What are some recommended textbooks for graduate level mathematics courses?

[Esran]

Which are the best graduate level textbooks for the following subject areas?

Applied Nuclear Physics
Electricity and Magnetism
Classical Mechanics
Quantum Mechanics
Unified Field Theory
Statistics (both applied and theoretical)
Analysis (real and/or complex)
Modern Algebra and Linear Algebra
Differential Equations (with partial differential equations would be preferable)
Topology and Metric Spaces
Combinatorics, Probability
General Discrete Mathematics
Graph Theory
Number Theory

Thank you for your recommendations.

 

[zpconn]

Here are the graduate-level textbooks I like for the subjects in your list that I have studied (I'm not a physics major, for instance):

Real analysis: I like the second volume of Elements of the Theory of Functions and Functional Analysis by Kolmogorov and Fomin for measure theory.

Complex analysis: Complex Analysis by Ahlfors is the only book I've ever found that gives an adequate treatment of both the geometric and the analytic halves of the subject.

Conway and Lang are both close runner-ups. Neither builds geometric intuition the way Ahlfors does. Conway has no spirit. Lang has no command of English.

Abstract algebra: For a graduate-level introduction to plain old algebra I like Algebra: A Graduate Course by Martin Isaacs. It's not perfect, though. It doesn't discuss category theory, which anybody studying advanced mathematics should have a passing familiarity with, and it doesn't discuss tensor products. For the latter, Introduction to Commutative Algebra by Atiyah and MacDonald is a nice textbook, although it's not perfect either. Commutative Algebra: with a view towards algebraic geometry is a more lively book covering a lot of the same topics.

Topology: Here the two books by Lee are the clear winners, without any doubt at all: Introduction to Topological Manifolds and Introduction to Smooth Manifolds. Behold nearly perfect mathematical exposition. For the algebraic side of things, a decent exposition is Algebraic Topology by Hatcher.

Combinatorics: Look no further than the definitive Enumerative Combinatorics by Richard Stanley.

Metric spaces: I'm not sure what you mean by this. I wouldn't consider metric spaces graduate-level material. But nonetheless a good book covering them is either the undergraduate textbook by Rudin (Principles of Mathematical Analysis) or the first volume of the excellent and already mentioned Elements of the Theory of Functions and Functional Analysis. (The latter book is fantastic but does unfortunately use outdated terminology. You shouldn't discount it just because of that, though.)

Discrete mathematics: I haven't read this one myself, but I have heard wonderful things about Concrete Mathematics by Knuth.

 

[jtbell]

There are threads discussing textbooks in many of these subjects, in the science books forum to which this thread has been moved.

 

[Esran]

Thanks for finding an appropriate forum for this thread.

Okay, I will check out the recommended books so far. I look forward to seeing what other opinions there are on this.

 

[Landau]

No book is perfect, and I always use several books for a different perspective. Some books I like:

Real/functional analysis: Hewitt & Stromberg's Real and Abstract Analysis, Pedersen's Analysis NOW, Rudin's Functional Analysis, Knapps' Basic and Advanceed Analysis.

Complex analysis: I am currently using Lang's book in a course, and I like it, although it's a bit strange. Alfhors' is the classic, I plan to check it out too.

Topology: As for point-set, Munkres is the well-known book, but it is a bit long-winded (and expensive). Dugundji, Willard and Kelley are great. Also, I'd like to check out Jänich's book, which gets very good reviews. As for algebraic toplogy, I don't know, haven't studied it yet :)

Algebra: For linear Algebra, look no further than Roman's Advanced Linear Algebra, which is the most comprehensive reference about the subject I found. For 'abstract' algebra: Lang contains a lot, but is mostly a reference. I very much like Knapp's Basic and Advanced Algebra (him again), Aluffi's Algebra Chapter 0 (all basic algebra using categories from the ground up, very interesting!), and Dummit and Foote.

 

[DrMik]

Calculus, James Stewart, ISBN 0-534-13212-X, 2nd Ed. Primer for partial diff, (Chapter 12)

 

[Fredrik]

Electricity and Magnetism: I don't know, but I know it's not Jackson.

Classical Mechanics: You said "graduate level", and you seem to like math, so I suggest you check out "Mathematical methods of classical mechanics", by V.I. Arnold. (I haven't read it myself yet, but I'm buying it the next time I order some stuff from Amazon). And whatever you do, don't get Goldstein. There's nothing in physics that I hate more than the 2nd edition of that book. The 3rd is likely better, but how good can it be if it's based on the 2nd?

Quantum Mechanics: Ballentine. Also read Isham, if you haven't already. (It's an easy read that you can finish pretty quickly, and it will help your understanding a lot).

