the runniness of the egg actually facilitates the identification of distinct cauchy sequences in the same equivalence class
#serious-discussion
static loom
although Cauchy didn't have access to chopsticks at the time, he had sausage links and bacon which were effectively the same, even if it isn't the modern construction
fair mural
i can’t tell if my jokes are funny or annoying
neat lintel
they are funny
the joke above your inquiry was unfunny as an example
not necessarily annoying
mint patio
I think Mero is very funny 
fair mural
static loom
I'm fairly ignorant about it but I think it's playing off a silly meme of saying everything is 1984 around here lately lol
static loom
i hate being called elitist lol
that's exactly what an elitist would say 
leaden torrent
cold needle
that's exactly what an elitist would say <:catshrug:563818735085944843>
lmfao ok fair point 😭
tall badge
the runniness of the egg actually facilitates the identification of distinct cau...
pls elaborate
honest veldt
pretty self explanatory, I think
velvet dagger
that's exactly what an elitist would say <:catshrug:563818735085944843>
That's what a level 1 elitist would say
The correct thing is to simply acknowledge that you're the elite and it's okay to be elitist as a result
wild lantern
What exactly does high level elitism look like?
pale orchid
you're too far below it to even recognize it
wild lantern
The truth hurts lol
cold needle
damn lmao
pale orchid
i thought it was gonna be funny as a self-fulfilling example, but i guess not
wild lantern
No, I found it pretty funny lol.
real sigil
what is the value of i?
bronze pelican
Value?
fair mural
i guess just $\sqrt{-1}$
fathom swallowBOT
quantum
fair mural
but i don’t know a rigorous formal definition
cold needle
one way to construct C is to take R^2 and define a multiplication on it
and then you can introduce a notation which leads to i^2 = -1
scenic narwhal
R[x]/(x²+1) 😌
fair mural
scenic narwhal
i = equivalence class of x
cold needle
i am read dummits and footes in spring in class
bronze pelican
The chapters on field theory and galois theory
cold needle
ic
bronze pelican
And watch my lectures
https://youtube.com/playlist?list=PL6V-xJKDlCJh0SxgmwQ0ryXu_JnOlt2Rk
cold needle
😼
neat lintel
i need to learn field extensions 😔
no now
fields are rings where every element has an inverse
commutative also
all ring homomorphisms between fields are injective
so you can think of each field homomorphism as being domain imbedded in its target
think of Q-> R for example
field extensions are just this rly
but usually extending by polynomials
when i say extending by polynomials
scenic narwhal
¿does metal know ring theory? 
neat lintel
i mean quotienting a polynomial ring by a polynomial with a root that isnt contained in the base field
dont you need to know ring theory to be mod?
scenic narwhal
I doubt it, moth is a mod too
cold needle
i don't know ring theory yet
neat lintel
are you in college?
cold needle
i have just completed learning basic group theory
yes
second year undergrad
going to spring 😼
neat lintel
what type of math do you like?
neat lintel
😎
scenic narwhal
going to spring 😼
Allow me to flex, I learned ring theory in Fall of second year 
cold needle
wowza
neat lintel
Allow me to flex, I learned ring theory in Fall of second year <:smugsmug:812884...
same
but im third year
and ur an adult or smnth now
surreal sapphire
is fall semester before spring semester
cold needle
yes
scenic narwhal
He has ninja in his name
scenic narwhal
You can, just explain ring theory
Metal, come back here, we need to explain the first isomorphism theorem (for rings) to you 
It will be discussion-2's greatest moment
surreal sapphire
pls ping me when you get to noether normalization, i needed a refresher recently but forgot to look it up
velvet dagger
Noether normalization is coo
I forget the proof but I remember my AG prof basically explained it as part of the proof of the Nullstellensatz
That every affine variety is a branched cover of some A^n
neat lintel
bronze pelican
That every affine variety is a branched cover of some A^n
That sort of makes sense
But branched over some subvarieties I guess
surreal sapphire
i think i needed krull dimension = transcendence degree
i decided to just believe it
neat lintel
hi i have questions
say im trying to convince a science enjoyer of the type of math i like
what topics do i talk about
i think I like AG
and i get general gist of the goals of AG
i also like AT, but a lot of stuff is unmotivated
whats the best pop sci thing i can say besides coffee cup is donut
leaden torrent
nullstellensatz
neat lintel
im talking about laymen
anything fun i can tell them about similar to donut and coffis cup
velvet dagger
27 lines on a cubic surface?
mortal igloo
An algebraic isomorphism from the unit circle over Q to the projective line over Q
And its connection to Pythagorean triples
jovial ember
Say a continuous stream of incoherent math buzzwords to appear smart at the cost of making them feel stupid
chrome goblet
whats the best pop sci thing i can say besides coffee cup is donut
Banach-Tarski paradox
neat lintel
isnt this diff top tho?
