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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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  • $\begingroup$ This is borderline, but I think it is a legitimate question of interest to math instructors. I think its better as a community wiki though. $\endgroup$ – David Zureick-Brown Oct 16 '09 at 17:28
  • $\begingroup$ OK... Do I need to do something to make it a community wiki or is it already like that? $\endgroup$ – person Oct 16 '09 at 18:04
  • $\begingroup$ @person: Normally, to make a question or answer community wiki, you click the "community wiki" checkbox in the lower-right when you're composing or editing. But moderators have the power to convert a post to community wiki, which is what David has done here (actually, three moderators independently thought it should be converted to CW ... it's not something we do willy-nilly), so you don't need to do anything. $\endgroup$ – Anton Geraschenko Oct 16 '09 at 18:15
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    $\begingroup$ 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. $\endgroup$ – Scott Morrison Jul 11 '10 at 13:30
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    $\begingroup$ See discussion on meta: tea.mathoverflow.net/discussion/499/… (and remember to vote this comment up, so it is visible to others) $\endgroup$ – Victor Protsak Jul 14 '10 at 10:34

95 Answers 95

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A Field Guide to Algebra by Antoine Chambert Loir. Covers a surprising amount of material considering the short length and minimal prerequisites

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"TOPOLOGY WITHOUT TEARS" by SIDNEY A. MORRIS is a very nice introduction in topological spaces theory, I think.

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Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.

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  • $\begingroup$ That sounds like a nice book. It's pretty expensive, unfortunately. $\endgroup$ – Ryan Budney Jan 1 '10 at 1:39
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"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.

The author is a prof at the Indian Statistical Institute (Kolkata)

This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms.

I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.

"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it.

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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 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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  • $\begingroup$ I really like Munkres,too. My one complaint with it is I just wish Munkres had included some physical applications of this material,such as in differential equations or electromagnetism. $\endgroup$ – The Mathemagician Jul 11 '10 at 16:11
  • $\begingroup$ Spivak's book is somehow too concise for undergraduate. I prefer Munkres. And Janich&Brocker's introduction to diff. geometry might serve as a reading material if we want to make the flavor. $\endgroup$ – Henry.L Aug 1 '13 at 12:37
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Atiyah and MacDonald, Introduction to Commutative Algebra

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    $\begingroup$ Do you really think this book is undergraduate level? $\endgroup$ – Qiaochu Yuan Jul 10 '10 at 18:48
  • $\begingroup$ I feel that in the case of AM, this criticism misses the mark: although not everyone is an algebraist, commutative algebra is a valid subject of undergraduate study (i.e. not "too deep")! The book is very clearly written (incomparably better than van der Waerden) and is good for self-study for problem-oriented people. Nonetheless, I wish that at the time I had the courage to read Zariski and Samuel instead: in spite of being 2 volumes, it is so much more relaxed because the authors take care to $\mathit{explain}$ the material, in multiple ways, and offer both breadth and depth of perspective. $\endgroup$ – Victor Protsak Jul 11 '10 at 8:15
  • $\begingroup$ @Victor I totally agree on Zariski and Samuels.But if you're going to invest that much time in a tome that lengthy,then you may as well get Eisenbud.In any event,all these books will be too difficult for any but the best undergraduates,I'm sorry. $\endgroup$ – The Mathemagician Jul 11 '10 at 9:05
  • $\begingroup$ Right. I definitely think Reid's book is at a much more manageable level than A&M. $\endgroup$ – Qiaochu Yuan Jul 11 '10 at 9:18
  • $\begingroup$ The second paragraph of the introduction begins, "This book grew out of a series of lectures given to third year undergraduates ..." The authors also point out that their book is not intended as a substitute to the Zariski-Samuel or Bourbaki books. $\endgroup$ – Kiochi Jul 11 '10 at 14:33
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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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Metric Spaces by Mícheál Ó Searcóid

https://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/ref=sr_1_1?ie=UTF8&s=books&qid=1256082496&sr=1-1

It's an exhaustive introduction analysis at the level of the metric space that's well worth reading.

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Introduction to Analysis by William R. Wade

https://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_1?ie=UTF8&s=books&qid=1256082707&sr=1-1

This is a good transition from undergraduate calculus to analysis.

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Godement "Analysis" (I,II,III,IV) https://www.amazon.com/Analysis-Convergence-Elementary-functions-Universitext/dp/3540059237

"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.

