Do you guys know of any good resources on dimensional analysis (applied math). I'm mostly searching for problem sets and some explanation of how to reduce PDEs to ODEs?
@GTR: LOL, no. This is just a course about multivariable analysis and linear algebra. We don't even do Baire (usually) in our undergraduate analysis or topology courses, although I certainly have a few times.
It is so sad that Marsden and Weinstein's Caculus I, II, III are so seldom used today compared to Stewart's Calculus when the former costs a small fraction of the latter.
It's very well written, but has some nonstandard things which make some people not like it. Although they will probably be removed by the co-author in the third edition.
Basically, all calculus books are isomorphic, some more hand-holding than others. Edwards and Penney was my favorite, but after the second edition it got worse, too.
(In particular, this frustrated my students in a section he has on integrating things of the form $\int sin^{2n} \cos^{2m}$; despite having the entire section be about a reduction formula for this, he doesn't give any indication as to how one might do it. The integration by parts involved isn't trivial, either!)
@Jasper, FYI, at this point a lot of schools (e.g., UC San Diego and Georgia) have a math degree (called applied) which require students to write hardly a single proof.
@TedShifrin I must say that after my time and before my time, the math course I took was much more rigorous, so basically it was a shit time for me, lol.
At UGA, I now have approximately 20 students in my super-hard multivariable math class, and I won't have quite that many in the second semester. All the other hundreds of math students take the standard course.
@TedShifrin I find such downvotes offensive because they make correct answers look incorrect. (This is also, apparently, Mhenni's objection, but his answers suck.)
I agree, @Jasper. My book is better for students than Spivak's ... unless they already know the subject. Munkres rewrote Spivak's Calculus on Manifolds to make it a bit more readable.
@TedShifrin After reading more, Hatcher's notes/book on characteristic classes is quite good. But it's probably written more towards the homotopy theory-minded than Hirzebruch's or Milnor's.
@TedShifrin: And what do you exactly mean by uniform? Uniform across the whole country? Across all students, no matter whether they are studying maths or engineering or CS?
I know you haven't, @Ted, hence the review :D I'm going to read Hirzebruch afterwards. You said it's a differential viewpoint? i.e., I'll be integrating characteristic classes and such? (I'm thinking of the first chern class of a line bundle, which I remember being equal to some integral that I've forgotten.)
@TedShifrin: It usually takes me a day to write an exam for my high school students. The most previous one I wrote today and it is really easy, I hope. :)
@TedShifrin: I'm sorry to hear. But still a good reputation of the top-universities. Some friend of mine will go to MIT soon. I'm eager to hear whether he thought that he learnt more or less than over here.
Chern's original curvature definition is not prominent in the literature, although Milnor talks about it. But the proofs that you get duality to Schubert cycles by actually integrating curvature forms are not to be found in the literature other than in Chern's papers (or my courses) :P
@Huy: MIT's standard calculus course for the masses is the same as everyone else's, but just much faster. There is a Calculus with Theory course that uses Apostol. Most great schools have a super course for freshmen that does my book or Hubbard-Hubbard or something harder (e.g., at Chicago or Harvard) ... no one does a rigorous single-variable calculus course any more other than the tiny class at MIT, and a few schools — few — that still have a Spivak course.
They "use" it in the sense that they define things that way. But no one proves things that way, other than me :P
Yeah, @Studentmath, I made it easy. Well, some of them will manage too well, others probably not. I've decided to put a few "proofs"/derivations on the final.
@Huy: France, Italy, Switzerland, England
Oh, and, @Studentmath, I did put your expectation of exponential of linear normal r.v. on the next problem set.
Well, I think I want to make the A students write me some sort of derivation/proof, so I'll eventually announce that. But it won't be anything too hard (I did prove weak law of large numbers yesterday, modulo the assumption of finite variance).
basic high school algebra, yes, @Studentmath, and working with indices in summations, etc.
@TedShifrin: I did not expect Germany to be missing in that list. Is there a particular reason for that? Where exactly in France, Switzerland and England have you been?
I am annoyed with what I am trying to prove. I think I managed to prove the result I need to use in the proof itself, but I can't bring myself to formulate it. There are so many 'bad scenarios' that I need to explain why they don't matter, I fear I will end up with as complex proof as the usual proofs..
@TedShifrin If I have the homology groups with $\Bbb Z$-coefficients of a space I can easily compute the homology groups with different coefficients via the UCT. Given the cohomology groups, can I "change coefficients" easily in some similar manner?
@TedShifrin: I really disliked the amount of people in London. I think on holidays I am the kind of person who likes to relax and that doesn't work for me in London.
If $H_n(X;\Bbb Z)$ is finitely generated, then I can do this by a silly computation by this same theorem (since $\text{Hom}$ and $\text{Ext}$ behave nicely in this case). But if $H_n(X)$ is bad I see no way.
@TedShifrin: Maybe I have an abnormally sensitive nose, but basically, most of the times I I had found a quiet place, it was a bit spoiled by the smell.
