Books Added: Differential Topology

Introduction

Putting out the following few books has been far harder than I expected, and has taken a lot more time. There are 6 of them: 3 texts, 1 reference, and 2 small sets of notes.

The fundamental problem is that I haven’t worked thru these books yet. Simply put, I’m effectively a grad student trying to figure out which books to read in order to introduce myself to a new field.

To put it more fancifully, I feel a bit like a wide-eyed urchin looking in a bakery window, trying to figure out what the different pastries will taste like, and I’ve picked out a few of them to try.

That simile fails, of course, because I’m not just looking at the pastries; I’ve held them in my hands and looked inside. I own these books, I’ve read each preface and table-of-contents, and I’ve read further into them. I’ve seen every one of them in other bibliographies; I’ve just read some of the reviews on Amazon….

The problem is, I haven’t gone into these books and come out the other side.

That matters, because my opinion of books has been known to change after I finished them, though usually from bad to good. Sometimes what I thought was incomprehensible and poorly written ends up seeming a nice presentation after I have a global framework. (That was true of Dugundji, and of Ahlfors’ complex analysis, and of Dean’s elements of abstract algebra: they are all very nice books, looking back on them, now that I understand more about the subjects.) OTOH, I’ve also found books that seemed inviting and appropriate, only to run into insuperable obstacles. (Hamermesh’s group theory comes to mind: even today his book is still opaque to me; I have trouble following him even on things I understand.)

So, I am encouraged by the other reviews of these books, particularly in Bloch, on Amazon, and among themselves: Guillemin & Pollack recommend all but Hirsch – which hadn’t been published yet! Bloch recommends all but Guillemin & Pollack; Milnor Topology lists all but Hirsch and Guillemin & Pollack, which didn’t exist at the time.

book reviewers on Amazon

Let me point out that there are two especially detailed Amazon reviewers all of whose reviews seemed worth reading; there’s a button for doing that. One reviewer is “Malcolm”; he appears to be an expert in differential manifolds. In fact, I’ve just ordered two books based on his reviews. If you want to see his reviews, start with any of these four: Hirsch, Guillemin & Pollack, Munkres, or Milnor Topology. He did not review either Milnor Morse Theory or Wallace.

The other reviewer seems far more trustworthy than his name, “mathwonk”, might suggest. He’s a college math professor. To find him, look under the reviews of Wallace.

What is it?

So just what is differential topology? “That sounds rather formidable,” said a physicist friend yesterday.

If nothing else, it’s a very reasonable search string on Amazon.

Seriously, it ought to be the study of differentiable manifolds (smooth manifolds). But differential manifolds are, by intention, what we can do calculus on. And the calculus is a lot of machinery.

So I took every “differentiable manifolds” book off my shelves and went thru them, just to help put the differential topology books in perspective. And that was a lot of books.

One way to think of differential topology is: the topological properties rather than the calculus-related ones. More generally, one can think of it as differential manifolds without any additional structure “… such as lie groups, Riemannian manifolds, symplectic manifolds, vector bundles, foliations….” which Lee (I need to add his Smooth Manifolds to the bibliography) says he would call differential geometry.

More importantly, I have seen the subject, whatever it may be, referred to as “differential and algebraic topology” (by Jean Dieudonne). The math arxiv (front end) lists general topology, algebraic topology, and geometric topology, and the last includes “manifolds”. Bloch says that geometric topology more narrowly defined would be topological and piecewise linear manifolds, but not differential.

I think five of these six books can be characterized as: the topological properties of differentiable manifolds without much regard to additional structure, and without using algebraic topology (that’s important for these books if not for the subject).

And I’m in trouble already. These books are going to discuss vector fields. Surely that counts as additional structure. Well, that’s why I allowed for some regard of additional structure. The fact remains that these books look different from my differentiable manifolds books, which in turn look different from my differential geometry books. I hope that as I add more books to the bibliography, I can make some sensible distinctions.

Overview

I started out with 6 books in hand. I wondered if I should add some differential manifolds books; I wondered if I should drop the Milnor “Morse theory”. In the end, I stayed with the original 6, having seen no compelling reason to add or subtract from that list.

Ah, one last point. Munkres specifically uses the phrase and title “elementary differential topology” to mean those theorems in differential topology which do not require algebraic topology for their proofs; similarly, as he says, a theorem in number theory is said to be elementary if its proof does not involve functions of a complex variable. (And that doesn’t mean the proof is easy.)

Oh, another last point. Guillemin & Pollack use an interesting rating scale, for difficulty. There is a perfectly straightforward one used by the MMA (Mathematical Association of America). Since North America uses phrases like “grade 10″, the MAA uses numerical grade levels, extending grades 1-12 up to 18 for “2nd year graduate studies”. A book suitable for seniors and first-year graduate students, then, would be marked “16-17″.

(A very little kid asked me what grade I was in in school, when I was a college freshman. Turned out, he didn’t recognize the word “freshman”, so I told him I was in the 13th grade.)

Guillemin & Pollack took another rating scale that we are familiar with: movies. G = elementary, with only analysis and linear algebra as prerequisites. PG means it requires something more, whether that’s some abstract algebra, or some topology, or whatever. R denotes graduate level mathematics, and X would be “hard going for a graduate student, meant to be read more for inspiration than for comprehension.”

I doubt that I will use either of those systems. I will comment that Guillemin & Pollack labeled Milnor Morse Theory and Wallace as PG; they suggested Milnor Topology as collateral reading with them, but did not rate it. Since Guillemin & Pollack was designed as a “leisurely first year graduate course” predominantly accessible to juniors and seniors, I would guess Guillemin & Pollack and Milnor Topology would have PG ratings.

Ok, on to the books.

Munkres is a reference book. It’s only 109 pages. Part I proves “the folklore theorems” of the subject: those things that almost everyone else says “go look somewhere else for a proof”. Skimming it, I get the impression that it would be a good place to start sharpening ones tools by working out the exercises (and Malcolm’s review concurs). Part II proves that every differentiable manifold has a unique smooth triangulation. That’s pretty major. Both Milnor Topology and Wallace cite this book.

I’ve repeatedly said that TOP, PL, and DIFF (topological, piecewise linear, and differentiable) structures are not equivalent in higher dimensions. And I also don’t understand the distinction between simplicial, piecewise linear, and triangulations.

Munkres gives me a precise definition of triangulation as he used it for the theorem. That’s a big step.