chrome goblet
Oh nvm
chrome goblet
I didn’t read AG
leaden torrent
nullstellensatz is totally approachable to a layman
not using the formal terms obviously
but i think you could introduce the ideas in 30 minutes if youre experienced
not as visual as ONE DONUT EQUALS ONE COFFIS CUP, certainly
sterile hatch
idk man everything is a buzzword to me
honest veldt
is riemann hypothesis the string theory of math?
surreal sapphire
string theory is the string theory of math
honest veldt
crap
No, no, too application-y
No one would think of math before physics when they hear string theory
surreal sapphire
a physicist might
honest veldt
I suppose so yeah
But we're talking veritasium-sauce-gezagt level stuff right now, so that's a bit beyond the target demographic
spring wyvern
No one would think of math before physics when they hear string theory
i can guarantee that the first thing that they think of is strings.
bold ferry
No one would think of math before physics when they hear string theory
Not true
sly mason
Sorry to ping you @carmine bane , but when I tried this: #help-2 message It doesn't work:
icy forge
The first argument is x, of which you provided none.
You provided a named argument k, which goes into ..., and so you have nothing for required first-position argument x
carmine bane
add == instead of =
sly mason
ohh, thanks lol
faint gale
What is the definition of a unit group?
sly mason
add == instead of =
The simulation does not seem to match this formula. I've triple checked I typed everything corretly
sly mason
carmine bane
remove the 1-
sly mason
Starts giving me strange stuff like this
Could it be because I'm approximatiing the infinite series?
Wasn't a problem on the last one, I checked
carmine bane
$\sum_{n=10}^\infty \frac{25^n {n \choose 10}0.07^{10} 0.93^{n-10}}{e^{25}n!}$
fathom swallowBOT
ScapeProf
carmine bane
**ScapeProf**
This is P(N>=10) * P(k=10 | N>=10). Since if N<10 then P(k=10)=0.
sly mason
So that should give me the correct formula for finding the probability of getting a specific number of ice cream?
What does the " | " mean in this context?
"and" maybe?
bold ferry
Again, in the context of pop-physics, pop-math stuff, it's probably true
in that context ya. definitely
carmine bane
given
sly mason
oh, yeah that makes sense
carmine bane
So that should give me the correct formula for finding the probability of gettin...
and yes
carmine bane
,w sum from n=2 to infty of (25^n(n choose 2)0.07^(2)*0.93^(n-2))/(e^(25)n!)
fathom swallowBOT
sly mason
I didn't expect it to be so close to a binomial distribution that assumes p to be certain (the green)
real sigil
what are some not very common yet understandable logarithmic properties
fair mural
a+bi
real sigil
a complex?
fair mural
oof
sterile hare
No this is pretty common
most people here are high school level in case you didnt know
fair mural
better than geometry
real sigil
geometry is pretty neat ngl
deep mango
It's a pretty sunset right now
real sigil
guys my desmos website is crashing when i try f(x) = x^x^x^x^x^x^x^x^x^x^x^x^x^x^x^x^x^x^x^x
and then f(n) for 1/e^e < n < eth sqrt of e
fair mural
how surprising
real sigil
i am testing out the properties of e
but its lagging
does anyone know what n converges to?
fair mural
this is actual trash i’m not sorry
real sigil
what is trash?
static loom
type in y=x^y into desmos
real sigil
whats that
static loom
it's easier to work with x(y) instead of y(x)
what x^x^x^... will converge to
you can find the rightmost point on y(x) by thinking of it as the minimum of x(y) and use a lil calculus
real sigil
how does this prove that sqrt 2 converges to 2
static loom
it doesn't prove convergence
only shows that if y=x^x^x^... then it must be true that y=x^y
so (sqrt(2),2) satisfies this, which is a good thing to know going in to trying to prove it
real sigil
make sense
static loom
I'm guessing you can use the contraction mapping theorem to prove it, might be other ways
real sigil
in the first cell i wrote y=x^y
and it shows the same graph that x^x^x^x^x shows ( i think)
static loom
well y=x^y is not a function, fails the vertical line test
real sigil
now how do i get the y's of numbers in other cells
static loom
it turns back around to the left after going right
static loom
you can't invert a function in desmos
real sigil
sad
static loom
and there's no lambert W function either, but you could approximate either
real sigil
yeah when i did 1 converges to 1
sqrt2 converges to 2
one last thing, are the numbers between (1/e^e and eth root of e) going to converge to something specific?
because i read that only the numbers within this domain can coverge in the x^x^x^x^x^x... function
while the rest will be undefined
nimble shuttle
most people here are high level in case you didn't know
sharp mulch
Jon was responding to an earlier comment
static loom
helps if you know some calculus first
real sigil
i mean, i agree with you but, i expected an expression with respect to x as an answer to x^x^x^x^x^x...
because this guy didnt say what it was and just said that sqrt 2 converges to 2
crystal stream
there's a simple test to know if a given function f(x) has a fixed point
if you can find a given interval [a,b] such that the image of that interval under the funcion f is contained within it (i.e, f maps points in [a,b] to other points in [a,b]) and if |f'| < 1 on [a,b]
then you are guaranteed a fixed point in [a,b], as in, there will be at least one point satisfying f(x) = x
if I remembered the theorem from my numerics course correctly...
mint patio
@crystal stream I'll move what I said here since I don't think it's appropriate for that channel
Do you mean +-?