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Geroch, Mathematical Physics

Don't be scared by the title: it teaches algebra, topology and measure theory, using category-theoretic language.

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  • $\begingroup$ Indeed, that's a nice one! $\endgroup$ – Peter Arndt Mar 19 '10 at 1:11
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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 obfuscate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.

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  • $\begingroup$ I second this choice.It balances rigorous theory and applications better then any book on the subject I've seen. Charles Curtis's old classic is also excellent and it has a much better discussion of the Jordan form and decomposition algorithm. $\endgroup$ – The Mathemagician Mar 27 '10 at 21:45
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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 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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  • $\begingroup$ You are being incredibly optimistic here,Amitesh. See my comments below. Why don't you recommend Lang's ALGEBRA to sophomores without linear algebra while you're at it? "Undergraduate texts".At Yale or Harvard,MAYBE-and even THAT'S a stretch. $\endgroup$ – The Mathemagician Jul 11 '10 at 9:02
  • $\begingroup$ However, even if, by current standards, undergraduates are not expected to be able to read a book of this sort, I believe that it is something that is readily accessible if you are willing to put in the appropriate hard work. For instance, as I noted above, the only prerequisites for this book lie within 200 or so pages of mathematics "above calculus". (Rudin's earlier textbook.) But perhaps I am being optimistic now that I think about it ... $\endgroup$ – Amitesh Datta Jul 11 '10 at 9:17
  • $\begingroup$ For UNDERGRADUATES,Yes,you are,Amitesh.Then again,if you're a first year graduate student and you CAN'T read Rudin-not find it difficult,literally it's beyond your ability-you need to seriously consider a change of career. $\endgroup$ – The Mathemagician Jul 11 '10 at 16:09
  • $\begingroup$ But just out of interest, in case you know, is Rudin's Real and Complex Analysis actually taught at many universities at the graduate level? My perception is that Royden's and Bartle's texts are the "classics" but I have not actually read them. (I am reading Rudin, instead.) And it indeed seems that some universities use Royden for an introductory real analysis course, and Alfhors for an introductory complex analysis course. However, Royden is, in my opinion, too basic if you wish to purse analysis in some depth so if Rudin is not used, do you know what is? $\endgroup$ – Amitesh Datta Jul 12 '10 at 9:48
  • $\begingroup$ I apologize: I meant Ahlfors instead of "Alfhors" in the most recent comment. $\endgroup$ – Amitesh Datta Jul 12 '10 at 10:09
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There is a good list here (Wayback Machine), divided by subject, that also contains many links to freely available textbooks and lecture notes.

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  • $\begingroup$ can't see anything here - where is the list? $\endgroup$ – vonjd Oct 30 '09 at 13:03
  • $\begingroup$ ...perhabs it is because I don't speak chinese... $\endgroup$ – vonjd Oct 30 '09 at 13:04
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Linear Algebra: With Applications Otto Bretscher

Used at Carleton College – nice explanations, and quite a few proofs. Presents information primarily by providing examples, definitions/axioms and then proofs.

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  • $\begingroup$ Taught out of it several times, didn't like it at all. Also, the smugglers used to "motivate" (hah!) linear transformations in the first edition turned into evildoers in the second. I dread to even think who they've metamorphized into now. $\endgroup$ – Victor Protsak May 24 '10 at 5:55
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I had many trials and these are in my opinion the best for an introductory, undergraduate level:

ODEs: Holzner: Differential Equations for Dummies
PDEs: Farlow: Partial Differential Equations for Scientists and Engineers

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G. J. O. Jameson: Topology and Normed Spaces for an introduction to functional analysis from a topological point of view.

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Strichartz, The Way of Analysis

Herstein, Abstract Algebra

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  • $\begingroup$ I also like Herstein's Abstract Algebra. $\endgroup$ – Pandora Jul 10 '10 at 21:44
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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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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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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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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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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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Since Numerical Mathematics has not been covered, I would recommend the following

Introduction to Numerical Analysis by Stoer et. al. https://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X/ref=sr_1_14?ie=UTF8&s=books&qid=1255807973&sr=8-14

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lindsey childs a concrete introduction to abstract algebra

here maybe.

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  • $\begingroup$ It's a bit obsolete by now. $\endgroup$ – Victor Protsak May 24 '10 at 5:56

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