Best Dover Math Books for Mathematician

In summary, the author prefers books that are modern, easy to read, and focuses on examples. He recommends the book Algebra by MacLane, which is expensive but has good exercises. He also recommends the book Ash's Algebra book, which is temporary replacement for Lang. Finally, he recommends the book Hungerford, which is bad when he has Jacobson.
  • #1
I am looking for a dover book on differential geometry,
but is hard to decide which ones to buy.
I am trying to get a (ONE) book NOT for PHYSISTS but for MATHEMATICIANS.
For now I am paying attention to:
- Tensor Analysis on Manifolds by Bishop
- Differential Geometry by Kreyszig
- Tensors, Differential Forms, and Variational Principles by Lovelock
- Differential Geometry by Guggenheimer
- Differential Geometry by Graustein

My book preference is:
- Book mainly for MATHEMATICIANS
- Has lots of contents, but not too hard for beginning advanced mathematics undergraduates
- Does not require too much point-set Topology (did not finish Williard Book Yet)
- Techniques used are generalized instead of being too specific
- Good notations (unlike Stuik's book)
- Has great learning value

If you have more suggestions (from dover, or anything below $25 US dollar),
can you help me? Thank you.
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  • #2
Does it have to be dover? Also in what flavor do you want? :)
There are books that do the general theory by using Banach spaces (harder ones, they include infinite dimensional) and those that take the casual approach by using R^n.

For R^n I think Lee's Introduction to Smooth Manifolds is great. For the infinite dimensional case they say Klingenberger, Riemannian manifolds is good. Didn't read it. Lang's Differential Geometry is also very technical but recommended.
  • #3
I found much better books that are at least 3 times more expensive (John Lee, Lang, Boothby, ..., etc),
so I am planning to buy those later.
But I would like to buy one cheap but good book.
Oh, I think Banach space is going way too far for me.

I guess I should tell you where my level is:
- Currently doing 2nd year analysis course (book for that is Spivak's Calc on Manifolds)
- Studying Topology on my own, but did not go that far (book is General Topology by Willard)
- Learning Abstract Algebra (book is Dummit & Foote, but I don't like it)
(I recently bought Jacobson for it)

So I guess I am kinda of a noob yet...
Hope this helps.
  • #4
I am learning differential geometry now and find Struik's book full of classical examples that are easy to understand and are rich and varied. I also use Singer and Thorpe's book which I love an which covers basic topology of differentiable manifolds, de Rham's theorem. and differential geometry of surfaces using connection 1 forms on the unit circle bundle.

If you like learning from examples then generalizing on your own and posing problems for yourself, these books will be great when read in combination. I have found differential geometry to be a tangle of formalisms and have tried desperately get past them and to get a real taste of it in my mouth. These books have worked form me when read in combination. The downside is that they only discuss surfaces but personally I like that.
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  • #5
I do not know any Dover books on diff geom that I recommend because Dover books tend to be old, and old diff geom books tend to be out of date in a somewhat harmful way. It will not hurt you to look briefly at them, but you do not want to be limited by the out of date notation I presume is found in most Dover books.

Ahem, some physicists tend to be especially handicapped in learning diff geom because they seem to revere Einstein so much that they try to learn the same outdated math that he was forced to use in 1910 or so, rather than the modern version that physicists use today.

Here is a free diff geom book that I recommend highly. It is modern and well written. The author was a student of the great geometers Chern and Griffiths, and is a spectacularly gifted teacher. It is at a beginning level.

If you can read books by people such as Bishop. you may be ok, but I recall his books tend to be advanced and abstract, although modern. [Broken]

and try michael artin for abstract algebra, or shifrin if you want an easier one. I also have several free ones on my web site.
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  • #6
Thank you for your responses but Algebra is fine.
I don't like Dummit & Foote simple because he does poor job at exposing
categorical algebra (compared to MacLane, Jacobson, Ash, etc), and his explanation on
uotient group sucks so much.
(I don't have problems understanding his book, but I prefer my other books)
My most preferred book is actually Algebra by MacLane, but it is expensive
so it just stays in ... u know.
I bought Jacobson and found out it has good exercises and has categorical algebra.
I might also buy Ash's Algebra book (temporary replacement of Lang for near future)
(Grillet, Knapp books are really good but are expensive...)
(Hungerford is bad when I already have Jacobson (better explanation for free groups))
Unfortunately, Artin doesn't have what I am looking for (I guess my taste is different?).

I guess I will keep my options open and look for non-Dover books (maybe for borrowing).
Fortunately, I know what I need to look for when I do not restrict to Dover
(Do Carmo (not enough motivation, but good), Spivak, Boothby, John Lee, etc)
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  • #7
do you mean maclane and birkhoff, algebra?

here's one for $15.

Algebra. Second Edition. (ISBN: 0023743107 / 0-02-374310-7)
MacLane, Saunders;Birkhoff, Garrett (Mac Lane, Saunders)
Bookseller: Science Book Service
(St. Paul, MN, U.S.A.)

Bookseller Rating:
Quantity Available: 1
Book Description: Macmillan Publishing Company, New York, New York, U.S.A., 1979. HARD COVER. Book Condition: VERY GOOD PLUS. NO JACKET. Second Edition. VERY GOOD PLUS/NO DUST JACKET. Sturdy blue cloth covers are clean and bright with only minor external wear; about 50 pages out of 586 total pages have some pretty neat pencil notes or underlining which is not very offensive--otherwise inside pages are bright and solidly bound throughout. Bookseller Inventory # 012299

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heres one that claims to be by just maclane, but it may be the same book:

Maclane, Saunders
Bookseller: InstaShip Books
(Charlotte, NC, U.S.A.)

Bookseller Rating:
Quantity Available: 1
Book Description: Macmillan, 1968. Hardcover. Book Condition: Good. Dust Jacket Condition: Unknown. Like New Fine. Some underlining. Corners slightly bumped. Very nice copy. Third printing. - --InstaShip Books Normally Ships within 24 Hours, often, Same Day. Bookseller Inventory # suz082

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  • #8
As for categorical algebra, I recall what Hermann Weyl said: "Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor."
  • #9
Thank you for your efforts!
You motivated me to search in abebooks and I found one
with $42, $11 for shipping.
(Unfortunately, his 3rd edition is masterpiece, which is more expensive)

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