Wallace is where I want to start. I’m not sure I’ll make it all the way, because his chapter on “spherical modifications” looks rather intimidating. But Wallace seems to have been a significant researcher in the great breakthroughs of the 50’s and 60’s, and – more importantly for a first text – he is a good writer and expositor. (I own two other books by him.) More importantly for the subject, Wallace culminates in a proof of the classification theorem for compact connected smooth surfaces – but using critical points of functions on manifolds (this is the beginnings of Morse theory, I think) instead of using simplicial complexes. Since it’s a Dover paperback, you could buy it just for the classification theorem, and you shouldn’t care that it’s only 130 pages. Everyone else I’ve been reading proves the classification theorem for triangulable surfaces. (Ok, I haven’t really been reading Hirsch, but he also does it using the differentiable structure rather than simplicial; but I’ll get there sooner in Wallace.)

So I’ll be picking up Wallace first.

But he’s a special purpose text: one might say, a set of notes culminating in an important theorem.

Guillemin & Pollack is a full-blown text, aimed at graduate students but much of it accessible to undergraduates. Its four main sections are manifolds & smooth maps, transversality & intersection, oriented intersection theory, integration on manifolds. Within them, it lists immersions, submersions, and embeddings; transversality, intersection theory mod 2, and winding numbers; the poincare-hopf theorem, the Euler characteristic, and triangulations; the gauss-bonnet theorem.

I’ll be picking up this after Wallace. (Sadly, Guillemin & Pollack is out of print; the other 5 are still in print.)

Well, not quite. I’m going to do something a little strange, even for me.

Both Guillemin & Pollack and Hirsch are introductions to the subject, aimed at beginning graduate students. Guillemin & Pollack is the more elementary of the two, but Hirsch has a considerable amount of discussion.

I say, a considerable amount of discussion. Hirsch appears to be more detailed, and more advanced, so I would work thru it after Guillemin & Pollack. but I intend to read Hirsch first, and take notes on his remarks, before I start working thru Guillemin & Pollack. That’s how insightful his commentary seems.

Then I can work thru Hirsch. It looks more like an introduction to the key non-algebraic techniques of the subject: transversality (of course); vector bundles and tubular neighborhoods; degrees, intersection numbers, and the Euler characteristic; Morse theory; cobordism; isotopy; and he closes with the classification of compact surfaces using Morse theory.

This brings us to the two Milnor books. Milnor won the fields medal in 1962, and for that alone I’m very interested in what he chooses to say about a subject. Furthermore, he is famous for his lectures. The beginning of the Topology looks a lot like the beginning of Wallace, so I’ll keep it handy from the get-go. Heck, I should quote Malcolm’s review on Amazon: “… too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.”

The best I can do for the Morse theory is to quote an unknown reviewer (“A Customer”, which appears to be the generic name) from Amazon. He said that his advisor gave him this book, saying, “You’re not ready for this yet, but you should have it — it’s the best piece of mathematical exposition there is.” I am pretty damned sure that any bibliography of differential topology is flawed which does not include this book.

In summary, then, 3 textbooks (which I will do in order Wallace – Guillemin & Pollack – Hirsch), 1 reference (Munkres), 1 collateral reading (Milnor Topology), and 1 for dessert (Milnor Morse Theory).

The Books Added

Wallace, Andrew H. Differential Topology: First Steps. Dover 2006 (orig 1979).
ISBN 0 486 45317 0.
[differential topology; 20 Dec 2008]
Looks like a marvelous undergraduate introduction. From the preface: “… in this field, as indeed in any branch of topology, the first steps should be geometric.” He uses differential methods to obtain the classification of 2D manifolds. Short intense guide to further reading. Structured exercises, but no solutions. Epilog.

Hirsch, Morris W. Differential Topology. Springer, 1991 (corrected 4th printing).
ISBN 0 387 90148 5.
[differential topology; 20 Dec 2008]
“Graduate Texts in Mathematics”. The discussion and remarks throughout the text are worth reading first, in my opinion. Uses Morse Theory to get the classification theorem for surfaces.

Munkres, James R. Elementary Differential Topology. Princeton, 1966 (revised ed.).
[differential topology; 20 Dec 2008]
Reference. Proves the “folklore theorems”; culminates in the proof that any smooth manifold has a smooth triangulation.

Milnor, John W. Topology from the Differentiable Viewpoint. Princeton, 1997 (orig 1965).
ISBN 0 691 04833 9.
[differential topology; 20 Dec 2008]
Notes on a few selected topics. Legendary might be more appropriate than “a classic”. Brief guide to further reading.

Milnor, John W. Morse Theory. Princeton, 1969 (orig 1963).
ISBN 0 691 08008 9.
[differential topology; 20 Dec 2008]
A compact presentation, but also legendary. I think of it as dessert after Hirsch.

Guillemin, Victor; Pollack, Alan. Differential Topology. Prentice Hall, 1976.
ISBN 0 13 212605 2.
[differential topology; 20 Dec 2008]
Graduate / advanced undergraduate. A “leisurely first year graduate course”, predominantly accessible to juniors and seniors. Guide to further reading.

Books Added: general topology 2

OK, I said I wouldn’t go buy more books because mine were old. I didn’t. I bought two more old books. I was looking on the internet for more about the “separation axioms” and came across these two. One was a familiar title that I probably should have gone looking for (“Counterexamples”), but I didn’t know of the other.

(Any discussion of the separation axioms must cope with the fact that there are two distinct sets of terminology. These books were cited as the epitomes of the two terminologies.)

They’re both very well reviewed and, it seems to me, excellent. Quite apart from that, they are also Dover paperbacks, which means they are quite affordable.

Willard is in the same class as Dugundji and Kelley: a textbook which is exhaustive enough to serve as a reference. Like Kelley, it has lots of problems, and many of them investigate auxiliary material. Oh, unlike the other two, Willard has a few pictures.

It is also fun to read. No, he’s not trying to be a stand-up comic, but every once in a while he phrases something nicely. “In the next (and obvious) step to normal spaces, we find ourselves confronted with the real bad boy among the separation axioms.”

Steen and Seebach is a compact presentation of topology (40 pages), beautifully organized counterexamples (120 pages), a summary of metrization theory (24 pages), and a collection of charts and tables for finding a desired example (20 pages). I would think, speaking as an onlooker, that this is an indispensable reference if you do much topology.

Need a reference text? Unless you need something specific from Dugundji or Kelley, I suggest you get Willard.

Doing topology beyond your first course? Get Steen & Seebach on general principles.

Books Added

Steen, Lynn Arthur and Seebach, J. Arthur Jr., Counterexamples in Topology, Dover, 1995 (orig. 1978),
ISBN 0 486 68735 X
[general topology; 17 Nov 2008]
Reference. Very well organized, with many charts of relationships.

Willard, Stephen. General Topology, Dover, 2004 (orig. 1970).
ISBN 0 486 43479 6.
[general topology; 17 Nov 2008]
Textbook and reference. Well-written. Copious historical references and notes.

books added: general topology (point set topology)

Introduction

Let me discuss my favorite general topology, i.e. “point set topology”, books. I have already discussed “algebraic topology” here.

