/sci/ - Science & Math » Thread #11606246

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Anonymous Sun 26 Apr 17:39:36 2020 No.11606246 View Reply Original Report

Quoted By: >>11606267 >>11606273 >>11606277 >>11606322 >>11606328 >>11606389 >>11606406 >>11606434 >>11606577 >>11606609 >>11606724 >>11606736 >>11606902 >>11606949 >>11606959 >>11606963 >>11607876 >>11608877 >>11612251

Is this the most beautiful equation in all of mathematics/physics?

Anonymous

Anonymous Sun 26 Apr 2020 17:48:08 No.11606267 Report

Quoted By:

>>11606246
no

Anonymous

Anonymous Sun 26 Apr 2020 17:50:08 No.11606273 Report

Quoted By:

>>11606246
Faradays law

Anonymous

Anonymous Sun 26 Apr 2020 17:53:31 No.11606277 Report

Quoted By:

>>11606246
Superior form is
$\nabla_\mathbf{q}\mathscr{L}=\mathcal{D}_t\nabla_{\dot{\mathbf{q}}}\mathscr{L}$

Anonymous

Anonymous Sun 26 Apr 2020 18:09:09 No.11606322 Report

Quoted By: >>11606406

>>11606246
No, but
$d^2 = 0$
is

Anonymous

Anonymous Sun 26 Apr 2020 18:10:51 No.11606328 Report

Quoted By:

>>11606246
>Leibniz notation
Hard no.

Anonymous

Anonymous Sun 26 Apr 2020 18:27:45 No.11606389 Report

Quoted By: >>11612278

>>11606246
x=x

Anonymous

Anonymous Sun 26 Apr 2020 18:34:01 No.11606406 Report

Quoted By: >>11606565

>>11606246
I wouldn't know about physics, but the most beautiful in mathematics is $J \circ du = du \circ J$
>>11606322
Very nice, too.

Anonymous

Anonymous Sun 26 Apr 2020 18:46:21 No.11606434 Report

Quoted By: >>11606519 >>11606815 >>11606953

>>11606246
69 + 420

Anonymous

Anonymous Sun 26 Apr 2020 19:23:58 No.11606519 Report

Quoted By:

>>11606434
Kys degenerate.

Anonymous

Anonymous Sun 26 Apr 2020 19:36:57 No.11606565 Report

Quoted By:

>>11606406
Based pseudoholomorphicanon

Anonymous

Anonymous Sun 26 Apr 2020 19:45:42 No.11606577 Report

Quoted By: >>11612276

>>11606246
$\int_S KdA = 2\pi\chi(S)$
Either that or generalized Stokes theorem.

Anonymous

Anonymous Sun 26 Apr 2020 19:58:18 No.11606609 Report

Quoted By: >>11607354 >>11609519

>>11606246

Only correct answer

$e^(i \pi) = -1$

Anonymous

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Anonymous Sun 26 Apr 2020 20:26:48 No.11606687 Report

Quoted By: >>11606723 >>11606953

Clearly it's the Colebrook-White equation
$\frac{1}{\sqrt{f}}=-2\log\Bigg(\frac{\epsilon}{3.7D_\text{H}}+\frac{2.51}{\text{Re}\sqrt{f}}\Bigg)$

Anonymous

Anonymous Sun 26 Apr 2020 20:41:24 No.11606723 Report

Quoted By: >>11610962

>>11606687
>it's real
Hahahahhahahaha this is the ugliest equation I've ever seen.

Anonymous

Anonymous Sun 26 Apr 2020 20:42:28 No.11606724 Report

Quoted By: >>11610999 >>11612269

>>11606246

What about this masterpiece?

$R_{\mu \nu} - \tfrac{1}{2}R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}$

Anonymous

Anonymous Sun 26 Apr 2020 20:46:54 No.11606736 Report

Quoted By: >>11606817 >>11606869 >>11606956

>>11606246
F=ma

Anonymous

Anonymous Sun 26 Apr 2020 21:20:54 No.11606815 Report

Quoted By:

>>11606434
not an equation

Anonymous

Anonymous Sun 26 Apr 2020 21:21:36 No.11606817 Report

Quoted By:

>>11606736
this is a specific instance of OP's tho...