You're not going to call two numbers approximate unless the error is relatively small to their magnitudes anyways, so at that point I don't think it really matters. It's just space-consuming. And if the error is small but you still need to care like in a scientific setting then why not just write a = b +- epsilon anyways? That's standard. Or if you're making an approximation you will probably have justified why you made that approximation anyways and have provided error bounds earlier anyways
Or just write a \in [b - epsilon, b + epsilon] or b - epsilon <= a <= b + epsilon
crystal stream
yeah not really advanced lol
I was thinking of having the $\approx_{\epsilon}$ as compact notation for "everyday use", to quickly communicate how much you're truncating a result
fathom swallowBOT
random variable
crystal stream
it's really not that useful, just seemed cute to me tbh
sharp mulch
I've seen people wrote say $a=b+O(0.0001)$ for this
fathom swallowBOT
陆景和
sharp mulch
Of course this is an abuse of notation
crystal stream
if someone wrote that to me I would understand it but yeah
sharp mulch
That's fine, it gets the meaning across and notation was invented to be abused
crystal stream
heck i'm pretty sure $f(x) = O(1/x), x \to 0$ is abuse of notation since $O(1/x)$ is a set
fathom swallowBOT
random variable
crystal stream
most of the time you can manipulate $O()$ expressions and get meaningful results though so its justified in some sense
fathom swallowBOT
random variable
sharp mulch
Yeah but like no one cares because writing f=O(1/x) mirrors how we say this
We say that f is asymptotically 1/x, we don't say that f is a member of the functions that are asymptotically 1/x
static loom
but you can do f(y)
so what
if you're looking at y=x^y it's simple to write x(y)=y^(1/y) but there's no inverse to easily get y(x) so that you can just plug in x values to get y values in return
real sigil
f(y) = +-x
real sigil
ok so
uhh
i know how to do derivatives so some extent
now i, learned that we have secant lines that pass through the curve within 2 points ( when f(x) = f'(x) if i am not wrong)
and we have tangent line, which touches one point of the curve
my question is, what is the relation between tangent lines, secant lines, and derivatives
like, does the derivative of a given function represent anything on its own?
or not until we use to calculate the tangent line
dire mulch
Secant line is just using the difference quotient for slope of the line, the tangent line uses the limit of the difference quotient aka derivative of the function for the slope of the line.
real sigil
im not sure i fully comprehend that
acoustic surge
a straight line is a function like ax+b, here a is what represents the "inclination" of the line.
the derivative f'(x) of a function f gives you the a of the straight line tangent to f at the point x of your chosing
dire mulch
when i did f'(x) on desmos it gave me the secant line
Because that is a different function than the line you want.
acoustic surge
i'd assume desmos doesn't have built in diferentiation does it?
real sigil
Because that is a different function than the line you want.
so f'(x) gives the secant line?
fair mural
i'd assume desmos doesn't have built in diferentiation does it?
it does
real sigil
dire mulch
To get the tangent line y = f(a) + f’(a)(x-a)
acoustic surge
so f'(x) gives the secant line?
no
real sigil
no?
real sigil
does every point have a tangent line
dire mulch
does every point have a tangent line
Depends if the function is differentiable on a given subset of it domain.
acoustic surge
real sigil
every place where f(x) is defined
i see
real sigil
Depends if the function is differentiable on a given subset of it domain.
assuming that its defined to be real
as long as the given argument is real the function is valid
dire mulch
tangent.
acoustic surge
i mean, you can say that but uhh
real sigil
but im still confused
coarse grotto
desmos is needlessly powerful
but also not at the same time
dire mulch
For sectant line exchange f’(a) with a new function g(a) = $\frac{f(a+h)-f(a)}{h}$. f’(a) would be the limit as h$\to$ 0 for g(a).
fathom swallowBOT
Plegasus
acoustic surge
but im still confused
since the tangent line extends to both sides indefinitelly it is bound to hit another part of the function if it allows for that, you can call it a secant sure, but that's not the intention of the construction
real sigil
**Plegasus**
got it
real sigil
since the tangent line extends to both sides indefinitelly it is bound to hit an...
