Recent content by sammycaps

Hello all, I'm going through Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger, and I read a curious line that I was hoping someone here could clear me up on (perhaps I'm thinking too much into it). The begininning of the section on the Lebesgue integral introduces a...

So I might have worked it out (or at least thought about it a bit more), so I have another question (answers to the above are also very welcome). It is possible to find a 2-cube which has boundary = c1,n-c1,m if n≠m, if you allow the 2-cube to map to (0,0)? Or, does no 2-cube exist with such...

I understand (or maybe not), but the solution that I posted asks that you first take as given that ∫c*dθ is n/2∏. I don't see this (especially if the goal of the entire problem is essentially to prove this). Am I completely missing the point?

I think I sort of understand, but I'm not sure how to see that it must integrate to 2\pin, for some n, explicitly. I'm on a pretty basic level, since I just started learning about this stuff.

Maybe I'm just not used to the computations with pullbacks. I get ∫c*d\theta=∫f1(c)det(c')+f2(c)det(c'), where f1 and f2 are the component functions of d\theta. Is this right? Why does this get you an integer (once you normalize it by 2∏) And for it to be change of variables, don't we...

Also, in the link I posted at the beginning of the thread, can anyone explain why n in problem 4-24 is necessarily an integer?

Hmmm, I'll have to review covering spaces. So, then you would just fill the closed curve in and use that to get the 2-cube with boundary the difference of the two curves?

I've seen a bit of covering maps, but I don't know too much about them. I took a course that studied the fundamental group (but oddly skipped covering maps). Is your solution online? As an aside, is the c* in the link I posted the pullback operator, or something else (once I know that I can...

Yes, sorry about that.

So I'm working on Little Spivak and I found a solution to problem 4-24 at http://www.ms.uky.edu/~ken/ma570/homework/hw18/html/ch4c.htm. My question is, is the integral defined there the integral of a differential form over a chain, or something else? I only ask because the book did not reach...

Ok, thanks very much.

Is it necessary to finish Spivak's little book to move on to Spivak's Differential Geometry I, or is the material on differential forms and integration on manifolds in Chapter's 4 and 5 of Spivak's little book covered in Differential Geometry I?

I was trying to show that any ball in ℝn is a Jordan region, and this amounts to showing that its boundary has volume zero (Jordan content 0). My TA proposed that you break the circle up into pieces of lines, and then adapt the following proof. [ Pf: A Horizontal line of length 1 has content 0...

So the beginning of Rudin's Real and Complex Analysis states that the Riemann integral on an interval [a,b] can be approximated by sums of the form \Sigma\stackrel{i=1}{n}f(ti)m(Ei) where the Ei are disjoint intervals whose union is the whole interval. At least when I learned it, the Riemann...

Thank you, but that is not quite my question. I know that different metrics may induce different topologies. But, some properties are properties of the metric and not the topology (like boundedness), as in two metrics can give the same topology but the space may be bounded in one and not the...