Like so much other pure mathematics that I do not use professionally (for modeling power plants), topology is not on the tip of my tongue. But it’s fun, so I do it once in a while. And it’s fundamental, so I often have to go back to it when I’m playing with other mathematics.

This is a discouraging review in one respect: 4 of these 10 books are out of print: Kasriel, Dugundji, Sieradski, and Seifert & Threlfall. Heck, if you want Seifert & Threlfall, you should buy it in German! And for two of the books (Naber, Chinn & Steenrod) that Amazon claims to have in stock, there are multiple listings, many of which say the books are not available.

But I’m not going to go buy more books just because the ones I have are out of print. This is what I like, of what I have.

Buying Used Books

Let me point out that I buy used books online via abebooks.com. I have no connection with them except as a mostly satisfied customer. I think I have bought 10-15 books from them, and only one was not as advertised: it did not have “clean unmarked pages”. OTOH, it was so rare and hard to come by that I chose to keep it anyway: the student who marked it up stopped after page 51 and the rest of the book, the good stuff, was clear. (I could have returned it at the bookstore’s expense with a full refund, but I chose not to.)

I also buy used books online from powells.com; it’s a huge physical complex of bookstores (it really is huge!) in Portland Oregon, and wandering through them is almost a religious experience for bibliophiles. I have no connection with them except as a completely satisfied customer. Abebooks includes books available from Powell’s, but I buy such books directly from Powell’s because I’ve been there and I like them.

Two First Courses

First, there are two fine textbooks, intended and quite suitable for, a first course. Munkres came highly recommended, and from my own reading of it, I would imagine that it is the standard text. Well-written, with lots of examples. Good enough that I bought everything else he wrote; fortunately for my wallet, that’s only 3 other books, one of which is already in the bibliography.

I have no idea how I got the other book, by Sieradski. It is used, worn but unmarked. I suspect I bought it in person, but I have no recollection of where. It’s a little denser than Munkres, by which I mean only that it seems to have fewer words between definitions and theorems, but it still has plenty of descriptive text, and it has lots of examples and drawings.

If I feel like reading topology, I grab both of these and see what catches my fancy.

Sieradski is out of print, but if you’re interested in topology and you ever see a copy, you should investigate it.

Three References

Then there are three classic texts, all of which I consider to be references, although two of them do have exercises, and hence they are in principle, textbooks. (In fact, both English-language books declare themselves to be both text and reference.)

The first is Kelley. Famed for its exercises (is that a good thing?), and like so many math books of its era, it has no drawings whatsoever. He said he thought of it as “What Every Young Analyst Should Know.” originally published in 1955, it was republished by Springer in 1975 and is still available.

The second is Dugundji. It was allegedly the textbook for my first-year graduate course, but the instructor didn’t use the book at all. To be honest, I don’t know that there even was a “textbook” for general topology back then, other than Kelley – which I bought at the end of the year from another grad student who was bailing on math. When I looked through Dugundji years later, I liked its organization; good enough to make up for its lack of handholding. It’s probably the first place I look to see whether something is true or false. It includes chapters on set theory, ordinals and cardinals. Oh, it has no drawings either.

If you know enough to be looking for reference books in topology, you probably know better than I whether you want to find a copy of Dugundji.

(I am not sure where or when I learned what I know of topology. it might very well have been the combination of Dugundji & Kelley. both are reasonable places to look if you’re tripping over topology in, say, functional analysis. That is, they don’t have a lot of motivation: just the math, ma’am. But when I brought my own motivation, they were fine.)

The third is Seifert & Threlfall, but I list it not because I know it, but for its language. No, not for its exposition, literally for its language. It is available only used, and last time I looked, the cheapest English translation was $300. The book is frequently referenced for its counterexamples, and it is highly praised in books that do more than just list references.

I found a German reprint by Chelsea, in excellent condition, for $35. Having recently read some Harry Potter in German, I think my high school German is good enough for a math book. Since it’s an order of magnitude cheaper, I encourage you to buy the German rather than the English, if you think you can handle the language.

Three Proto-topology

The next group is three books which spend a lot of time on proto-topology, as it were. The first is Kasriel. I love this book because it has, in one volume, and in order, the topology of the real line, of R^n, of metric spaces, and then it does general topology. No running for an advanced calculus book to compare and contrast a metric space theorem with the corresponding R^n theorem. It is typical that I love seeing topology and metric spaces in the context that justifies the generalizations. In contrast to the next two books, you could learn topology out of this book.

Kasriel essentially gives us the historical path to point-set topology.

Unfortunately, Kasriel is out of print. If it sounds interesting, either look for a used copy, or look for another book that has the same organization: lots of R^n, lots of metric spaces, and topology, in one binding.

The second is Naber. This book has a lot of applications of topology, all in R^n, without ever even defining “topological space”! As I like seeing the context of topology in Kasriel, I enjoy the Naber book as a chance to see topology applied. I would not have wanted this book before I knew general topology per se, which defeats its purpose, but I look forward to working thru it someday. It covers simplicial complexes, homology, homotopy, and vector fields, all within R^n.

Be careful when you look for it: the 1980 paperback is out of print, but the book was reprinted in 2000.

The third is an ambitious little book by Chinn & Steenrod. (If you know enough to wonder, yes, that Steenrod.) It’s a lot like the Naber book, trying to do topology without defining it per se. More to the point, this book has two parts: the first wishes to motivate and prove that a continuous real-valued function defined on a closed bounded interval has and attains its extrema, and that y = f(x) can be solved for each y in between those extrema; the second part moves to 2D, where just stating the theorem is an adventure. It has a much simpler goal than Naber, but it’s still a fairly sophisticated goal.

There are things in the second part that I’ve seen but cannot say I’ve understood: specifically, winding number of a curve and index of a vector field. (Did I miss that in Apostol II? Yes! But not by much: it’s only a page or two. I suppose I should be embarrassed but I’ve gotten used to not knowing everything.) Still, I need to work thru this book Real Soon Now. The truly terrifying thing about this book is that its intended audience was “high school students and laymen”. (Was it accessible to high students before the “new math”?)

Be even more careful: although Amazon says it has it in stock, as I write this, it has so many conflicting additional entries saying it’s not available, that I’m a little uncertain.

Two Others

Then, there is a little workbook by Adamson. Too sparse to be a text, but lordy, what a wonderful way to review! 70 pages of definitions and exercises, followed by 78 pages of proofs. You don’t have to hold a piece of paper over the proof to try working it out for yourself. (While checking to see if it’s in stock, I learn that he has also put out a workbook in set theory. hmm….)