i mean
deep mango
🤨
acoustic surge
it approximates the function near p
real sigil
not necessarily
deep mango
Convexity moment
real sigil
the line doesnt have to touch the curve twice
acoustic surge
yup
real sigil
but even if it does its considered a tangent?
deep mango
It could touch it as many times as it wants, but locally (if you zoom in far enough) it's just once as long as the function isnt just linear itself
Maybe locally tangent is a better way to describe it
fair mural
i would say that’s a better term
deep mango
A derivative has no idea what's going on anywhere outside a little ball around the point where you're taking the derivative
real sigil
so the concept of a tangent is respective to a certain area
deep mango
Yeah
fair mural
a tangent line could actually touch like 20 other points on the graph
it’s just tangent at that point
deep mango
Tangent line to cos(x) at 0
real sigil
so each point has its own tangent
deep mango
Yep!
fair mural
oof
real sigil
but some points can share the same tangent?
deep mango
Yeah
real sigil
got it
deep mango
I like to think of little tangent line segments
acoustic surge
Maybe locally tangent is a better way to describe it
should one meantion that when teaching calculus at all? i wonder if this confusion can really trouble ppl
deep mango
So that I dont mess up where my tangent is supposed to be
coarse grotto
I honestly didn't even consider it until you brought the confusion up
deep mango
should one meantion that when teaching calculus at all? i wonder if this confusi...
Yeah it doesnt really warrant mentioning imo. If a student is clever enough to ask then I'd definitely explain it to them this way though.
real sigil
should one meantion that when teaching calculus at all? i wonder if this confusi...
personally, im not sure, but there must be some people who didnt grasp the idea of tangents and secants, hence they might get this confusion
wicked ore
should one meantion that when teaching calculus at all? i wonder if this confusi...
I think it is confusing
Yeah if you ask about it, then you should answer like this
But it really doesn’t have anything to do with the derivative
deep mango
Like if you go "wait a minute, this is not the same kind of tangent that I learned about in geometry..." then I would talk about this with them. But not otherwise.
real sigil
what about the secant line? does every curve accept one secant line only? (for example f(x) =x^3 )
wicked ore
Secant lines require exactly two points on the curve. If it intersects more points, so be it, but it must intersect the first two
deep mango
Well, distinct pair
Yeah. In that way some secant lines are also tangent lines.
(if the line through one of the two points just happens to be the tangent line through that point too)
wicked ore
Yeah, the secant between sin(0) and sin(2pi) is also a tangent line
real sigil
tangent to what?
deep mango
Yeah, the secant between sin(0) and sin(2pi) is also a tangent line
Well, maybe you want cosine
wicked ore
Tangent to sin(0)
real sigil
oh
wicked ore
Right
real sigil
is there a symbol for tangent lines and secant lines
deep mango
But only in another context
wicked ore
But we should move away from thinking about derivatives as lines and more as slopes
It’s not about the line
Nobody really cares about the line
deep mango
People in "differential geometry" talk about "tangent spaces" to things like curves or surfaces. So in that context you would call the tangent line to a graph "T_x Graph(f)" where f is your function and x is the point where you want the tangent line.
wicked ore
Ah
neat lintel
fathom swallowBOT
Kanga Gang Drug Mule RYC
coarse grotto
"differential geometry" (derogatory)
deep mango
And youd probably use a gamma instead of Graph
wicked ore
Graph(f) is {(x,y) | y = f(x)} ?
fathom swallowBOT
Kanga Gang Drug Mule RYC
deep mango
Cause its cool
deep mango
Graph(f) is {(x,y) | y = f(x)} ?
Yeah exactly! Just the relevant subset of the plane.
Some might say the function IS its graph
wicked ore
So in that case the function is really a set, like everything else
real sigil
soooo
what is a derivative
like, by itself
is it just a calculation used to calculate tangent and secant
or something else
acoustic surge
it tells you the local behaviour of a function
that of a line with "inclination" f'(what ever point)
if it exists lol
neat lintel
it tells you well, the slope of whatever curve at a certain point, or the slope of a line tanget to it
tall badge
did anyone mention best linear approx yet
deep mango
I like to think of it as telling you the velocity of the function / the rate of change of the function at that point.
Other people like to think about it as the best linear approximation starting from some point.
Actually I like to do that too.
rich bay
Let’s say u have function f(x) and two x values x1,x and x1 is bigger than x but by an amount infinitesimally small the value found by (f(x1)-f(x))/(x1-x) is derivative
acoustic surge
did anyone mention best linear approx yet
i mentioned approx only 
rich bay
Let’s say u have function f(x) and two x values x1,x and x1 is bigger than x but...
