Munkres' Analysis on Manifolds and Differential Geometry

Question

Will Munkres' Analysis on Manifolds prepare me for a text like John Lee's Introduction to Topological Manifolds and his Introduction to Smooth Manifolds text? Would one be able to successfully tackle Spivak's Differential Geometry series after Munkres'?

Answer 1

Smokeypeat - years ago I was able to slog through Spivak until I got to the chapter on integration on chains, where I got lost in the abstractions of multilinear algebra, alternating k-tensors, wedge products, differential forms and all that. This was unfortunate because it prevented climbing to the heights of the generalized Stokes Theorem, which is the climax of Spivak.

Would Munkres' Analysis on Manifolds help fix this? One might guess yes because of Munkres' great clarity, e.g. in his Topology text; and I seem to have gathered the rumor that Munkres wrote his book on manifolds partly to explicate Spivak. This seems vindicated by a glance at Munkres' treatment of differential forms (I have the book), which seems to be more user friendly than Spivak's.

You could also try books like Hubbard and Hubbard's "Vector Calculus Linear Algebra, and Differential Forms"; Harold Edwards'"Advanced Calculus: A Differential Forms Approach"; and David Bachman's short and sweet "A Geometric Approach to Differential Forms."

Spivak is very terse, Lynn Loomis' old Advanced Calculus book is very abstract, and I found R. C. Buck's use of forms for vector calculus too concise and unmotivated.

As far as Lee's book on Smooth Manifolds, I have it, and as the saying goes, it's not for the faint hearted. It is both more adult and much more encyclopedic than Munkres, and requires a pulling together of much more topology, linear algebra, and even some diff. eq's. But certainly reading Munkres could do no harm in preparing for it, for then at least the idea of a manifold, and concepts related to differential forms, would not be new.

Of Lee's other book, on Topological Manifolds, I know nothing.

I wonder how many would-be mathematicians have hit the wall, among other places, at differential forms. Comments, anyone? Perhaps, as with tensors, I should just try to take the rules of manipulation on faith, and practice with them until they make more sense?

Answer 2

Analysis on Manifolds by Munkres is one of the finest books on the subject ever written, it is the subject matter for the second semester of Advanced Calculus at MIT. There are also lecture notes by Prof. Victor Guilleman available for download, which supplement and improve the text. Go to ocw.mit.edu and find the Mathematics course 18.101, you will see in detail what is in the course. The book compares well with the second half advanced calculus by Loomis and Sternberg of Harvard University. Yes, it will prepare you well for graduate courses in Manifolds. John Lee's book on smooth manifolds is good and well used. An alternative which I like is An Introduction to Differentiable Manifolds by William Boothby.