Having said that, I must admit that the author intended this as first course, for “independent study based on material carefully prepared by a teacher who otherwise gives minimal assistance.” (If you’ve ever heard of R. L. Moore and his method, you understand.)

Finally, there is a Dover paperback by McCarty. I probably picked it up because it was a Dover, but I might have seen it in a list of references somewhere. It presents just the topology required to get to topological groups, but along the way it has a very nice introduction to commutative diagrams, and the entire book is extremely readable. I sat down to see what was in it, and didn’t put it down until I’d read the whole thing!

The Books Added

Adamson, Iain. A General Topology Workbook. Birkhäuser, 1996.
ISBN 0 8176 3844 X.
[general topology; 10 Nov 2008]
Upper division. From the introduction: “This book has grown from my attempts to provide a self-learning introduction to general topology for several generations of students….” Answers. Guide to further reading.

Chinn, W.G. and Steenrod, First Concepts of Topology. L.W. Singer & Random House1966.
[general topology, algebraic topology; 10 Nov 2008]
From the introduction: “… to show how topology arose, develop a few of its elements, and present some of its simpler applications…. Our presentation … will be centered around two existence theorems… {in one and two dimensions respectively}.” Answers. Guide to further reading (3 books). Epilog.

Dugundji, James. Topology. Allyn & Bacon, 1966.
[general topology; 10 Nov 2008]
Out of print. Reference and text. If they reprint it, i’ll call it a classic. Some homotopy theory.

Kasriel, Robert H. Undergraduate Topology. Krieger, 1977.
ISBN 0 88275 444 0.
[general topology, metric spaces, euclidean spaces; 10 Nov 2008]
Out of print. From R and R^n to metric spaces and then to topology. From the preface: “… it is to {the graduate student’s} advantage to have taken a course in general topology before beginning his graduate program…. essentially self-contained except for elementary calculus.”

Kelley, J. L. General Topology. Springer, 1975 (was van Nostrand, 1955).
[general topology; 10 Nov 2008]
Graduate Text in Mathematics. Classic reference and text. From the preface: “… a systematic exposition of the part of general topology which has proven useful in several branches of mathematics.” Epilog and guide to further reading – in some of the exercises!

McCarty, George. Topology: An Introduction with Application to Topological Groups. Dover, 1988 (orig 1967).
ISBN 0 486 65633 0.
[general topology, topological groups; 10 Nov 2008]
Upper division. How can I not like a book which says, in an exercise, “… make the assumption that every subgroup in sight is closed.” A very readable book. Guide to further reading. Epilog. (Both at end of chapters.)

Munkres, James R. Topology. Prentice Hall, 2000 (2nd ed).
ISBN 0 13 181629 2.
[general topology, algebraic topology; 10 Nov 2008]
A very well written senior and first-year graduate textbook. Lots of words between the theorems provide lots of motivation. If this isn’t the premier textbook, I’d really like to know what is.

Naber, Gregory L. Topological Methods in Euclidean Spaces. Dover, 2000.
ISBN 0 486 41452 3.
[algebraic topology, differential topology, euclidean spaces; 10 Nov 2008]
From the preface: “… to persuade students… that the evolution of topology from analysis and geometry was natural and, indeed, inevitable; that the most fruitful concepts and most interesting problems in the subject are still drawn from independent branches of mathematics…. an ambituous agenda of topics has been included….” Guide to further reading. Answers.

Seifert, H. and Threlfall, W. Lehrbuch der Topologie. Chelsea, 1945.
[general topolgy; 10 Nov 2008]
Out of print. Classic reference. I bought it for the counterexamples; I bought it in German for the price.

Sieradski, Allan J. An Introduction to Topology & Homotopy. PWS-Kent, 1992.
ISBN 0 534 92960 5.
[general topology, algebraic topology; 10 Nov 2008]
Out of print. A very well written senior and first-year graduare textbook. From the preface: “Most topics are developed slowly in their historic manner, in order that a newcomer not be overwhelmed by the ultimate achievements of several generations of mathematicians.”

Books added: no, but additional information added

I have made a major edit to the bibliography. I have tried to mark 3 pieces of information for each book. To be specific, I have tried to mark affirmative answers for 3 questions. (A search should find only affirmative answers.)

Answers. Does it have answers or hints for “many” problems? If so, that is one more reason why the book may be suitable for self-study without a teacher.

Guide to further reading. Not just a bibliography; not just detailed references; but “if you want to know more about this, consider these books, and for more about that, look at those books….” This was prompted by the magnificent guide in Bloch. I wish it were always so clear-cut. O’Neill’s “Elementary Differential Geometry” (2nd ed.), for example, has a short bibliography (8 books), which leads me to say, “I could afford to own all these.” (I own all but one; maybe I should rectify that just for completeness. I like short bibliographies.) Furthermore, he has a sentence about 2 of them. Technically, he has a guide to further reading, and I said so; but I qualified it as only two books. OTOH, a few books break things down into subcategories, but unless they said something about individual books, I didn’t view that as a guide. (The same effect is achieved by end-of-chapter bibliographies, and it’s not what I’m trying to flag.)

Epilog. What might we find in the next book? Not appendices, but for example, a final chapter entitled “advanced topics” – specifically because he said he couldn’t cover them but he wanted to mention them. This was prompted by a marvelous epilog in Massey’s “Algebraic Topology: an Introduction”.

I have tried to set it so that searching the bibliographies page for “guide”, “answers”, or “epilog” will find no extraneous occurrences of these words, except for titles which contain those words.

Finally, the edit date for every book affected by any of these changes is 5 Nov 2008, so you can search the bibliography for 5 Nov to find these changes. That is, until I have some reason to make another edit to any one of the books affected today.

No new books were added today.

I tried to be careful, both looking thru books and editing the bibliography, but don’t bet the farm on my tags; and remember that some of my editions are not current.

books added 9 Aug 2008: Algebraic Topology

introduction

Let me begin by citing a site: here you will find, among other things, a free downloadable version of an algebraic topology book, offered by its author. It looks pretty good.

http://www.math.cornell.edu/~hatcher/

Someday I’d like to write an introduction to topology (a post! not a book!), but trying to do it now is taking me too far out of my comfort zone: I am reasonably familiar with general (also called point-set) topology, but I am rather ignorant of algebraic topology; I am reasonably familiar with differential geometry, but differential topology is a different and unknown beast. I have opinions about how topology hangs together, but when I try to be precise, I find that I’m not sure I can justify my opinions. I’d rather get it more right later than get it wrong now.

Remember that the path from ignorance to knowledge in any subject is not straight and true, but is almost always rather zigzagged. One seems to learn things by a method of successive approximations to the truth.

William S. Massey, Algebraic Topology: An Introduction. p xiii.