So this would be basically what I said in equation form
coarse grotto
i mentioned approx only <:sully:651816820122189834>
It's not the best approximation bc differentiable functions can be arbitrarily approximated by polynomials
But linear approximation, yes
fair mural
So this would be basically what I said in equation form
these are the same thing
rich bay
these are the same thing
Yeah I was just providing visual representation of my explanation of derivative
Like the equation and stuff
neat lintel
@real sigil if you're just getting into calc
real sigil
yeah?
fair mural
yeah i used pauls notes
neat lintel
try searching up professor leonard, he has a lot of lectures posted with well explained stuff
rich bay
Paul was a homie from calc 1 to diff eq
real sigil
alrighty
neat lintel
its a little long but you get the idea
rich bay
try searching up professor leonard, he has a lot of lectures posted with well ex...
I second this his diff eq
Videos were p good
neat lintel
its how i learned calc
YOO PAUL my boy got me through intro to ODEs
i mean im just 14 so i didnt rly have any other option to learn calc
real sigil
you learnt calc and youre 14?
neat lintel
yeah, got a bit too attached with maths
rich bay
Damn u must be cracked to start calc at 14 lol, I started at 16 ur fast
real sigil
well i was 14 two months ago so hopefully ill get it together
neat lintel
Damn u must be cracked to start calc at 14 lol, I started at 16 ur fast
i thought it'd be harder but honestly its not too impressive since i have much free time
rich bay
Damn
real sigil
i thought it'd be harder but honestly its not too impressive since i have much f...
do you study trigonometry too?
neat lintel
i really only have identities and unit circle stuff memorised, idk much about trig itself
rich bay
U must be fast lol, I think I’m bit behind curve for math cuz I’m starting real analysis 2nd sem of sophomore yr of college lol
real sigil
i used to hate it at first, but just after 8 hours of binge studying it, i started liking it
lol
neat lintel
i know the soh cah toa
ofc
but ehh
trig isnt too much my thing
but after seeing how much trig is in calc 2 i def will do more
/calc 1
real sigil
yeah
rich bay
Lol it’s weird but I haven’t used too much trig in my upper level math classes
real sigil
cause theres like Sec, Csc Cot, in addition to their hyperbolic functions
neat lintel
yea
real sigil
Lol it’s weird but I haven’t used too much trig in my upper level math classes
this year is my first year studying radians, im using this 3 weeks winter vacation to learn it
ngl switching from degrees to radians is kinda weird
but its not really that hard when you look more to it
rich bay
Ye but u get used to it
real sigil
yeah
real sigil
lmao yeah
modern oasis
Hm, do you think this question is suitable for 12 years old?
Wait I think it's not suitable to post in here
modern oasis
Oh, I see
modern oasis
Hm, do you think this question is suitable for 12 years old?
It's pretty easy for me tbh
fair mural
i would just guess and say 38
real sigil
take fof x= diameter
we have
x+12 + x + 12 +x = 22 +x + 16 +x + 22
12 + 12 +x +x +x = 22 + 16 + 22 +x +x
24 + 3x = 60 +2x
24 +x = 60
x = 60 - 24
x = 36
fair mural
huh
wicked ore
How do you know the circles are all the same size
real sigil
lol
fair mural
it says identical
real sigil
yeah
fair mural
take fof x= diameter
we have
x+12 + x + 12 +x = 22 +x + 16 +x + 22
12 + 12 +x...
oh i see
that’s smart
modern oasis
take fof x= diameter
we have
x+12 + x + 12 +x = 22 +x + 16 +x + 22
12 + 12 +x...
Yep, correct
fair mural
well i couldn’t have thought of it
fair mural
obviously
most if not all people you show this to will get the answer wrong
or will have the wrong process
real sigil
yeah
modern oasis
Hm, do you think this question is suitable for 12 years old?
This is an psle exam question in Singapore
for 12 yo
real sigil
smh
fair mural
probably for advanced 12 year olds
fair mural
iq means nothing
fair mural
sure
real sigil
if you had to hire one guy with 180 iq or, someone with 90 iq
real sigil
would you take a random blind choice or rely no the iq
wicked ore
Here’s a pretty hard question for a 12 yo
real sigil
white triangles = 15
grey triangles = 10
total = 25
wait what the heck
how is question b and c for kids
????
wicked ore
Given two dots with a horizontal line below them, what is the shortest path between the two dots that touches the horizontal line exactly once?
