/sci/ - Science & Math

[Anonymous No.7371288]

What's you favourite little bit of maths/physics?

Be it a formula, an expression, or a method

Pic related

[Anonymous No.7371313]

>>7371288
The canonical ensemble (specifically, the probability distribution) and its derivation, in stat mech.

[Anonymous No.7371321]

N = NP

[Anonymous No.7371333]

>>7371313
This
It feels so trivial but its not

[Anonymous No.7371365]

>>7371288
Newton-Rhapson

[Anonymous No.7371396]

>>7371313
isn't just a taylor series expansion?

[Anonymous No.7371412]

>>7371288

<span class="math">E^2 = (mc^2)^2 + (\rho c)^2 = (\gamma mc^2)^2[/spoiler]

Although I don't know how to apply this to quantum mechanics.
I'm aware replacing <span class="math">E[/spoiler] and <span class="math">\rho[/spoiler] with their thingymadoobiwotsits would give the correct equation for particles with no spin but it doesn't work for particles with spin.

[Anonymous No.7371429]

I find it stunning that it is useful to consider fields <span class="math"> F_{q^n} [/spoiler] as vector spaces over <span class="math"> F_{q} [/spoiler] instead of a larger field containing it. Gaining insight by losing structure. Not very difficult to understand but still surprising.

[Anonymous No.7371439]

>>7371412
Why the hell are you using rho for momentum?

[Anonymous No.7371442]

The monotone convergence theorem.

Also probably the fact that on their domain all holomorphic functions are infinitely differentiable/represented by a power series.

[Anonymous No.7371471]

>>7371288
Lagrangian Formalism. Modern physics would probably be lost without it.

[Anonymous No.7371474]

>>7371288
Gödel Incompleteness Theorem

It's like being given some rules that are supposed to prevent cheating in a game and then proceed to cheat like a motherfucker by using these exact same rules.

[Anonymous No.7371526]

>>7371429
Why do you artificially restrict this to finite fields?

[Anonymous No.7371554]

>>7371526
probably because I worked exclusively with finite fields for the past year or so. Obviously it's the same for general fields, as you mentioned.

Another interesting thing with finite fields: all functions from a finite field on itself are polynomials (and you can just construct it with a formula akin to the Lagrange representation on the real numbers)

[Anonymous No.7371598]

>>7371554
Of course, polynomial interpolation with <span class="math">\#\mathbb F_q[/spoiler] supporting points :-). Quite nice nevertheless. In what context do you work with finite fields?

[Anonymous No.7371610]

>>7371598
I do combinatorics with algebraic methods and finite geometry and in both of them you pretty much exclusively work with finite fields.

[Anonymous No.7371612]

>>7371288
I saw a neat proof of the quadratic reciprocity law using a clever counting argument, really short and elegant

[Anonymous No.7371613]

i think the diagonal argument is pretty cool

[Anonymous No.7371624]

>>7371288
scalar products and projections of about anything. pic related

[Anonymous No.7371927]

>>7371396
gr8 b8 m8

[OHP No.7371928]

>>7371474
In what contexts have you used the first incompleteness theorem? Honestly, Gödel's completeness theorem is far more remarkable. And in the vein of model theory, the Löwenheim-Skolem theorem is wild.

In response to OP, I think cohomological algebra in general is very beautiful and profound. Assigning algebras to each point in a space, and then manipulating chains of these algebras (cochain complexes) is not only novel and useful, but also a taste of where mathematics will be a hundred years from now.

And as I always say, I think the Yoneda lemma, while a rather obvious result in the right environment, is incredibly useful almost any time you are working with categories.

[Anonymous No.7372015]

Maxwells relations in thermodynamics.

Very simple and easy stuff.

[Anonymous No.7372042]

>>7371927
im serious. i've seen two sources where it's just a taylor series in the entropy.

[Anonymous No.7372047]

Barnett's Identity

[Anonymous No.7372052]

>>7371321
thats cool, I like P=NP

[Anonymous No.7374059]

>>7372042
Its not a Taylor series but you assume some things remain constant, which were the temperature and the number of particles if I remember correctly

[Anonymous No.7374096]

>>7371928
>Assigning algebras to each point in a space
It's more just cohomology if you're working with spaces. Homological algebra applies that to algebraic things.

[Anonymous No.7374118]

>>7371928
Hard to say which is "more remarkable". As you say, the incompleteness theorem is more of a result than a tool, but an extremely strong one at that.

