Okay. I'm trying to explain in my own words what I understood from Wikipedia's "Classical definition". A field is any set, such that on any random pair of elements if we apply the operations "addition" and "multiplication" then we get an element belonging to the same set. Furthermore, the operations are to satisfy the properties of : Associativity, Commutativity, Distributivity and also Additive/Multiplicative identity/inverses. Makes sense? @JasperLoy @Semiclassical I think I got it now
Suffering for a purpose can be a form passion, we like to think of our emotions as complicated things, but they're just chemical interactions in our brains.
Just because we don't fully understand the mind (and consciousness) yet, doesn't mean we'll lean back to supernatural explanations, there are good reasons to believe that only the physical exists.
per defintiion: $\mathrm{grad} f(x) = (\partial_1f(x), \ldots, \partial_nf(x))$. Is $\mathrm{grad} f = (\partial_1f, \ldots, \partial_nf)$ a correct notation, too?
@LeakyNun yes, a slider. I have already uninstalled Geogebra though.
@LeakyNun From the 'patience' entry: Late 12c., "sufferings of Christ on the Cross," from Old French passion "Christ's passion, physical suffering" (10c.), from Late Latin passionem (nominative passio) "suffering, enduring," from past participle stem of Latin pati "to endure, undergo, experience," a word of uncertain origin.
Is there any sense in which $\dfrac{\sin\pi x}{\pi x}$ is the most natural extension of $\Bbb N\to\Bbb N:x\mapsto\begin{cases}1,&x=0\\0,&\rm else\end{cases}$ to the reals?
Oh yeah, if anyone wants to try and make a really large number/function, please support my proposal: https://codegolf.meta.stackexchange.com/a/13524/58880
Does somebody know this guy: $\lim_{h \to 0, h \ne 0}\frac{r(h)}{h}$ and his brother $\lim_{h \to 0 h \ne 0}\frac{\mid r(h) \mid}{\mid h \mid}$? They come from the formula $f(x+h) = f(x) + f'(x)h + r(h)$. What are they?
in one-dimensional analysis I could imagine this approximation as the limit of the secant line going towards the tangent line. Here we have derivatives of $f$. Where do they come from?
@LeakyNun this formula is not ok for $\mathbb{R}^n$ for $n\ge2$.
f(x+h)=f(x) is a 0th order approximation f(x+h)=f(x)+f'(x)h is a 1st order approximation
@Kirill remainder of taking away the approximation
with division, if you want to approximate 1776 with a multiple of 10, you have 1770, and the remainder (difference between estimate and true value) is 6
f(x+h)=f(x)+f'(x)h+f''(x)h^2 would be a 2nd order approximation
its' version for a one-dimensional space is quite ok, but for others is quite strange. It says, "there is a value, so, that the UNequality holds". I haven't even thought about connecting these two theorems. Can one derive Taylor from MVT?
I still hope that Mr. Taylor wrote his formula not randomly and had proved it by induction, so that there is a logical argument, why derivatives can approximate a given function.
A good question would be to ask whether or not anyone knows a GAP implementation to search for rationally rigid triples of conjugacy classes of certain finite simple groups.
@alan2here Well, the point of Euler's theorem more generally is that it tells you about the space in which your graph lives. For instance, you could embed a planar graph on the surface of a torus; in that case, V-E+F wouldn't be 2.
But if you've got a single node then you you haven't got enough data to say anything about the space in which your graph lives.
I guess you've got to have edges loop around the space for this to work, on the plane I guess this means going off to infinity then reconnecting on the other side.
maybe it's true to say that if you need to have enough information in the graph to understand the space it's on then you don't need nearly as much information for a plane so it's nice and simple
Well. It's got half of those edges identified, so that's E=2g. And there's still one face, so F=1. So V-E+F=V-2g+1, which should be 2-2g. So there's only one vertex?
Better exercise, @Semiclassic, is to cut the 8-gon in half and see how that's putting two 1-holed tori (each minus a disk) together. That's the connected sum all the topology people in here talk about.
@TedShifrin A guy in the workshop kept drawing a hyperbolic 9-gon instead of an octagon when discussing conformal vs. hyperbolic structures on surfaces
Reminds me of the novel Watership Down, which is about rabbits. They've only got four digits on their paws, so their counting goes: one, two, three, four, many.
It's funny. I gave a pretty precise hint on a multivariable analysis question and then afterwards two advanced people with lots of rep gave complete proofs by contradiction. I put on both "Why is everyone so enamored of proofs by contradiction?" ... Another occurrence of people with rep doing someone's complete question rather than leaving it to the OP to think through.
I actually want to see like a GIF of a ball traveling on an octagon, like a billiard ball, but instead of bouncing off the wall it obeys the quotienting of the sides and comes out one of the other sides
@AkivaWeinberger No but I'd try doing binomial expansion. I think there's a short proof from considering (a + bsqrt(n))^k + (a - bsqrt(n))^k or something
@BalarkaSen Yeah, it's an integer iff it equals its conjugate, so we want $(a+b\sqrt n)^k=(a-b\sqrt n)^k$, but that's impossible since $|a+b\sqrt n|=|a-b\sqrt n|$ implies either $a=0$ or $b=0$
