What Are Your Thoughts on These Mathematics Books?

[Diffy]

Contemporary Abstract Algebra. by Joseph A. Gallian

I own this book, and personally love it. I just want to know if others share the same feelings.

Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics) by Walter Rudin

Read great reviews. I need a good refresher.

Algebra by Michael Artin

This one seems to be a classic. But is it any good?

Topology (2nd Edition) by James Munkres VERSUS Algebraic Topology by Allen Hatcher

I have them both, but I am not sure which one I should be reading first. I am not strong in Topology and after reading through the first few pages of each I think that munkres might be the right choice.

Complex Analysis (Graduate Texts in Mathematics) by Serge Lang

I know Lang has written a few other classics, but I fear this may not be great for beginners. Is there another option?

Thanks for your input.

 

[mathwonk]

i have a lot of recommended books on my thread who wants to be a mathematician, and we have a thread solely dedictaed to books.

it is ahrd to know what you mean by "good". i never heard of gallian, but michale artin is a world famous algebraicst and algebraic geometer, so to me his book is more likkely to be good. and i have read and taught from it aND CONSIDER IT THE BEST intro ALGEBRA
BOOK in existence. but best for what? for whom? i like it because the explanations are so clear and authoritative, and the material is chosen so intelligently, and he tested it in the classroom and rewrote it for several years.

but it is too hard for some people to begin with, so for them it is notso good, until they are ready for it.i like langs complex book a lot, it may be my favorite. i do not like rudin at all, as it does not explain anything to me, and has no geometry.

hatcher is a favorite with our expert topologists and i think it is ok, but a little tedious for my taste. everyone seems to believe munkres writes well.

 

[Diffy]

Sorry for posting in the wrong place, perhaps someone can move it to somewhere more fitting.

Thanks Mathwonk, I was hoping you would reply because I have read a few of your recommendations here, and in fact, reviews on amazon.com as well.

I am kind of taken aback that you have not heard of Gallian's book, but then again, perhaps it is not as well known as I thought. I like the hundreds of examples he gives, and the illustrations, and figures are very nice. At times he is witty, and to me always interesting. The biggest criticism I agree with, is that he seems to avoid matrices whenever possible. Some say he lacks rigor too, but I do not see it (or perhaps am not well versed enough to see it).

I have been eying that Artin book for some time now, I think I will go ahead and grab a copy of it for my growing collection.

 

[Cincinnatus]

Eh, I don't like Gallian's book. Sure quoting the Beatles a lot is kind of fun, but does it really make for a good abstract algebra book? There seems to be much more material in Artin, though I haven't read it myself...

 

[morphism]

I can't let myself not mention Herstein's Topics in Algebra.

 

[Dissonance in E]

While on the topic, what do you think of Robert Adam's Calculus: a complete course (6th e)? It's the only text I use for calculus self study.

 

[mathwonk]

you misunderstand me. i have heard of gallian's book, i.e. of gallian the textbook author, but not of gallian the mathematician. i am more impresed with authors who are famous researchers. i.e. textbook authors should have some credentials in research before they write a book. e.g. artin, vander waerden, jacobson, sah, lang,...

but i will check him out. maybe he is a top researcher and i just do not know him, since i am not an algebraist, or because i am so old i am out of the loop.

ok he is at the university of minnesota, duluth, and claims over 100 journal articles. his publications do include research papers in journals such as archiv, journal of algebra, proceedngs,... these are not top research journals, but they are good journals, and he is obviously an expert. he surely knows more algebra than i do, but i suggest he is nowhere near the level of an author like artin or van der waerden, or lang or jacobson or say milne.

he has spent much or most of his time interested in and active in education especially for undergraduates. while this polishes his undergraduate teaching and exposition sklills, it takes away time from his research in algebra itself.

take what you can from him i suggest, and try to graduate to van der waerden or artin or lang.

 

[mathwonk]

i do not know adams' book on calculus.

herstein is a famous early text in algebra which atempted to bring significant and deep topics down to undergraduate level, sophomore i believe originally, as waS THE CASE WITH ARTIN.

as a young grad student i read herstein trying to learn what i needed for grad school. i never learned anything from it, except by doing his many wonderfully fun and challen ging problems (and the construction of an extension fielkd where a polynomil has a root). his proofs are the kind that explain every logical step so anyone would grant the truth of the theorem, but do not give any insight at all into why the theorem is true, or what is really going on.

perhaps since he was writing for sophomores, he talks as "a missionary talking to cannibals", as g.h. hardy put it, i.e. he talks down to you.

hence as a person wanting to understand, herstein was almost worthless to me. far better was lang, algebra, and van der waerden. herstein is for the very young or beginning student who aprpeciates clear statements and patient proofs, but has not graduated to the level of really getting down inside the proofs and theorems to where he can do such things himself.

herstein is still on the "i show you, you listen to me" level. it is almost as if he does not expect anyone reading his book ever to be on the level of writing such a book. but the problems are useful.

one other fatal flaw is the notation. almost no one today multipies their matrices backwards as he does. one nice feature found only there is the decriptiion of all orthogonal matrices in terms of blocks of sins and cosines. that is a nice fact.

the three special topics at the end are also beautiful, but again they are written in an expository way, whereas one really wants to be shown how to do such things oneself, rather than how they were done.

artin is vastly superior from this point of view.

 

[mathwonk]

since math is so large a field, one wants to learn general methiods that can be applied again and again to many situations, and to elarn when to use them. i never got this from herstein. his book is what it says: topics in algebra, not methods or insights.

i guess it takes a while to realize this and as a young student it may seem appealing for its simple sounding explanations.

 

[mathwonk]

by the way there are three free algebra books (class notes for math 4000, 843-4-5, and 8000) on my website.

(ugly fonts but clear explanations and sparkling wit. worked examples but no answers to your exercises.)

http://www.math.uga.edu/~roy/