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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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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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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. – David Zureick-Brown Oct 16 '09 at 17:28
    
OK... Do I need to do something to make it a community wiki or is it already like that? – person Oct 16 '09 at 18:04
    
@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. – Anton Geraschenko Oct 16 '09 at 18:15
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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

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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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. – The Mathemagician Jul 11 '10 at 9:02
    
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 ... – Amitesh Datta Jul 11 '10 at 9:17
    
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. – The Mathemagician Jul 11 '10 at 16:09
    
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? – Amitesh Datta Jul 12 '10 at 9:48
    
I apologize: I meant Ahlfors instead of "Alfhors" in the most recent comment. – Amitesh Datta Jul 12 '10 at 10:09

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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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. – The Mathemagician Jul 11 '10 at 16:11
    
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. – Henry.L Aug 1 '13 at 12:37

Real Analysis, by Frank Morgan. The chapters are short and very directed. The proofs are written well. The exercises are well-selected. The book is written at a level accessible for most students.

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Vinogradov's Elements of Number Theory - the problems more so than the text itself.

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