I cannot resist saying that all topology is divided into three parts. It may not be true, but it was too good a line to pass up. (If you’ve never had Latin, forget it.) Within topology, we find general, algebraic, and differential. Within algebraic topology, we find homotopy and homology; within homology, we find simplicial, singular, cubical, and possibly more, and there’s cohomology, too, mixed in. (Oh, I’m not counting topological groups and lie groups among “topology”. Too specialized. Frankly, I wonder if algebraic topology and general topology shouldn’t be considered completely separate; but most of my general topology books do include an introduction to algebraic topology.)

Anyway, trying to get more comfortable with simplicial complexes, I wandered thru my algebraic topology books, and found some fun stuff. Then I went looking thru just about every book (oops, it occurs to me that I have some physics books that I didn’t check) with topology (or homology) in the title – about 30 all told – and ordered a few more, of course. If all I wanted to do was add a pile of books to the bibliography, I could do that; but I’d like to talk about the subjects, too, and I’m foundering for lack of precise knowledge of their areas and boundaries.

I’d like to get to the point where I can write that overview / introduction, but I’m going to call it a longer-term project, instead of trying to get it done now: there’s more specific math I feel a need to get to.

Nevertheless, what started out as a search for more about simplicial complexes got all wound up in counting vertices, edges, and faces. It was fun, and I have one post ready to go, and a draft that may turn into more than one post.

I used a couple of books already in the bibliography. The Firby & Gardiner “Surface Topology” was invaluable; the Biggs et al. “Graph Theory 1736-1936″ provided a lot of auxiliary material. (That, in fact, was why the graph theory books were added a while ago.)

general

The following four books, by three authors, have been added to the bibliography. They are all “algebraic topology”. They are being added primarily because they are cited in the next post to come. As it happens, they are also high on my list of what to do next in topology after Bloch, but there are a couple of other books high on that list, too.

I know that both Fulton and Massey are reputed to be good expositors, and I can confirm that; I don’t know Rotman’s reputation, but I myself find him to be good at talking about what we’re doing and why. Reference books are invaluable, but textbooks should not be reference books. These four books are textbooks to learn from.

Massey

Note that both Massey books are still in print: the older was republished by Springer, which is also the publisher of the newer book.

For the older of the Massey books, let me quote his preface: “undoubtedly some experts will be shocked that a textbook purporting to be an introduction to algebraic topology does not even mention homology theory.”

His first chapter is two-dimensional manifolds; his second is the fundamental group (homotopy; e.g. that there are two distinct kinds of loops we can draw on a torus, those collapsible to a point and those which are not). His third and fourth chapters are focused on free groups and the Seifert-van Kampen theorem, respectively, but he uses these two chapters to show us “universal mapping problems” and, especially, to discuss why we use them. (A simple example, not one of his, is: “the function f is 1-1 if and only if there exists g (a left inverse) such that the following diagram commutes”).

(Before I got to Carnegie-Mellon, I had never even heard of a commutative diagram; my very first class meeting was general topology, and as I recall, I’ll bet the very first theorem was that diagram.)

His fifth chapter is covering spaces. And these five chapters are the beginning of the newer Massey book. The newer Massey also has the same appendix B as the older (but not the same appendix A!).

The older book then does the fundamental group and covering spaces of a graph (but you can find that elsewhere); the fundamental group in higher dimensions; and an appendix on the quotient space topology (but that’s general topology); and an appendix on transformation groups – which is also in the newer Massey.

The newer Massey picks up homology theory – the stuff he left out of the older book. Actually, it’s pretty much all of a 3rd book by him (“Singular Homology Theory”). About all I’m going to say is that the initial homology chapter (ch. 6) is 10 pages of background and motivation in homology theory. Worth reading even if I don’t continue in Massey.

Let me clarify that. There are actually three books involved here, two of which I own. Most of Massey’s “Algebraic Topology: An Introduction” and his “Singular Homology Theory” were combined into “A Basic Course in Algebraic Topology”.

If you are going to buy a new copy of Massey, you might as well buy the newer book, “A Basic Course….”but if, instead, you simply stumble across a cheap copy of either older book, hey, buy it, understanding that it’s about half of the combined book.

Fulton

Fulton is divided into parts, each of a few chapters, and some of the parts are divided. We find:

• I calculus in the plane
• II winding numbers
• III cohomology & homology (then again parts V and VIII)
• IV vector fields
• VI covering spaces and fundamental groups (then again part VII)
• IX topology of surfaces
• X Riemann surfaces
• XI higher dimensions

He says (p. viii), “To achieve this variety at an elementary level, we have looked at the first nontrivial instances of most of these notions: the first homology group, the first De Rham group, the first Cech group, etc…. We have tried to do this without assuming a graduate-level knowledge or sophistication.”

Finally, I should point out that the book encourages one to work along with it. For example, the section from pages 43 to 47 defines winding number, degree, and local degree; it has two lemmas and one proposition, 3 exercises, and 9 problems (requiring more ingenuity than exercises). All interspersed, suggesting things for the reader to play with.

Rotman

From the table of contents of Rotman, one might think that it was very similar to Massey or Fulton; not so. Yes, we see the usual suspects: simplexes, fundamental group, singular homology, covering spaces, etc; but we also notice chapters on “long exact sequences” and “natural transformations”. In the preface, he says, “I am an algebraist with an interest in topology.” indeed: the word “functor” first appears on page 3.

I like that. I don’t want to do it first, but I do want to do it this way eventually. Rotman’s book calls to me.

the books added

Fulton, William. Algebraic Topology: A First Course. Springer, 1995.
ISBN 0 387 94327 7.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. “To achieve this variety at an elementary level, we have looked at the first nontrivial instances of most of these notions: the first homology group, the first De Rham group, the first Cech group, etc…. We have tried to do this without assuming a graduate-level knowledge or sophistication.”

Massey, William S. A Basic Course in Algebraic Topology. Springer, 1991.
ISBN 0 387 97430 X.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. The first 5 chapters of this appear to coincide with the first 5 chapters of “Algebraic Topology: An Introduction.” In particular, it includes the excellent introduction to universal mappings. OTOH, it has a substantial introduction to the “why?” of homolgy.

Massey, William S. Algebraic Topology: An Introduction. Harcourt, Brace & World, 1967 (Springer, 1977).
[algebraic topology, 8 Aug 2008]
He is a wonderful author, and I love his style. Both versions of this text have the most readable introduction to universal mappings which I’ve seen. This book covers Surfaces and homotopy; it does not cover homology.

Rotman, Joseph J. An Introduction to Algebraic Topology. Springer, 1988.
ISBN 0 387 96678 1.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. In the preface, he says, “I am an algebraist with an interest in topology. The basic outline of this book corresponds to the syllabus of a first-year’s course in algebraic topology….”

