Any time @Ted. I always find it very interesting that while e.g. we only talk through some largely anonymous chatroom, I've gained great respect for you.
I consider it one of the best things of the internet.
Just like the course you are teaching these undergrads.
I have enough distinctive problems in my undergrad diff geo text that I can surprisingly often tell when someone's asking things from it. They usually get all defensive/coy, too.
@TedShifrin Hmm, I just noted I proved #15 only for $n = k$. The idea is that $\omega_i$ live in $(\Bbb R^n)^*$, so I can write $\omega_i = \sum_j a_{ij} dx_j$, and $\sum_i dx_i \wedge \omega_i = 0$ says $\sum_{i, j} a_{ij} dx_i \wedge dx_j = 0$. If both $i, j$ move from $1$ to $n$ (which it is for $n = k$), then I can just switch things around to get $i < j$ and coeffs of $dx_i \wedge dx_j$ in the sum be $a_{ij} - a_{ji}$. That expression being zero forces these to be zero, nope?
@Ted: I just learned from Dick Gross that there's a new heuristic from Poonen, Stoll, Wood, that would imply there are only finitely many elliptic curves of rank greater than 21.
I say it's important because it's called the Cartan lemma. It proves, among lots of other things, that the second fundamental form of a hypersurface is a symmetric bilinear map.
@user19405892: I'm asking you to propose what you think something should be. All notions of an integral that I know are for integrals of functions, and one uses those to compute areas/volumes/masses, etc., of more general regions.
We have that $\displaystyle \int_{a}^b f(x) dx = \lim_{\text{max} \Delta_{k} \to 0} \sum_{k=1}^n f(x_k^*) \Delta x_k$ and a function $f$ is said to be integrable on a finite closed interval $[a,b]$ if the limit exists and does not depend on partitions or on the choice of points $x_k^*$.
@AkivaWeinberger Do you have to participate in every conversation? Talking about weird ideas when someone is struggling with integrals themselves is actively damaging pedagogy.
Ponder what you said to me, btw. If a function is discontinuous only on a nowhere-dense set, can you prove the Riemann integral exists in the first place?
Now I know what that guy actually means. Funny enough, my intuitive explanation of Green's theorem sounds exactly like that. A bit more dignified, probably.
HMM I don't understand something Our code C correspond to ideal $(x^n - 1)F_q[x] \subset C \subset F_q[x]$ why is this true ? a code C is of the form $C = a_0 + a_1x + ... + a_{n - 1}x^{n - 1}$