Anonymous

Anonymous Sun 26 Apr 2020 21:30:14 No.11606834 Report

Quoted By:

$\mathrm{ card }( {\mathbb R} ) = \mathrm{ card }( \omega_2 )$

Anonymous

Anonymous Sun 26 Apr 2020 21:30:52 No.11606837 Report

Quoted By: >>11606843 >>11606874

Can somebody explain this to a brainlet like me?

Anonymous

Anonymous Sun 26 Apr 2020 21:33:10 No.11606843 Report

Quoted By:

>>11606837
Yes. You have an integral and you want to minimize it. What you get is the equation in the OP.

Anonymous

Anonymous Sun 26 Apr 2020 21:48:11 No.11606869 Report

Quoted By: >>11606956 >>11607641

>>11606736
Assumes constant mass 0/10 would not touch.

Anonymous

Anonymous Sun 26 Apr 2020 21:50:44 No.11606874 Report

Quoted By: >>11606878

>>11606837
Basically, you have an integral $\phi(y) = \int_a^b L(x,y,y')dx$ and you want to find a function y(x) minimizing it. That equation is called the Euler-Lagrange equation.
Under certain (but rather general) conditions, the y function is a solution to the differential equation in OP's post.
Classical mechanics can be rewritten to be all about solving this kind of integral equations, and some deep results are much simpler to see in this form. Indeed the fancy L in the pic stand for Lagrangian, which is a term from physics.
There are also other applications in geometry, engineering, etc.

Anonymous

Anonymous Sun 26 Apr 2020 21:51:47 No.11606878 Report

Quoted By:

>>11606874
*Op's equation is the Euler-Lagrange equation

Anonymous

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Anonymous Sun 26 Apr 2020 21:53:57 No.11606885 Report

Quoted By: >>11606940

Anonymous

Anonymous Sun 26 Apr 2020 22:00:19 No.11606902 Report

Quoted By: >>11606914 >>11606962

>>11606246
$\Xi\in\mathbb{C} \implies \frac{\overline{\Xi}}{\Xi} = \frac{\overline{\Xi}^2}{\lvert\Xi\rvert ^2}$

Anonymous

Anonymous Sun 26 Apr 2020 22:05:25 No.11606914 Report

Quoted By: >>11606962

>>11606902
Fuck anyone who unironically uses Xi in their equations.

Anonymous

Anonymous Sun 26 Apr 2020 22:15:26 No.11606940 Report

Quoted By:

>>11606885

{F, H} = dF/dt

{A, B} = -{B, A}

{H, H} = dH/dt = 0 (conservation of energy)

Anonymous

Anonymous Sun 26 Apr 2020 22:17:54 No.11606949 Report

Quoted By: >>11606972 >>11606999

>>11606246
Einstein-Hilbert action is one of my favorites:
$S = \int R \omega$
All of general relativity falls out of that simple equation.

Anonymous

Anonymous Sun 26 Apr 2020 22:18:58 No.11606953 Report

Quoted By: >>11610962

>>11606434
Retard
>>11606687
>having actual rando constants in your equations
Literally repulsive

Anonymous

Anonymous Sun 26 Apr 2020 22:19:59 No.11606956 Report

Quoted By: >>11607363

>>11606736
>>11606869
Constant mass and only one force acting on that mass

Anonymous

Anonymous Sun 26 Apr 2020 22:21:04 No.11606959 Report

Quoted By: >>11606966

>>11606246
$2^\kappa > \kappa$ for all cardinals $\kappa$
Potential form of the Maxwell equations in the Lorenz gauge:
$\box A = J$
where $A$ is the 4-potential, $J$ is the 4-current, and $\box$ is the d'Alembertian.

Anonymous

Anonymous Sun 26 Apr 2020 22:21:48 No.11606962 Report

Quoted By:

>>11606914
>>11606902
Imagine writing that on a black board, where the Xis become just three lines stacked on each other.