The blue line is an example of such a path
sharp mulch
Snell's law
wicked ore
You definitely don’t need that
sharp mulch
Finding the optimal bouncing point is equivalent to rederiving snell's law
wicked ore
Sure, but find a way to do it without snell’s law
And convince me that it’s the shortest path
modern oasis
we will have 63001 triangle in figure 250
How did you calculate it?
real sigil
summation
a summation from n=0 going until m for equation of 2n +1
where m is basically the number of figures
modern oasis
Totally not for 12 yo
real sigil
a summation is the way to write it out
you can sumplify it to (1 + 3 + 5 + 7 + 9 .... ) 250 nomial
so just grab your calculator and sum the first 250 odd numbers up
or use a sigma
wicked ore
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
…
Adding up n odd numbers gives you n squared.
modern oasis
What’s PSLE
Primary School Leaving Examination
wicked ore
Primary school?
modern oasis
yes
wicked ore
God that’s difficult for primary school lol
modern oasis
Exactly
real sigil
why?
real sigil
^^
wicked ore
So going from 0 to 250, you’re adding up 251 odd numbers
real sigil
yeah
wicked ore
Given two dots with a horizontal line below them, what is the shortest path betw...
Ok back to my problem
Can anyone figure this out
mild nebula
Call the point where the path hits the line "x". Get a distance formula for the blue line, which will include x. Use derivative to optimize
That sound legal? @wicked ore
coarse grotto
😎
real sigil
who speaks arabian here?
honest veldt
امونج اص
pallid thorn
just curious, is there any way to plot a function whose input is another function like derivatives of polynomials as a 3d graph or is it just stupid
static loom
I think I see what you're asking
pallid thorn
like d/dx x^n = nx^(n-1)
as a function you can plot in 3d space or some higher dimension
static loom
like instead of the x and y axis, you have differentiable functions for the x and y axis
and somehow you're graphing the derivative as a function
pallid thorn
^
static loom
kind of difficult to do I think in any way that is easy to visualize
pallid thorn
but i'd imagine it would be a smoothly changing function
which should have a nice shape in high dimensions
static loom
it might help to try to imagine what makes two functions nearby as points
bronze pelican
and somehow you're graphing the derivative as a function
oh
pallid thorn
n is the parameter and somehow that should map to the derivative
static loom
if you define distance as something like $$d(f,g):=\sup_{x\in X} |f(x)-g(x)|$$ then your "x axis" and "y axis" are quite strange
fathom swallowBOT
Merosity
static loom
sort of like an infinite branching tree at every point I guess
and that's just describing the "axes" not the graph of the derivative itself yet
pallid thorn
and this is just for polynomials, i can't think of a way to connect all functions to one big graph, like what would be separating cos(x) from e^x in that space
bronze pelican
if you just consider the space of linear polynomials ax+b indexed by points in the plane (a,b), then the derivaive sends the point (a,b) to the number a. And the graph of this will be a plane.
static loom
I'm saying the domain is all differentiable functions on some compact set I guess, so that the sup is going to be defined, I think
static loom
if you just consider the space of linear polynomials ax+b indexed by points in t...
yeah maybe I'm being a bit extreme, that sounds nicer to think about lol
mortal igloo
Function spaces mmmm….
mortal igloo
Uncountably dimensional vector spaces mmmmm
pallid thorn
pls send help
bronze pelican
If you have any two parameter family of functions you can visualize the derivative operator on this family of functions as a graph in 3d space
pallid thorn
visualizing this monster
mortal igloo
It maps to (0,2a,b)
bronze pelican
yeah okay, you can think the polynomials of degree d as being indexed by d-tupes of real numbers:
so for example the quadratic ax^2 + bx + c corresponds to the point (a,b,c)
Then derivative operator will act as a linear transformation of d-dimensional real space
The graph of this function will be a subset of R^d x R^{d-1}
very hard to visualize for d > 2
but its just a linear transformation on a finite dimensional vector space, so if you know linear algebra this is very easy to understand this as a linear transformation
in general the graph will be a d-dimensional subspace of R^d x R^{d-1}
When you're thinking about operators on function spaces, i dont know if its so helpful to try to visualize the graph.
because the dimensions get really big and at that point you're not going to be able to visualize much
static loom
in that case like I was describing earlier, you can kinda pretend it's like a big nasty web where two differentiable functions are close by if they are only slightly wiggled from another, not really visualizeable yeah lol
then try to imagine some representative f(x) from the set of functions of the form f(x)+C for constant C as partitioning this web apart, and so each of these sections map to the same point in the codomain 😬
fathom swallowBOT
Result:
69
gotten
thorny peak
I didn’t do too well for my first semester, having said that next semester I have to take calc 2 and matrices. Are there any resources you guys would recommend to study ahead for these two classes?
thorny peak
With how bad I’m doing, anything that’ll teach me something about calc 2 or matrices is helpful. Having said that,thank you very much!
ancient flame
no problem :)
mint patio
I didn’t do too well for my first semester, having said that next semester I hav...
a lot of people might flame me for this but I like bprp for calc 2 stuff
though he says some headass shit sometimes lulw
but I like the way he explains stuff
also if you haven't already checked out 3b1b's playlist for calculus
it's p goated
thorny peak
Alright, thank you!