Books Added 1 Aug 2008: Graph Theory

I have had a long-standing question: why is the usual triangulation of a torus so large? It is a square with 18 triangles. Why so many?

While looking through algebraic topology books for more information about simplicial complexes – which is supplementary material for Bloch (“A First Course in Geometric Topology….” – I found the answer. Well, I found a clue. I found an inequality, which implied that the minimum number of triangles was 14 in any possible triangulation of the torus.

What I didn’t find was a proof of that inequality, and I couldn’t work it out for myself.

Eventually, in another algebraic topology book, I found another clue, and it was enough to let me work out the first inequality.

Along the way I found some pretty interesting things, and I want to chatter about them. The result will not really be math, but more in the nature of a travel guide.

Well, no, not even that detailed. More in the nature of some really cool pictures from a foreign country.

The search took me into both algebraic topology and graph theory. Before I put out the posts (there will be at least two), I need to put out some bibliography.

The graph theory is easy. I only own 4 books on graph theory, and I acquired one of them just this week.

So, the following books on graph theory have been added to the bibliography.

There are two old classics. Biggs et al. is a collection of 37 original papers that helped found graph theory. Where necessary, they have been translated into English, and the authors have written whatever introductory material was necessary before each paper.

Here you will find Euler’s foundational paper on the Konigsberg bridges, of course; but also Kirchhoff’s paper using graph theory for electric circuits. (Yes, the use of graphs for simplifying the application of Kirchhoff’s laws goes back to Kirchhoff himself!)

Among other things you will find Euler’s polyhedral formula (V – E + F = 2), chemical graphs, the four-color problem, and planar graphs.

The other old classic is Bondy & Murty “Graph Theory with Applications”. Every chapter ends with “Applications”. I like that. In fact I would be unhappy without them.

I love the terminology of graph theory: an acyclic graph is also called a forest; each component is called a tree (so a tree is a connected acyclic graph). But after a while I begin to feel like I’m just proving random facts about random things. I’d like to do better.

The other two books? One is a brand new book by Bondy & Murty. They say in the introduction that it began as a second edition. It grew too much, and the different title reflects that growth. (It wouldn’t have been the first “second edition” to be a different book.)

Realistically speaking, I could have mentioned the book, said I didn’t own it, and left it up to you to decide whether to buy it. OTOH, if I try to order it myself and find that you all have exhausted Amazon’s supply… well, that would be terrible.

I just had to buy the book. (That happens a lot, for one reason or another.)

It is a much larger book than it’s predecessor. (Have I ever seen a second edition smaller than the first? Only when the book split into multiple volumes.) It still contains applications, but you’ll find them listed under “problems” in the index. OTOH, you’ll also find a detailed sublist of “proof techniques” in the index. I think that’s way cool. (You will find elementary techniques such as induction and contradiction, intermediate techniques such as the pigeonhole principle, as well as techniques specific to graph theory.)

The book can be used as an undergraduate text: just read the first few sections of most of the chapters; or as a graduate text: read it all!

One major caveat: there is a second printing under way. According to the web site (here) it will, of course, correct errata, but it will also be “slightly reorganized”. I’m not sure what that bodes, but you might want to wait for the second printing.

Finally, I have a book by Bollobas. I can tell from the price sticker that I bought it during a Springer “yellow sale” in 2003, for about 1/3 off. I am sure I bought it just because I wanted a more modern book than the old classics.

I have no regrets. Quite the contrary. The first chapter, fundamentals, is a rather compact presentation: graphs, trees, Euler & Hamiltonian cycles, and planar graphs. Whew! But it’s well written, and it’s nice to have what amounts to: if I only get to show you one chapter of graph theory, well here it is.

The second chapter is electrical networks. Good, we have some applications.

But then section 2 of the second chapter uses electrical networks to search for “squared squares” and “squared rectangles”. Hey what? An example of a squared rectangle is the following (where each integer is the length of a side of the square in which it sits, and the whole thing is a 33×32 rectangle):

I know some of those! I played with them several years ago.

I am looking forward to Bollobas.

I’m getting excited about doing graph theory. I’m pretty sure I’ll go through the old Bondy & Murty first, but with the history book handy. (If I didn’t have the old Bondy & Murty, I would work the first few sections of each chapter of the new one. Having the old one, I view the new one as a reference book. Perhaps I’m wrong to do that.)

After I had worked both of those, I would hit Bollobas.

Or maybe I’ll do the first chapter of Bollobas, and then the other two….

I’ll let you know.

Bollobas, Bela. Modern Graph Theory. Springer, corrected printing 2002.
ISBN 0 387 98488 7.
[graph theory;1 Aug 2008]
“Graduate Texts in Mathematics”. I haven’t worked thru it yet, but its first two chapters look spectacular, and it has lots of exercises. The first chapter is an introductory overview, possible because we can get powerful theorems early on. The second chapter is a delightful looking digression on electrical networks and subdivisions of rectangles.

Bondy, J.A. and Murty, U.S.R. Graph Theory with Applications. North Holland, 1976.
ISBN 0 444 19451 7.
[graph theory;1 Aug 2008]
This is the old classic. Every chapter ends with applications. Just looking thu it makes me want to curl up and work thru it.

Bondy, J.A. and Murty, U.S.R. Graph Theory. Springer, 2008.
ISBN 978 1 84628 969 9.
[graph theory;1 Aug 2008]
“Graduate Texts in Mathematics”. Looks as though it could be used as an undergraduate text, or as a second-year graduate text getting one ready for research; just decide how far into each chapter to read.

Biggs, Norman L, Lloyd; E. Keith; Wilson, Robin J. Graph Theory 1736–1936.
ISBN 0 19 853901 0.
[graph theory;1 Aug 2008]
This seems a very gentle introduction, although it does have proofs. It’s a collection of the original papers about graph theory – translated into english, with substantial commentary before each.

books added 21 June

The following books have been added to the bibliography.

The Ashley book is a welcome addition to control of flight vehicles (Bryson; Blakelock): it’s got a lot more detail about the underlying dynamics. I have no idea when I bought it, but I eventually remembered that it was somewhere in my library, and was delighted to find its more detailed explanation – and excellent drawing – of the various coordinate systems in use for aircraft and missles. This is material which the control theory books assume you’ve seen in more detail.

The Ideals & Varieties book is an introductory text which I am working thru with a friend. The third author, O’Shea, is the author of a recent book on the Poincare conjecture which is what got me started on the geometry of surfaces.

The 3 mechanics books (Marion, Symon, and the Berkeley) were additional references (cf. Goldstein) for acceleration in rotating coordinate systems. I have listed the Berkeley text twice, for the same reason I list Schaum’s Outlines twice. I’ve always heard it called “the Berkeley mechanics book”, and that’s how I searched to see if it – and the rest of the series – were in print (no) and available used (yes).