Anonymous

Anonymous Sun 26 Apr 2020 22:21:50 No.11606963 Report

Quoted By: >>11607000

>>11606246
Seems pretty obvious

$\frac{\partial \mathscr{L}}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t} (\frac{\partial \mathscr{L}}{\partial \dot{q}}) = \frac{\partial \mathscr{L}}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t} (\frac{\partial \mathscr{L}}{\partial \frac{\mathrm{d}q}{\mathrm{d}t}}) = \frac{\partial \mathscr{L}}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t} (\frac{\partial \mathscr{L}{\mathrm{d}t}}{\partial \mathrm{d}q}) = \frac{\partial \mathscr{L}}{\partial q} - \frac{1}{\mathrm{d}t} (\frac{\partial \mathscr{L}{\mathrm{d}t}}{\partial q}) = \frac{\partial \mathscr{L}}{\partial q} - \frac{\partial \mathscr{L}}{\partial q} = 0$

Anonymous

Anonymous Sun 26 Apr 2020 22:22:20 No.11606966 Report

Quoted By:

>>11606959
* $\square A = J$
Also Euler's product formula
$\zeta(-s) = \sum_{n > 1} n^s = \prod_{p \text{ prime}} \frac{1}{1-p^s}$

Anonymous

Anonymous Sun 26 Apr 2020 22:23:59 No.11606972 Report

Quoted By:

>>11606949
Except that relativity isnt a science. >>>/X/

T. Chemist

Anonymous

Anonymous Sun 26 Apr 2020 22:28:35 No.11606987 Report

Quoted By:

$<br />S_\text{Einstein-Hilbert} = \int R_{\alpha\beta} \wedge \star (e^\alpha \wedge e^\beta)<br />$

Anonymous

Anonymous Sun 26 Apr 2020 22:33:56 No.11606999 Report

Quoted By: >>11607006

>>11606949
Version with matter fields:
$S = \int (R + \mathcal{L}_\text{M}) \omega$

Anonymous

Anonymous Sun 26 Apr 2020 22:35:07 No.11607000 Report

Quoted By: >>11607012

>>11606963
Nothing to the right of the second equality is a valid manipulation

Anonymous

Anonymous Sun 26 Apr 2020 22:36:38 No.11607006 Report

Quoted By: >>11607392

>>11606999
What is omega here? I take it R is the curvature 2-form
What book did you encounter this in? I'd really like to have a look at it.

Anonymous

Anonymous Sun 26 Apr 2020 22:39:02 No.11607012 Report

Quoted By: >>11607038

>>11607000
And yet it works. Can't argue with results.

Anonymous

Anonymous Sun 26 Apr 2020 22:47:07 No.11607038 Report

Quoted By: >>11607091

>>11607012
It works by coincidence and due to dimensional reasons. There are plenty of instances in which the simple Euler-Lagrange equation doesn't hold. For example when there are constraints or when $\mathscr{L}$ can depend on higher order derivatives of $q$ .

Anonymous

Anonymous Sun 26 Apr 2020 23:11:52 No.11607091 Report

Quoted By: >>11607370

>>11607038
I'm fairly sure that the manipulations he did are total nonsense (he cancelled the d of the dx notation keeping only the x at one point)

Anonymous

Anonymous Mon 27 Apr 2020 01:02:03 No.11607354 Report

Quoted By:

>>11606609
Yes, OBVIOUSLY.

Anonymous

Anonymous Mon 27 Apr 2020 01:04:57 No.11607363 Report

Quoted By:

>>11606956
Still applies with the sum of all forces and overall acceleration.

Anonymous

Anonymous Mon 27 Apr 2020 01:07:22 No.11607370 Report

Quoted By:

>>11607091
>cancelled the d
lmao didn't even notice that before

Anonymous

Anonymous Mon 27 Apr 2020 01:15:17 No.11607392 Report

Quoted By:

>>11607006
R is the Ricci scalar.
$/omega$ is the Riemannian volume form.
(Performing a volume integral of any kind requires not just a scalar, but a volume form.)
https://en.wikipedia.org/wiki/Einstein–Hilbert_action

Anonymous

Anonymous Mon 27 Apr 2020 02:31:23 No.11607641 Report

Quoted By:

>>11606869
>laughs in F = dp/dt

Anonymous

Anonymous Mon 27 Apr 2020 04:30:54 No.11607876 Report

Quoted By:

>>11606246
What is L?
Couldn't you add a factor of d/dt[A(q(t),t)] to L to get another solution?

