What's your favorite equation/theorem/science fact Veeky Forums?

Sure, so long as n=1

2+2=4. Not even joking.

Take an infinitely long wire, and put resistors distributed evenly across the wire.

The current is (up to Heisenberg certainty) 0.000002 ohms

Lol don't you know any algebra? You can prove this through induction over n.

The base case n=1 is trivial so now lets do the n+1 case, assumig it works for n

[math](a+b)^{n+1} = (a+b)^n(a+b) = (a^n + b^n)(a+b) = (a^na + b^nb) = a^{n+1} + b^{n+1} [/math]

QED.

Now, if it works for rationals and irrationals is still a mystery and it is actually tied up to the zeros of the Reman function. If the Reman Function has non trivial zero for x then this theorem works for x but no such 0 has been found in the entire universe yet.

[eqn]\int_0^\infty \prod_{k=1}^n \frac{\sin\frac{x}{k}}{\frac{x}{k}}=\frac{\pi}{2} \;\;\;\;\forall n[/eqn]

[math]\nexists \text{a solution to} \iiint f(x,y,z) dxdydz \forall f(x,y,z)[/math]

God's theorem: Every complete logical system will necessarily be consistent.

What about the space of Barnett-integrable functions?

[eqn]\exists x\in\mathbb{R}/\mathbb{Q}[/eqn]

if p is a prime, and (a,p) = 1, then a^p-1 = 1 (mod p), this shit and eulers generalized theorem with phi(p) is fuckin BEAUTIFUL

I got an A+ in my multivariable calculus course where we used Stokes' theorem. I don't understand it at all. Can someone explain the equation?

Does (a,p) means they are relative primes?

(a,p) is the gcd of a and p
so (a,p)=1 means relatively prime

K, thnx bro

Doesn't 1 mod p always equal p? Since p will always be greater than 1

>He doesn't know the binomial expansion of (a+b)^n+1

free energy!

This is cooler:

en.wikipedia.org/wiki/Borwein_integral

1 (mod p) = pk+1 for any k, pk+ 1 != p, u fuck

i like kronecker/diracs delta

it's just nifty and cute

I like the equation y=x+2

it gives me a reason to live

I really like those funny stuff in number theory.

I also proved something an user on Veeky Forums conjectured, so I like it (even though it was probably an already known rule)

Let A, B, C, D > 0.
If [math]\frac{A}{B} < \frac{C}{D}[/math], then we have :
[math]\frac{A}{B} < \frac{A+C}{B+D} < \frac{C}{D}[/math]

Hey, that's my line

[math](a^n + b^n)(a + b) = a^{n+1} + b*a^n + a*b^n + b^{n+1}[/math], though.

that's like the standard proof that the rationals are dense

I find the evolution of species quite elegant, poetic and inspiring.

Yeah, it's pretty trivial but I felt good when I attacked it and proved it though.

its a satire

Incorrect

t. Wildberger

i like how e*i + pi = 1

The first time I derived the wave equation from Maxwell's equations was pretty cool. I can't imagine what it must have been like the first time he did it.

I had a Borwein as a professor.

It is physically impossible to bend a pizza

wat

>where A(T*M)
Fount it.

I dunno, I guess I like the proof by contradiction that [math]\sqrt2 \in \mathbb{R}\setminus\mathbb{Q}.[/math] I also like the geometric proof that it's possible to construct [math]\sqrt{d}[/math] given that [math]d[/math] is a constructible number. I also like how [math]\frac{\text{d}}{\text{d}x} e^x = e^x.[/math] It also still blows my mind how [math]\int_{a}^{b}f(x)\text{d}x = F(b)-F(a).[/math] It shouldn't be this elegant, but damn it is.

Gershgorin circle theorem. no more determinants for computing eigenvalues people
en.wikipedia.org/wiki/Gershgorin_circle_theorem

gershgorin only gives estimates though, nothing exact

I thought this identity was pretty neat:

[math]
\int_0^\pi \frac{(1 + (-1)^m)\,\Gamma\left(\frac{1+m}{2}\right)\,\Gamma\left(\frac{1+n}{2}\right)}{2\,\Gamma\left(1 + \frac{m + n}{2}\right)}
[/math]

Typo, meant to write:

[eqn]
\int_0^\pi \sin^n\vartheta \cos^m\vartheta\;\mathrm{d}\vartheta = \frac{(1 + (-1)^m)\,\Gamma\left(\frac{1+m}{2}\right)\,\Gamma\left(\frac{1+n}{2}\right)}{2\,\Gamma\left(1 + \frac{m + n}{2}\right)}
[\eqn]

I'm bad at this.

[eqn]
\int_0^\pi \sin^n\vartheta \cos^m\vartheta\;\mathrm{d}\vartheta = \frac{(1 + (-1)^m)\,\Gamma\left(\frac{1+m}{2}\right)\,\Gamma\left(\frac{1+n}{2}\right)}{2\,\Gamma\left(1 + \frac{m + n}{2}\right)}
[/eqn]

Protip: if you're on Veeky Forums X, there's a TeX button you can press to preview what you've written before you post it. Also, the [ math ] tags were correct.

if you're not on Veeky Forums X it's still on the top left corner of the reply window

I know this sounds kinda stupid but I really like the equation to find the area of a triangle.

I remember as a kid I was like "o bingus this shit is hard!" and when i found the equation it was like someone hit me with a bar of chocolate.