neat lintel
hi
cold needle
minecraft
toxic schooner
warframe
toxic schooner
I could feel a tumor growing in my brain
mint patio
I enjoy League most of the time but that game legitimately hurt my soul
Everyone except for 3 people were aboslute utter trash
toxic schooner
the game itself or the people?
mint patio
That game specifically, I don't mind the community
Just mute the annoying people
And I enjoy League as a whole
toxic schooner
That game specifically, I don't mind the community
i see
Most people (including myself) are more annoyed of the community lol
mint patio
Yes
Idk I just got jaded I guess
Been playing for like 6 years now
And following for even longer
mint patio
Like if you're a League player you have some form of mental deficiency
That's the fact I've established
toxic schooner
i have been following fortnite since its inception (in 2017) and maganged to get the chance to play only in 2019
toxic schooner
That's the fact I've established
mint patio
And like once you accept that it becomes easier to deal with the degenerates
mint patio
i have been following fortnite since its inception (in 2017) and maganged to get...
fortnite 🤢
toxic schooner
i can see that working out yeah
mint patio
game is cool but community when it first came out
just ruined my perception of the game
toxic schooner
mint patio
comment on my typo
mint patio
uh
how do I open an email if I'm writing it to my advisor but then my advisor has a notice saying she's out of office so contact the general advising email for general questions
like is it just to whom it may concern?
She does specify "I will assist all registration emails when I return"
my question is just about currently I havbe an Incomplete in a class and I want to make sure it wont hinder my enrollment in classes for next semester
none of the classes require it as a prerequisite but some of the classes I took this semester (all of which I passed) have it as a corequisite
the university portal says I'm still enrolled in the classes for next sem so
modern oasis
Do you think this question suitable for 12 yo?
deep mango
That looks like a system of linear equations question, which could be suitable for a 12 year old taking algebra 1.
leaden torrent
Do you think this question suitable for 12 yo?
why do you ask?
fringe needle
||
||
neat lintel
Do you think this question suitable for 12 yo?
no, it s not very rigorous as stated
bronze pelican
Have u ever heard 'vulgarization' used in a positive light?
I like how vulgar both means obscene and having to do with common people
/laypeople
That feels like elitism is built into our language lol
Yes making something accessible is vulgar
light needle
Lmao thats so bad
vague lion
Have u ever heard 'vulgarization' used in a positive light?
i have never heard vulgarization used
static loom
I've heard people use it in a kind of tongue-in-cheek way to say they're simplifying something to explain it
sharp mulch
It's used in a historical context with regards to latin
fair mural
we have made progress
coarse grotto
fair mural
just change misato to ryc
i guess uh
just put a bird on there for the picture
random bird picture
deep mango
fair mural
i wasn’t kidding
coarse grotto
dude's just repurposing a misato meme into ryc 
fathom swallowBOT
neat lintel
Aight
wise rose
Do you think this question suitable for 12 yo?
I think it's fair.
wise rose
Have u ever heard 'vulgarization' used in a positive light?
Jerome's Latin translation of the Bible is commonly called the "Vulgate" from "vulgar", meaning "popular" or "common", because Greek was for scholars, Latin was the common tongue in the Roman Empire.
stray kite
That feels like elitism is built into our language lol
language is a reflection of the culture which speaks it lol what do you expect
neat lintel
,av @neat lintel
fathom swallowBOT
neat lintel
how beautiful
rotund steppe
What’s a kanga
deep mango
Kanga is a character from the winnie the pooh franchise
pale orchid
i had never heard of it referred to as a "franchise"
cold needle
cinematic universe, even
pale orchid
the winnie the pooh multiverse
surreal sapphire
its one of the highest grossing franchises, more than star wars
pale orchid
winnie the pooh multinational conglomerate, the powerhouse behind google, facebook, microsoft, and tesla
vague lion
i had never heard of it referred to as a "franchise"
well it's more of a government really
last oxide
well it's more of a government really
true
neat lintel
isnt some chinese president winnie the pooe
xi xing pi?
idk name
witcher s2e3
you see anything interesting on this screen shot?
pure sun
reddit account doxxed https://www.reddit.com/r/math/comments/rncdz9/is_this_penrose_tiling_in_witcher_s2_e3/
modern oasis
How do you do this
neat lintel
How do you do this
any more data?
modern oasis
any more data?
