ChapterZero

My latest mathematical struggle: differential forms. As an aside, it seems like once we hit Chapter 8 in “Principles of Mathematical Analysis”, the material is dimensions harder than it was before. I’m having trouble visualizing everything, and for me math is usually a very visual process. A lot of this higher analysis stuff (which I define as that which starts in Chapter 8 of Rudin ) is based on topological concepts/linear algebraic concepts— pull backs, rank of mappings, etc— that I’m just not comfortable with, yet. And all of it seems to have a more ‘natural’ home on manifolds.

Differential forms are, to me, abstractions of the change of variables formula which makes it possible to define integration over abitrary surfaces. I remember when we had to do volume or line integrals in my static EM class, we did a lot of hand-waving and suspicious sign fiddling— differential forms give a rigorous form to this whole process. It took me a while to get this, due to the new terminology, the use of the wedge product, and the hype surrounding it in general (e.g. generalized Stokes Theorem), plus the fact that the ‘natural’ setting for them arose in tensor calculus on manifolds, blah blah …

But they really aren’t that bad. Rudin has a weird definition though; if I were writing a book, I would try to come up with something a little less off-putting. Maybe that was his intent though, to get the reader paying close attention. I don’t think I would have paid much attention if he had explicitly pointed out the connection to the change of variables theorem.

Possibly relevant posts:

• Upcoming analysis test (3/27/2005)
• Change of coordinates on manifolds (9/20/2006)
• Chapter 9 of Principles of Mathematical Analysis (3/22/2005)