(1/2)bh gahhhhdamn it's just so clean

like i know theres a bunch more intuitive stuff out there but this really oils my eardrums for me

I like Duhamel's principle. Ergodic theorem is cool too.

Least upper bound property. The least-upper-bound property states that every nonempty set of real numbers having an upper bound must have a least upper bound (or supremum) in the set of real numbers. S = { x ∈ Q. x 2 < 2 }

some god type shit right there

Weierstrass functions

>tfw Veeky Forums is too dumb to understand your meme thread

These are all mandatory "this is how fucking great humans can be" relationships

MANDATORY

noetherian rings

[eqn]\frac{d}{dt}\bigg(\dot{q}\frac{\partial L}{\partial\dot{q}}-L \bigg) = 0 [/eqn]

it looks like a benis

What's the black-scholes equation? The wikipedia page went a bit over my head with all the finance stuff.

I was about to ask the same thing. Wikipedia didn't define the variables and I just started digging into it.

Integrate from omega to what?

lol

>"Calculus"
>not "definition of the derivative"

>∫ e^-2πixw dx

I have to integrate that shit every time in DiffEqs?
Fuck engineering, I'm out.

Back in college, I was in a modern physics class and the professor introduced us to the time independent Schroedinger wave equation. He was explaining its significance and how it is related to conservation of energy and yada yada yada. After the lecture, one of the students in my class raised his hand and asked, "Why don't you just cancel the psi's on both sides of the equation?" The professor set the chalk down slowly, grabbed his coat, and calmly walked out of the room. After he left, the kid came to his senses, answered his own question, and felt really stupid and embarrassed.

Johan?

>Universality Theorem
does it only approximate it at a point, so at that point the function + all derivatives are the same, or does it approximate it in a finite area sothat you can take a chunk of the zeta function and it looks just like another function?

is this how the pyramids were built?

kek i fucking love this guy

>it is actually tied up to the zeros of the Reman function
>Reman function
> If the Reman Function has non trivial zero for x then this theorem works for x but no such 0 has been found in the entire universe yet.
>the entire universe
kek my sides might as well not be found in the entire universe

Okay, I'll bite.

[eqn](a^{n} + b^{n})(a + b) = a^{n}a + b^{n}a + a^{n}b + b^{n}b[\eqn]

...

baby's first LaTeX

[eqn](a^{n} + b^{n})(a + b) = a^{n}a + b^{n}a + a^{n}b + b^{n}b[/eqn]

>attributing ""Pythagoras's Theorem"" to Pythagoras

ancient egyptians knew this in thousands of years before Pythagoras

What a shitty professor.

>current
>ohms

He was great. It wasn't like we were in the middle of class. Class was over and he had to leave. Our class had like 10 students in it and he made sure to make fun of all of us equally. But, yeah he could be rude and clearly had his favorites.

It represents a surface or volume, the limits in each coordinate direction are inserted later. It's just for brevity. Omega is commonly used for potential energy or solid angle.

Favourite equation, there are too many to pick one. Among them would be Euler's theorem, Dirac's Equation, Mass-energy equivalence, Schrodinger's Equation, Einstein's Field Equations, Entropy. Probably missing out an absolute ton of them though.

Favourite theorem, would be a clash between General Relativity, Quantum Mechanics (no pun intended), evolution and Thermodynamics. Though all of vector calculus and complex analysis are fascinating too.

Favourite facts, We all Stahdoost, We existed alongside several different species of hominids at the same time that were at least of equal intelligence and we only out-competed them because of voiceboxes and rotating forearms. Also, there's the heat death. Not as nasty as people think, it's actually themost elegant and natural solution to the Friedman Equation, and things would be a lot worse for the universe if it weren't the case. Kinda gives you more certainty to physics in general.

When you take a smooth projective variety over C cutout by polynomials with integral coefficients, then you can said polynomials mod p and get a smooth variety over the closure of F_p.

The l-adic cohomology of the mod p variety tensored with C is isomorphic to the usual zariski cohomology of the variety over C.

[eqn]A = abc[/eqn]

No idea what this all means, but I know it's differential geometry.

1 + 1 = 2

[eqn]F=-kT\ln Z[\eqn]

12 is fucking beautiful.

did you mean??
[math]e^{i \pi}+1 =0 [/math]

>The current is (up to Heisenberg certainty) 0.000002 ohms
>current
>0.000002 ohms

only if your wires are straws that accelerate resistors at nearly c

My favorite Veeky Forums shit is propositional calculus. Yes, I'm a CS pleb, but you can't deny that it has an amazing charm. It's so neat, so perfect, our lifes circle around it.

it approximates it on any compact subset whose complement is connected

i.e. given a nonvanishing holomorphic function on that compact subset, you can find infinitely many vertical shifts of that compact subset where the zeta function behaves arbitrarily close to that function

>No mention of the Residue Theorem
Veeky Forums pls

This is the deepest shit I've read all week. Glorious.

>propositional calculus
thats a shitty mess compared to Homotopy Type Theory

how is that definition remotely attributed to newton wtf?

>implying black-scholes is comparable to the fourier transform or maxwells equations

why?

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model.
>mathematical finance
Dropped

fpbp

arxiv.org/pdf/chao-dyn/9406003v1.pdf

fugg it's the continous analog (no pun intended) of the library of babel