M A T H E M A T I C S

>what's poincare duality useful for?
computing (co)homology

>studying Group Theory from Herstein for the first time
>learn that every finite abelian group is isomorphic to the direct product of cyclic groups
>mfw

Is Linear Algebra the most mind-blowing branch of mathematics?

Favourite+Least Favourite topics in Maths, everyone?

Least Favourite:
L = O;

Favourite:
F = {∀x:xϵL^(c)}

Lol wait for it.

ignore the sperg above me

can confirm
i put on a dress today and finally understood what a hilbert space is

I thought this last year too
good times

thats not math

thats analysis

fuck off

>what's poincare duality useful for?
Establishing [math]H^{k}(M,\mathbb{Z}) \cong H^{n-k}(M,\mathbb{Z})[/math] as modules, which implies that for an [math]n=4[/math]-dimensional symplectic prequantizable manifold [math](M,\omega)[/math] with [math]\omega \in H^{2}(M,\mathbb{Z})[/math], [math]\ast \omega \in H^{4-2}(M,\mathbb{Z}) = H^{2}(M,\mathbb{Z})[/math] the polarization [math]P[/math] given by the coadjoint orbit of [math]Sp(N)[/math] acting on [math]M[/math] gives two inequivalent prequantum bundles [math]B\rightarrow M[/math], [math]B' \rightarrow M[/math] generated by [math]\omega[/math], [math]\omega' = \ast \omega[/math], respectively. The sections [math]\psi,\phi \in H_P[/math] of which satisfy [math] [\psi(x),\phi(y)] = \delta(x-y)[/math] for [math]N \equiv 0 \mod 2[/math], and the sections [math]\psi',\phi' \in H'_P[/math] of which satisfy [math]\{\psi'(x),\phi'(y)\} = \delta(x-y)[/math] for [math]N = 1 \mod 2[/math], for any [math]x,y\in \Omega[/math] where [math]\Omega[/math] is a Cauchy surface in [math]M[/math].
This is the spin-statistics theorem.

>green's
>stokes'
>divergence

What is the best field?

LMAO. Real analysis is generally babby's first "wow math is sick"

Are you retarded? :D

i love how calculus 3 classes throw those concepts into the last 2 week of classes in such a overly simplified form

"so uh yeah these are pretty much the most important theorems of vector calculus kind of a big deal lol final in about a week gl guys"

that is nearly verbatim from what my professor told us,,

he actually told us that the math required to fully understand such concepts is actually taught in a way higher level of a class than vector,,, logic in school never ceases to fail comprehension kek

how's middle-school treating ya?
>not [math]r^{\mathrm{th}}[/math]-order, [math]r\in \mathbb{R}[/math], fully-heterogenous, chaotic partial differential equations with hidden coefficients

Why do textbooks write [math]f^{-1}[/math] when they mean [math]\text{preim}_f[/math]?

Because it is shorter

textbook creators are lazy fucks that can get away with mickey mouse'd shit like that lol

i feel like the application of the big three theorems isn't as hard as the comprehension required to actually understand the theory behind them. you could learn how to use a computer without ever understanding why it works, which would take rigorous study to understand the ins/outs of computing.

>not even infinite-order
brainlet confirmed

Because not everybody follows the notation that makes the most sense in your head.

Yeah but then he takes it one step further in confusing notation by writing [math]f^{-1}(1)[/math] when he means [math](\text{Im}^{-1}\ f)(\{1_H\})[/math] (Im^{-1} f more aesthetically pleasing than preim_f)

>Too much typing
He could just write TeX commands for it.

It's just an appendix but still...

write the author a well-worded letter

That's literally standard straightforward notation. Just fucking kill yourself if you're confused about this.

This.

Doing Stokes' theorem properly takes a significant amount of machinery beyond a even what would be considered a "good" calculus 3 stream. However applying it in 2-3 dimensions is not very hard and quite useful so they touch on it.

literally nothing wrong with f^-1
kys u pedantic cunt

Reading analysis, and being completely blown the fuck out by this series of function of general term [math]f_n(x) = \frac{x}{n(n+x)}[/math].

I need to prove that lim f(x) = +infinity and lim f(x)/x = 0.

To prove the first thing, I said that, when n -> +infinity, we have, for any x :

[eqn]\frac{1}{2n^2} < \frac{1}{n^2*(1 + x/n} < \frac{1}{n^2}[/eqn]

As a result :

[eqn]x*(\pi^2/12) < f(x) < x*(\pi^2/6)[/eqn]
But it doesn't follow that lim f(x)/x = 0.

Indeed :

[eqn](\pi^2/12) < f(x)/x < (\pi^2/6)[/eqn]
Obviously, I fucked up [math]somewhere[/math], but where exactly ?

You're not making much sense.

Well, maybe it's my bad English, but I understand what I've said.

The limits are when x -> + infinity.
f(x) is the resulting function of the sum of all [math]f_n(x)[/math].

I tried to use the "Squeeze theorem", with the first inequality. When n -> +infinity, we have :

[eqn]\frac{1}{2n^2} < \frac{1}{n^2*(1 + x/n} < \frac{1}{n^2}[/eqn].

I multiply everything by x.

[eqn]x\frac{x}{2n^2} < \frac{x}{n^2*(1 + x/n} < \frac{x}{n^2}[/eqn].

Now I transform these into sums, since : [math]\sum_{n=1}^{\infty}1/n^2 = (\pi^2/6)[/math].

[eqn]x*(\pi^2/12) < f(x) < x*(\pi^2/6)[/eqn]

So, it shows that [math]\lim_{x\to\infty} f(x) [/math] = +infinity, but not that [math]\lim_{x\to\infty} f(x)/x = 0. [/math]

Which is not what I was supposed to prove.So obviously I made a mistake. I'm trying to know where.

How do you know that 1+x/n < 2 for all x? You're taking the limit as x-> infinity, but in the squeeze theorem with n you treat x as fixed

Ok, so that was where I was wrong. Thanks.

for fixed n:
[eqn]f_n(x)=\frac{x}{n(n+x)}=\frac{x}{nx(\frac{n}{x}+1)}=\frac{1}{n(\frac{n}{x}+1)}\rightarrow \frac{1}{n},\; \mathrm{when} \;\;x\rightarrow \infty[/eqn]
But then [math]\frac{1}{n}\rightarrow\infty[/math], so [eqn]\lim_{x\rightarrow\infty,n\rightarrow\infty}f_n(x)=0[/eqn]
Similary [eqn]\frac{f_n(x)}{x}=\frac{1}{n(n+x)}\rightarrow 0,\; \mathrm{as}\;\; x\rightarrow \infty[/eqn]

I'm studying discrete mathematics rigorously.

>I'm studying discrete mathematics rigorously.
>rigorously
So you're a CS major that sucks at math?

Nice meme my friend.

I don't like CS courses as much as I do math and I am studying for my own interest before I take actual courses in it. By the time I take data structures I'd have already studied it and by the time I take algorithms I will again have already studied it. My approach to CS is theorem, proofs over code monkey web design

R