Reddit reviews An Introduction to Manifolds (Universitext)
We found 7 Reddit comments about An Introduction to Manifolds (Universitext). Here are the top ones, ranked by their Reddit score.
i have three categories of suggestions.
advanced calculus
these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.
out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
smooth manifolds
out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
conceptual books
these books should be helpful as side notes to this material.
i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
conclusion
if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.
An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.
A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.
Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.
Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.
You have a long way to go. Here's a path that will get you to (smooth) manifold theory, with book suggestions:
Sorry, Lorring Tu. The book is http://www.amazon.com/An-Introduction-Manifolds-Universitext-Loring/dp/1441973990
Both Lee's and Tu's books are on my reading list. They both seem excellent.
However, my vote is for Professor Tu's book, mainly because it manages to get to some of the big results more quickly, and he evidently does so without a loss of clarity. In the preface to the first edition, he writes "I discuss only the irreducible minimum of manifold theory that I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly." Consequently, his book is roughly half the length of Lee's.
I'd rather hit the most essential points first, and then if I want a more expansive view, I'd pick up Lee.
Disclaimer: I may not participate very frequently, as I have some other irons in the fire, so you might want to weigh my vote accordingly. If your sub sticks around for a while, I'd definitely like to join in when I can.
When I was in your position I learned some representation theory of finite groups, from this book. It was at the perfect level for somebody who only has one semester's background in group theory. It'll gently introduce you to some things that you'll constantly need when you get further into algebra, like tensor products. Also, it's a topic which doesn't get covered at all in most undergrad abstract algebra courses, so it's a good thing to learn by yourself.
On the other hand, if you liked topology more than you liked group theory, you'd probably like Tu's Introduction to Manifolds.
It is algebra. But Lie algebras and the jacobi identity are standard subjects in any book on differential geometry.
Try Tu.