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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 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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Atiyah and MacDonald, Introduction to Commutative Algebra

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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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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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