Anonymous

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Anonymous Mon 27 Apr 2020 12:55:55 No.11608877 Report

Quoted By: >>11612311

>>11606246
$\sum \limits_{k=0}^\infty k = \frac{e^{\pi i}}{11.999...}$

Anonymous

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Anonymous Mon 27 Apr 2020 16:24:18 No.11609519 Report

Quoted By:

>>11606609
Why not Euler's formula, which is more general? I could just as easily say e^i2pi = 1, or e^ipi/2 = i, or any number of equally beautiful identities. Why do people get so hung up on e^ipi = -1 when the general formula is much more beautiful and useful?

Anonymous

Anonymous Mon 27 Apr 2020 16:27:50 No.11609530 Report

Quoted By:

oh god. you just knew some normie would bust into this thread and go MUHHH OILERS EQUASCHUNNNNNNN

imagine not knowing more than complex numbers and thinking you are hot shit cus of it i cant READ IT ANYMORE

Anonymous

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Anonymous Tue 28 Apr 2020 00:39:22 No.11610962 Report

Quoted By:

>>11606723
>>11606953
BEHOLD, the most beautiful equation in all of physics.
$v_{\text{ave}}=\frac{1}{n}R^{2/3}S^{1/2}$
Where R is hydraulic radius and S is head gradient (that is, slope in feet per feet). "When field inspection is not possible, the best method to determine n is to use photographs of river channels where n has been determined using Gauckler–Manning's formula"

Anonymous

Anonymous Tue 28 Apr 2020 00:56:53 No.11610999 Report

Quoted By:

>>11606724
Maybe if you set 1/G as 8 pi and c as 1

Anonymous

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Anonymous Tue 28 Apr 2020 12:02:04 No.11612251 Report

Quoted By:

>>11606246
fixed

Anonymous

Anonymous Tue 28 Apr 2020 12:11:42 No.11612269 Report

Quoted By:

>>11606724
based GR poster

Anonymous

Anonymous Tue 28 Apr 2020 12:15:04 No.11612276 Report

Quoted By: >>11612978

>>11606577
This. It completely destroyed my mind that the distribution inside a surface doesn't matter for the field it produces outside the surface. It's incredible.

Anonymous

Anonymous Tue 28 Apr 2020 12:16:32 No.11612278 Report

Quoted By:

>>11606389
this

Anonymous

Anonymous Tue 28 Apr 2020 12:32:50 No.11612311 Report

Quoted By:

>>11608877
>double dubs
confirmed convergence, RH BTFO

Anonymous

Anonymous Tue 28 Apr 2020 17:35:10 No.11612978 Report

Quoted By: >>11613238 >>11613265

>>11612276
The distribution does highly influence the field.
But the flux is conserved for all closed shapes
around the distribution.

Anonymous

Anonymous Tue 28 Apr 2020 18:49:35 No.11613238 Report

Quoted By: >>11613265 >>11613279

>>11612978
As far as I can remember, it does not. The field produced outside the surface is the same field you would have gotten if all the charge was concentrated at one point in the center of the surface.

Anonymous

Anonymous Tue 28 Apr 2020 18:54:33 No.11613265 Report

Quoted By:

>>11612978
>>11613238
forget it, I just thought harder about it and it was only like that because in EM you would exploit the symmetry of well behave distribution where the charges and in equilibrium.
It's a lot less counter-intuitive when you think like that, and it's actually something really easy to visualize and it's something "natural" mathematically. I was actually under the wrong impression that despite no symmetry in the distribution, the vector field outside would still exhibit symmetry as if the charge centered

Anonymous

Anonymous Tue 28 Apr 2020 18:59:09 No.11613279 Report

Quoted By:

>>11613238
You're thinking of harmonic potentials, no? Stokes relates boundary behaviour with inside behaviour

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