Definition: If there is a number L, such that for every positive number ε you give me, I can find a number N such that every partial sum with more than N terms is more than L-ε and less than L+ε, then the value of the infinite sum is L
yes, and the analogy is this. given a cw complex $X$, I can't define dimension of $X$ to be the $k$ such that $H_k(X)$ is nonzero, because that's nonsense. that being said, I think you can probably fix your definition for reasonable $X$.
@Balarka, @DogAteMy: For future reference, it's quite easy to draw $z=x^2-y^2$, less easy (for me) to draw $z=xy$. However, if we rotate the axes 45° in picture 1, we get picture 2, so that's the cheating way to draw it. :P
Because this was an entirely local computation, you then see you would have gotten the same result if you had wedged an $S^2$ and a $D^3$ on the interior of the latter.
@AkivaWeinberger There's a generalization of FTC, called Stoke's theorem. Mike's asking for a reason why a 2 dimensional version of Stoke's, called Green's theorem, is obvious, and I think the same reason should apply to FTC.
@TedShifrin I'm still stuck at the same point of Hatcher! (And I also have The Martian in two different foreign languages that need to be read at some point)
BTW, @Semiclassic, did I ever ask you if you'd looked at Bamberg & Sternberg (a Harvard text integrating "first year" physics and serious mathematics)?
Of course, @Semiclassic. And I've always adopted a serious amount of it. Which is why, 15 years ago, two chemistry majors in my (standard) multivariable calculus class raked me over the coals on evaluations for putting too much physics in the course. Not to mention that they didn't do very well. :(
I don't remember, @J.M. I thought it highly ironic that the chemistry majors complained and the math education and math majors did not. Obviously, the engineering students were quite happy.
@MikeMiller Can you sketch a walkthrough for what kind of obvious explanation you'd want? You said a few things about trying to see what happens to the corresponding vector fields, and then an obvious explanation for 1-dim FTC. I am confused how I should approach the question because it's vague up until now.
@TedShifrin So, do you want me to try to prove Green's along the little piece thing idea you mentioned? I am very confused because I don't have a precise question I can think about.
Well, I have a question. "What does $\partial X_2/\partial x = \partial X_1/\partial y$ mean when $X$ is a 2 dimensional vector field?". But I am not sure if that's what everyone wants me to think about.
Well, start with the 1-dimensional version of FTC that way, @Balarka. And, yes, then progress to Green's Theorem. On a tiny little rectangle $R$, where you can assume $d\omega$ is essentially a constant times $dx\wedge dy$, why is it that $\int_R d\omega = \int_{\partial R} \omega$?
I just made it more precise. No guarantees that Mike cares about what I'm saying.
I doubt that's the only way you know to prove it. The usual way is to prove that two primitives differ by a constant (assuming $f$ is continuous, but we needn't).
Also, think about how there are two statements for FTC, @Balarka. One assuming $f$ is continuous, one merely assuming $f$ is integrable but equal to $F'$.
@MikeMiller Ahh. $-x dy$ e.g. is also not closed, and the vector field $(0, -x)$ has also a "spinny" nature about it (if I place something round, it'll make a spin). I think it's the local swirling nature is what matters, not global. This is something to think about!
I have no idea how $\partial X^1/\partial y - \partial X^2/\partial x$ detects such a thing.
Maybe $(0, -x^2)$ would be more surprising.
I think that intuitively explains what's wrong with $1/\|\vec{x}\|^2(-ydx + xdy)$. There is no local spin to it.
@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ what would you choose between being a super expert in a small area of mathematics and knowing large area of mathematics, but without being able to excel in it?
Well, a cliché, of course, but it's good to recollect that time is a very limited resource and out there is so much math. I don't even count other things you might like.
The thing that might scare you a lot one day is when you realize how much you can obtain starting from a simple result in math.
Hope to have soon my team working on my stuff and do things much faster than I did alone. That perhaps after finishing my main project.
@Agawa001 btw, have you experienced the issue with the limited connectivity in win 8.1? I tried more solutions exposed on youtube, but they seem useless.