I bought the Basilevsky Factor Analysis book because I wanted something more about noise in factor analysis methods (cf. Malinowski). It looks like a good and interesting book (I wasn’t expecting to find the Kalman filter in it), although it is the specific text in which I found the mistaken assertion that we could always choose the eigenvector matrix orthogonal. As I said when I corrected that very same careless error on one of my own SVD pages, I am inclined to be tolerant of other people’s mistakes: I make mistakes, too.

Ashley, Holt. Engineering Analysis of Flight Vehicles Dover 1992 (1974).
ISBN 0 486 67213 1
[aeronautical engineering; 21 June 2008]
This has more detail about the coordinate systems. It includes rocket launches and re-entry into the atmosphere.

Basilevsky, Alexander; Statistical Factor Analysis and Related Methods. Wiley-Interscience, 1994.
ISBN 0 471 57082 6
[multivariate statistics, factor analysis, principal components, time series; 21 June 2008]
I’ve only skimmed this book. It looks like an attempt to apply modern statistical analysis to factor analysis and principal components. I was hoping it would provide some background for Makinowski’s treatment of noise in factor analysis, and I think it will.

Berkeley Physics Course – Volume 1: Mechanics Kittel, Charles; Knight, Walter; Ruderman, Malvin A. McGraw Hill, 1965.
[physics, mechanics; 21 June 2008]
This freshman text has a fair bit of math in it. In particular, it has the vector equation for acceleration in fixed and rotating frames, for which all of my other references are upper-division books. In general, I look in here for good examples.

Cox, David; Little, John; O’Shea, Donal. Ideals, Varieties, and Algorithms Springer, 1992.
ISBN 0 387 97847 X
[algebraic geometry; 21 June 2008]
An upper division introduction. I bought it because I didn’t know what a Groebner basis was. There is a 3rd edition, but I have heard that there are many typos, although errata are available.

Kittel, Charles; Knight, Walter; Ruderman, Malvin A. Mechanics (Berkeley Physics Course – Volume 1) McGraw Hill, 1965.
[physics, mechanics; 21 June 2008]
This freshman text has a fair bit of math in it. In particular, it has the vector equation for acceleration in fixed and rotating frames, for which all of my other references are upper-division books. In general, I look in here for good examples.

Marion, Jerry B. Classical Dynamics of Particles and Systems. Academic Press, 1965.
[physics, classical mechanics; 21 June 2008]
There is a 5th edition; it’s not cheap. I used this as my reference for vectors when I was an undergraduate.

Symon, Keith R. Mechanics Addison Wesley, 1960.
[physics, classical mechanics; 21 June 2008]
There is a 3rd edition; it’s not cheap. I must have acquired this from a friend: I had forgotten about it. Judging from his treatment of acceleration in fixed and rotating frames, he’s a good alternative to Goldstein.

books added – 25 May

References for generalized eigenvectors, the matrix exponential and for the SN decomposition.

one correction 30 Aug 2008, per Brian Hall’s comment below. see “edit” in this post.

First, let’s talk about books which are already in the bibliography. It’s pretty easy to find statements that for matrix exponentials,

exp(A+B) = exp(A) exp(B)

if A and B commute; edit: it’s a little rarer to find the full (and true) “if and only if” A and B commute.

Correct is:

(Golub & van Loan, “Matrix Computations”, 2nd ed. p. 559)

end edit

What’s very rare is the SN decomposition,

A = N + S,

where N is nilpotent, S is diagonable, and N and S commute.

Halmos’ “Finite Dimensional Vector Spaces” has very little about the matrix exponential, but it includes existence and uniqueness of the SN decomposition as an exercise! More generally, Halmos has 3 consecutive sections on triangular form, nilpotence, and jordan form, i.e. on the underlying nature of eigenspaces.

Strang’s “Linear Algebra and Its Applications” has an appendix on jordan form, but I don’t believe he mentions generalized eigenvectors. More generally, Strang illustrates the matrix exponential for solving matrix differential equations.

Stewart’s “Introduction to Matrix Computations” has a different definition of generalized eigenvectors.

I’ll be looking in all three of those to finally make sense of generalized eigenvectors. (It would be so easy if repeated eigenvalues always led to generalized eigenvectors, but they don’t; and I will plead that it all looks more reasonable after I know that generalized eigenvectors exist: I was long out of graduate school before I heard of them.)

In addition to linear algebra texts, we may find the matrix exponential in dynamical systems texts and in lie algebra / lie groups texts.

The following books have been added to the bibliography.

The classic text by Hirsch & Smale, “Differential Equations, Dynamical Systems, and Linear Algebra”, has the NS decomposition. This is one of “the” books. I’ll confess that their examples were a lot easier to follow than their theorems. Unfortunately, it’s out of print, so you need to seek a used copy; or go with Perko, below.

Be careful. There is a 2nd edition, with an additional author (i.e. Hirsch, Smale, Devaney) and a changed title: “Differential Equations, Dynamical Systems, & An Introduction to Chaos, Second Edition.” The back cover advertises “Simplified treatment of linear algebra”. It does not contain the SN decomposition.

Fortunately, there is another book on dynamical systems which has the SN decomposition, and it’s a book I like a lot: Perko’s “Differential Equations and Dynamical Systems”. In addition to having the NS decomposition, this has plenty of problems involving generalized eigenvectors, and I recommend it specifically for that, too.

Finally, one of my introductory texts on Lie stuff, Hall’s “Lie Groups, Lie Algebras, and Representations”, also has the NS decomposition. Any book on matrix Lie groups has to have the matrix exponential, but this is the only one in which I have found the SN decomposition.

Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Springer, 2003.
ISBN 0 387 40122 9.
[lie groups, lie algebras, representations; 25 May 2008]
This is a math book (for this subject, there are a lot of physics books). It is readable: it discusses its theorems and definiitons. It is primarily about matrix groups rather than lie groups, but it has lie groups in the appendices. As its title promises, it treats lie algebras, and representations. This is one of my favorite introductions to the subject.

Hirsch, Morris W., Smale, Stephen, and Devaney, Robert L. Differential Equations, Dynamical Systems & An Introduction to Chaos. Elsevier Acadameic Press, 2004 (2nd ed).
ISBN 0 12 349703 5.
[applied linear algebra, differential equations, dynamical systems, discrete systems, chaos; 25 May 2008]
This is a far different book from the 1st edition. It looks like a rather fine introduction to the first edition, and a good supply of examples.

Hirsch, Morris W., Smale, Stephen. Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.
ISBN 0 12 349550 4.
[applied linear algebra, differential equations, dynamical systems; 25 May 2008]
This is one of “the” books, even if it is out of print! (And Smale is a Fields medallist.) And yet, I find their examples a lot more informative than their theorems.

