Question

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.

Answer 1

Spivak, Calculus

Answer 2

Algebraic Topology by Hatcher (available online here).

Answer 3

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

Answer 4

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.

Answer 5

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.

Answer 6

Milnor, Topology from the differentiable viewpoint.

Answer 7

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".

Answer 8

Serre, A Course in Arithmetic.

Answer 9

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

Answer 10

Ordinary Differential Equations by Vladimir I. Arnold

Answer 11

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

Answer 12

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

Answer 13

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).

Answer 14

The Chicago undergraduate mathematics bibliography is a nice annotated list of books.

Answer 15

Complex Analysis, by Lars Ahlfors

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

Answer 16

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.

Answer 17

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.

Answer 18

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

Answer 19

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.

Answer 20

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.

Answer 21

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.

Answer 22

Guillemin and Pollack, "Differential Topology"

Answer 23

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.

Answer 24

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.

Answer 25

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.

Answer 26

Basic Algebra by Jacobson.

Answer 27

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.

Answer 28

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

Answer 29

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.

Answer 30

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