galois Theory by Ian Stewart is excellent. The third edition is quite different from the second and includes many more problems.


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General geometry: Coxeter, Introduction to Geometry.

Not so much a textbook as a collection of essays (in particular, it doesn't have exercises), but all of the essays are instructive and enlightening.

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