Unified Field Theory: What do you mean by this? If you mean quantum field theory, it's a good idea to get several books. I suggest Srednicki, Weinberg (vol.1) and Zee.

Linear Algebra: I really like Axler, but it's more of an introductory text than a "graduate" text.

 

[dx]

Electricity and Magnetism - The best I've found is "The Classical Theory of Fields" by Landau and Lifgarbagez

Analysis (real and/or complex) - "Calculus" by Michael Spivak (real analysis).

Modern Algebra and Linear Algebra - "Mathematical Physics" by Robert Geroch

Differential Equations - "Ordinary Differential Equations" and "Lectures on Partial Differential Equations" by V. I. Arnold.

Topology and Metric Spaces - "Mathematical Physics" by Robert Geroch (includes Homotopy and Homology theory)

 

[Landau]

Calculus, James Stewart, ISBN 0-534-13212-X, 2nd Ed. Primer for partial diff, (Chapter 12)

Analysis (real and/or complex) - "Calculus" by Michael Spivak (real analysis).

The question was about graduate level textbooks.

 

[Esran]

Right. I've already had courses in many of these subjects. I'm just looking to build a library for myself to use for review and also to just expand my knowledge (preparation for qualifying exams). I have this summer off pretty much, so I'm planning to dedicate it to a whole lot of hardcore studying.

Unified Field Theory-as a clarification, I meant a book that introduces the various attempts that have been made to reconcile quantum mechanics and relativity theory. You know, stuff beyond the standard model. I should have been more clear.

 

[qspeechc]

Classical Mechanics: ... And whatever you do, don't get Goldstein. There's nothing in physics that I hate more than the 2nd edition of that book. The 3rd is likely better, but how good can it be if it's based on the 2nd?

I don't know why everyone likes beating-up on Goldstein. I did the first three chapters of the first edition (my father's copy from his days as an engineering student) on my own a year ago. Yes it's difficult, and some places are not as clearly written as they could be (it is a grad book after all, grad students are supposed to suffer through unclear texts), but I enjoyed the book nonetheless. Okay, maybe I'm not qualified to judge the book having only read three chapters, but that's my two cents.
The preface to the third edition -- which has been revised by two other people, not Goldstein -- explains how the three editions differ considerably from one another, as an aside.

Anyway, sorry to further de-rail this thread, but since it has been brought up a few times already, could Landau and zpconn please tell me more about the complex analysis books by Alfhors, Conway and Lang?

Also, does anyone know anything about papa Rudin (real and complex analysis)?

 

[redrzewski]

I really enjoyed Rudin's Real and Complex Analysis. Definitely preferred it over Royden for real analysis, and preferred it over Ahlfors for complex analysis. But I would recommend going thru both Rudin and Ahlfors for complex since Ahlfors treats the applied aspects of complex analysis more directly.

 

[Esran]

Ah, yes, then I think I will choose Ahlfors.

Regrettably, I'm on a budget, so I can only choose one book in each category at this point. Pretty much. Unless there's one that's only $40.00. Then we could talk about doubling up or tripling up.

yeah...there's a hard existence proof for you. :rofl:

 

[Fredrik]

Unless there's one that's only $40.00.

Isham is just $20.70 at Amazon right now. (And Ballentine is $39.29. Hm, I wonder how they choose those prices).

I'm not sure what to recommend for an introduction to quantum gravity. I think Rovelli's quantum gravity book looks like a readable introduction to loop quantum gravity, and Zwiebach is supposed to be the best introduction to string theory. But I don't know a book that gives you a good overview. Maybe Penrose's "The road to reality", or Smolin's "Three roads to quantum gravity", but I have only skimmed the relevant sections of the former, and I haven't even done that with the latter.

 

[Esran]

What about the "yellow book" series in math?

 

[Landau]

What about them? There are lots of good books in that series. Some books named in this thread are yellow GTM's, such as Introduction to Smooth Manifolds, Advanced Linear Algebra, Kelley's Topology, Lang's Complex Analysis, Hemitt and Stromberg, Pedersen's Analysis NOW.

 

[fourier jr]

Regrettably, I'm on a budget, so I can only choose one book in each category at this point. Pretty much. Unless there's one that's only $40.00. Then we could talk about doubling up or tripling up.

yeah...there's a hard existence proof for you. :rofl:

in that case have a look at zimmer's essentials of functional analysis. it's just a small paperback so i think it's only ~$25, & it's definitely not an intro book. (the stone-weierstrass thm is given on p.3, in chap 0)

i'm also a bit surprised nobody has listed hungerford's algebra. i thought that was a fairly standard textbook. it's another one in that "yellow book" series. for linear algebra, greub's does it from a fairly advanced point of view but there could have been more problems, and more difficult ones too.