And I don't have hopes a sheaf-y perspective on a broader range of math will even be made simple enough to understand, so that it's taught early and brings out the structural beauty of things.
Then again, I believe sets will be turned turn in the future, in favor of "more computational" working enviroments.

[Anonymous No.7374119]

>>7371313
FIXED NUMBER OF PARTICLES

FIXED TOTAL ENERGY

FIXED VOLUME

[Anonymous No.7374120]

>>7374059
Temperature falls out of the derivation, you don't assume anything about it beforehand because you have no idea what temperature even is.

[Anonymous No.7374122]

>>7374119
>FIXED TOTAL ENERGY
That's the microcanonical ensemble m8.

[Anonymous No.7374126]

>>7374059
Microcanonical: N, V, E
Canonical: N, V, T
Grand canonical: µ, V, T

Constructing the microcanonical ensemble is trivial because each system is isolated. Each system in the canonical ensemble can exchange energy, so you specify a constant T. In the grand canonical ensemble, both particles and energy can be exchanged, so you additionally specify that each system has the same chemical potential.

[Anonymous No.7374128]

i remember finding the babylonian method alone and just having fun with it.

it was just one of those moments where you have to marvel on math

[Anonymous No.7374131]

>>7374126
Mean energy is fixed, not temperature. Sure, these turn out to be the same thing but that's what we find out later when we attach some meaning to this new parameter T we pick up.

[Anonymous No.7374218]

>>7374118
I actually hold the same sentiment with regards to sets. I think they are too synthetic for a foundational language. Homotopy type theory is interesting, but even that seems to fail at revealing the nature of the objects we study. Speculating about what the future of mathematics holds is a largely futile game, though.

Regarding the completeness and incompleteness theorems, I think the completeness theorem is more remarkable only because it defies my (and I would expect it to define others') intuitions far more. Even after reading the proof of the statement, it feels like some sort of magic is happening. With incompleteness, once I learned the proof, I no longer felt mystified at all. Just glad that we have the result under our belts as a community.

>>7374118
Eh, flip the arrows and what is the difference, right?

[Anonymous No.7374463]

i've been playing around with this action

<span class="math">
S = \dfrac{k}{4\pi}\int_{\Omega}Tr(A\wedgeA\wedgeA+\dfrac{2}{3}A\wedgedA)
[/spoiler]

[Anonymous No.7374467]

i've been playing around with this action

<span class="math">S=\frac{k}{4\pi}\int_{\Omega}Tr(A\wedgeA\wedgeA+A\wedgedA)[/spoiler]

[Anonymous No.7374470]

i've been playing around with this action

<span class="math">S=\frac{k}{4\pi}\int_{\Omega}Tr(A\wedge A\wedge A+ A\wedge dA)[/spoiler]

[Anonymous No.7374474]

i've been playing around with this action

<span class="math">S=\frac{k}{4\pi}\int_{\Omega}Tr(A \wedge A \wedge A + A \wedge dA)[/spoiler]

anyone recognize it?

[Anonymous No.7374502]

>>7374120
>you don't assume anything about it beforehand because you have no idea what temperature even is
>what is the First Law?
I think I meant volume
dU = TdS - P dV - mu dN
dN = dV = 0, etc

[Anonymous No.7374505]

[Anonymous No.7374507]

>>7371288
Rotations in 3 space.
quaternions, rotation matrices and angular velocity
Don't really know why.

[Anonymous No.7374510]

rimeann zeta because its the only not gay way we have of primes

>hurrdurr inb4 not proven conjecture
fuck off, Im not an autistic mathematician that needs to jerk himself off before using any equation

[Anonymous No.7374513]

Pick related, it reminds me of myself, intelligent, nihilistic and with a wicked sense of humor.

[Anonymous No.7374515]

not favorite, but weird: RP^2's fundamental group has a torsion element, and it's not even a very weird space (it's a manifold, even)

[Anonymous No.7374523]

>>7371442
monotone convergence is just plain pleasant

[Anonymous No.7374524]

Hamilton-Jacobi in classical mechanics comes to mind.

>>7374513
Am I on /a/ right now?

[Anonymous No.7374525]

>>7374505
>actual science and math
fuck off, /sci/ is for popsci and pseudoscience

[Anonymous No.7374560]

>>7374525

[Anonymous No.7374697]

>>7374474
Well if you add in a factor of 2/3 in front of A^A^A then you have the action of Chern-Simmons Theory.

[Anonymous No.7374699]

>>7374697
meant to but i'm poop with latex