Perko, Lawrence. Differential Equations and Dynamical Systems. Springer, 2001 (3rd ed).
ISBN 0 387 95116 4
[applied linear algebra, differential equations, dynamical systems; 25 May 2008]
I really like this book: it’s readable, it’s got lots of examples and pictures, and it covers a lot of ground. It is an upper division / beginning graduate math book.

books added

The following books have been added to the bibliography page. They both pertain to orbital mechanics. Prussing & Conway is my mainstay today, but I cut my teeth on the other, which has the advantage of being a Dover paperback. I still find both to be valuable.

Bate, Roger P., Mueller, Donald D., White, Jerry E.; Fundamentals of Astrodynamics. Dover, 1971.
ISBN 0 486 60061 0
[orbital mechanics,15 May 2008]
This is where I first played with orbital mechanics. It was good for mucking about with the parameters of an elliptical orbit. It has some fun examples.

Prussing, John E. and Conway, Bruce A. Orbital Mechanics Oxford University 1993.
ISBN 0 19 507834 9.
[orbital mechanics, 15 May 2008]
This is where i learned to send Mariner 4 to Mars, and the Voyagers to Jupiter. My simulations weren’t fancy, but they were good.

Books added – 4 May

The following books have been added to the bibliography.

Three are about regression, also called ordinary least squares. Johnston is my own reference for obscure special cases; I don’t know what the latest edition is like. Draper & Smith is “the” book for regression in general. I really like Ramanathan for applications to econometrics.

Two are about the control of aircraft. Bryson himself is a significant researcher in control theory, but the book is a bit terse, and at a graduate level. I think I own all his books; if not, I will. Blakelock is an upper-division / beginning graduate text. Where Bryson is predominantly state space, Blakelock is predominantly classical.

Kuipers’ book about quaternions includes “the aerospace (rotation) sequence”, which is the coordinate transformation from earth axes to aircraft axes. It’s also an excellent introduction to using quaternions for rotations.

While I’m at it, I added Goldstein. This is the 2nd edition of an ancient classic. I own the 1st edition, but I prefer the treatment of rotations in the 2nd.

There are two books about manifolds, Lee and Firby & Gardiner. I looked thru them after I finished chapter 2 of Bloch.

Make that three books about manifolds; if I add anything, I should add Thurston. He is a Fields medallist, and he was trying to write an accessible book about his own specialty. I think he succeeded, although it’s presumptuous of me to say so.

There are three references for quantum mechanics, Bohm, Messiah, and Schiff. I have some unfinished business with angular momentum, and they discuss it.

Blakelock, John H.;Automatic Control of Aircraft and Missiles. Wiley-Interscience, 1991 (2nd ed.).
ISBN 978 0 471 50651 5.
[controls, mostly classical but not all; 4 May 2008]
This looks like it’s right at my level: I know the tools, but I’d be happy to follow someone through the analysis of a comlex system.

Bohm, David. Quantum Theory. Dover, 1989.
ISBN 0 486 65969 0.
[quantum mechanics; 4 May 2008]
The story is that Bohm set out to write a text which would demolish “hidden variable” theories, and ended up their strongest supporter. I bought it because he was the author.

Bryson, Arthur E., Jr.; Contol of Spacecraft and Aircraft. Princeton University, 1994.
ISBN 0 691 08782 2.
[controls, mostly modern and continuous; 4 May 2008]
By a master of the field, but i’ll try after Blakelock, which uses mostly classical methods.

Draper, Norman R. and Smith, Harry. Applied Regression Aanalysis. Wiley-Interscience, 1998 (3rd ed).
[regression; 4 May 2008]
This is probably “the” book on general regression analysis; in contrast to “econometrics”, this considers experimental data, and there are some significant differences. In addition, he has a fuller treatment of diagnostics than, say, Ramanathan. There is a data disk.

Firby, Peter A. and Gardiner, Cyril F.;Surface Topology. Horwood Publishing, 2001 (3rd ed.)
ISBN 1 898563 77 2.
[surfaces, geometric topology; 4 May 2008]
In addition to introducing surfaces, it includes maps, graphs, tesselations, and surfaces with boundaries.

Goldstein, Herbert; Classical Mechanics. Addison-Wesley, 1980.
ISBN 0 201 02918 9.
[physics, classical mechanics; 4 May 2008]
There is a 3rd edition (2001). I own the 1st edition, but I am citing the 2nd because i like his expanded treatment of rotations. I have not read the 2nd ediiton anywhere near as throughly as the 1st; I have heard that the 2nd edition may be less accurate than the 3rd.

Johnston, J.; Econometric Methods. McGraw-Hill, 1972 (2nd ed).
[regression, econometrics; 4 May 2008]
There is a 4th edition. This is my reference book. For example, did you know that you could impose a linear constraint on the coefficients and still solve the estimation problem? Well, you can, and he has the solution, among many others. You might check to see if the 4th contains this specific problem.

Kuipers, Jack B. Quaternions and Rotation Sequences. Princeton University, 1999.
ISBN 0 691 05872 5.
[linear algebra, quaternions; 4 May 2008]
The book suffers from a lack of exercises, but it is still a marvelous introducton to rotations, using both linear algebra and quaternions.

Lee, John M.; Introduction to Topological Manifilds. Springer, 2000.
ISBN 0 387 95026 5.
[topological manifolds, geometric topology; 4 May 2008]
“This book is an introduction to manifolds at the beginning graduate level.”

Messiah, Albert. Quantum Mechanics Dover, 1999.
ISBN 0 486 40924 4.
[quantum mechanics; 4 May 2008]
Two volumes bound as one. “Simple enough for students yet sufficiently comprehensive to serve as a reference for working physicists….” I bought it because it was a Dover edition of a well-known book.

Ramanathan, Ramu;Introductory Econometrics with Applications Dryden Press 1995 (3rd ed.)
ISBN 0 03 094922 X.
[regression, econometrics; 4 May 2008]
There is a 4th edition, and there may still be data available for it on the internet. I picked up a used copy because the price was right and discovered that I had a well-written book with a lot of examples, and I enjoy running regressions on real data. I recommend this book for econometrics: it’s both fun and informative.

Schiff, Leonard I.; “Quantum Mechanics”. McGraw-Hill, 1968 (3rd ed.).
[quantum mechanics; 4 May 2008]
a graduate text. i probably bought it as the “standard text” although we were using a different text in the class.

Thurston, William P. (and Levy, Silvio); Three-Dimensional Geometry and Topology. Princeton University 1997;
ISBN 0 691 08304 5.
[geometry, 3D manifolds; 4 May 2008]
not just an intro. classification thm. this is a readable book by a fields’ medallist, and this is his field. don’t be confused by the notes available on the internet: this starts gently.