M A T H E M A T I C S

>>8546695
Reviewing differential equations for the study of dynamical systems.

The titles of my subsections are getting ridiculously large
> Second Order Homogeneous Linear Differential Equation with Constant Coefficients

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>>8546695
I'm going to crossdress and try to quantify its effectiveness on my productivity.

>>8546699
just wait for
> n-th Order non-Homogeneous non-Linear Differential Equation with Variable Coefficients

I was fucking off on projecteuler.net, but I guess I could shitpost here instead.

>>8546699
Would it be possible to teach myself-elementary differential equations in a week?

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prepping for algebraic topology next semester

what's poincare duality useful for?

>>8546715

Yes if you have some average background on derivatives and integrals.

Just pick up some book and start learning.

Preparing my anus for Calc 3, Calc physics 1, and C programming.

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>tfw probability final is just learning a dozen different distributions and their properties

this is the worst kind of studying

>>8546716
>what's poincare duality useful for?
computing (co)homology

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>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?

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Favourite+Least Favourite topics in Maths, everyone?

Least Favourite:
L = O;

Favourite:
F = {∀x:xϵL^(c)}

>>8546765
Lol wait for it.

ignore the sperg above me

>>8546703
can confirm
i put on a dress today and finally understood what a hilbert space is

>>8546765
I thought this last year too
good times

>>8546767

thats not math

thats analysis

fuck off

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>>8546716
>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.

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>green's
>stokes'
>divergence

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What is the best field?

>>8546765
LMAO. Real analysis is generally babby's first "wow math is sick"

>>8546779
Are you retarded? :D

>>8546787
i love how calculus 3 classes throw those concepts into the last 2 week of classes in such a overly simplified form

>>8546801
"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"

>>8546810
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

>>8546706
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]?

>>8546824
Because it is shorter

>>8546824
textbook creators are lazy fucks that can get away with mickey mouse'd shit like that lol

>>8546817
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.

>>8546819
>not even infinite-order
brainlet confirmed

>>8546824
>>8546824
Because not everybody follows the notation that makes the most sense in your head.

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>>8546839
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...

>>8546843
write the author a well-worded letter

>>8546843
That's literally standard straightforward notation. Just fucking kill yourself if you're confused about this.

>>8546832
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.

>>8546843
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 ?

>>8548310
You're not making much sense.

>>8548340
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.

>>8548359
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

>>8548405
Ok, so that was where I was wrong. Thanks.

>>8548310
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.

>>8548565
>I'm studying discrete mathematics rigorously.
>rigorously
So you're a CS major that sucks at math?

>>8548647
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

>>8546790
R