Nope
modern oasis
https://www.youtube.com/watch?v=-3T_NtTXx2A
oh thx
neat lintel
ahh it's same cone
sly vale
I swear this is the 2nd time a MindYourDecisions thumbnail has been posted, but not from the video
pure sun
strange
neat lintel
mind your cringe
fair mural
mindyourdecisions might be the worst youtube channel i can think of off the top of my head
light needle
Mindurdecision is actually good for like, quick mental maths. Its like getting a daily exercise in ykwim
neat lintel
Mindurdecision is actually good for like, quick mental maths. Its like getting a...
ye
actually sometimes really useful elentary math review
very clickbait and stereotype baiting
Can you solve this 7 yearolds chinese students problem
rotund steppe
no i cant
light needle
The nice thing is you dont even need to click on vid
Just watch the thumbnail and solve it
Thats good practice for the day
last oxide
I think you guys are giving the mind guy more shit than he deserves 
its just a puzzle channel
not even that bad
leaden torrent
yeah idk man
like its clickbaity but it seems like harmless clickbait
i think its stupid but i could see someone getting something valuable out of it
far better than your average youtube clickbait
sly thistle
I mean he’s fine; it is probably the type of thing I would have liked when I was younger
leaden torrent
hell, i wouldnt be surprised if showing 50 high schoolers that channel vs 50 high schoolers numberphile would produce better outcomes for the former group
since it at least vaguely attempts to reflect how mathematics is actually done
by demonstrating problem solving
last oxide
I have a problem with neither and I could see myself watching either nowadays (just not frequently, cause not that interested)
if anything, Id probably learn more things from the puzzle channel since numberphile never ever ever goes even slightly deep into something
leaden torrent
yeah man numberphile managed to make a youtube channel purely out of definitions
its almost impressive how far you can take nothing
last oxide
I dont know why math people are so prone to elistism
not targetting anybody specifically, but I see so many people complaining about harmless things
tbf that happens everywhere, not just in math
pure sun
Numberphile also did the -1/12 video
last oxide
in some other areas, its even more cringe because are sometimes elitist about things that are not even conclusive
leaden torrent
i will say that i dislike mindyourdecisions having a "quora culture" vibe to it
but i cant really formulate that in words
and something at the back of my head is saying i might just be being racist
last oxide
quora is ok in math
because of some great communicators like alon
step outseide of it and youre bound to see cringe
last oxide
yeah man numberphile managed to make a youtube channel purely out of definitions
I dunno if I could hold myself back from explaining things like the professors in that channel do
Ik there is heavy editting involved but even still
its not like they make it accessible
3b1b makes things accessible
they just vaguely refer to certain things
leaden torrent
in fairness though
i think "vaguely refer" actually works for a decent portion of numberphile's target audience
last oxide
well yes
leaden torrent
in that, if you give them a term to google, if they're interested they'll google it
last oxide
thats their audience
pale orchid
i dont think 3b1b makes stuff accessible
its largely useless if u dont already know the topic
last oxide
its people from adjacent areas that are not interested enough to learn things a bit deeper, from what it seems
last oxide
its largely useless if u dont already know the topic
Ill concede that
however, there is no denying thats more hands on
pale orchid
yes
last oxide
like, it at least shows the topic
numberphile has it being descontextualized added to it (probably due to the heavy editting)
I can barely follow the context half the times
neat lintel
i dont think 3b1b makes stuff accessible
big facts
tribal dove
This isn’t really math related but how what do scientists mean when they say the universe is flat? Doesn’t that imply that it’s finite and they reached the edge? I thought that for all we know it’s infinite in all directions.
Like wot
sharp mulch
Curvature 0
leaden torrent
to clarify, a priori it's possible that distances in the universe are slightly "bowl shaped"
at least on a macro scale
like things look euclidean on a small scale but "zoom out" enough and theres a very minor curving of the universe's geometry going on
pure sun
e.g. like the surface of the earth
leaden torrent
after all, this is how geometry on the surface of the earth is
on a small scale it looks euclidean
pure sun
sniped
leaden torrent
but zoom out enough and it becomes a sphere
this can "mess with" distances/geometry
for example, flight paths often arent very intuitive if you look at them on a 2d map
since they rely on the spherical nature of the earth to find a shortest path
(assuming theres no storms or anything in the way)
when we say "the universe is flat (or at least very very close to flat)", we're saying that there isnt that effect going on at a macro scale
at least at a macro scale we care about (ie observable)
a straight line is still a straight line no matter how far you zoom out
mortal igloo
leaden torrent
this might seem intuitive but its perhaps "surprising" in that we dont know how the universe, you know, came to be
so we have no reason to suspect it behaves "nicely" on a large scale - it seems like a hell of a coincidence
besides the fact that, well, it does based on our experiments
if you believe in a creator God then things are different i suppose
but thats irrelevant
tribal dove
But for something to be mostly flat doesn’t it have to have say a lot of length and width but relatively little height?
leaden torrent
thats not what flat means here
flat refers to the geometry
like the cartesian (x-y) plane has a "flat" geometry
pure sun
a piece of paper is "flat" no matter how you orient it
leaden torrent
as opposed to, say, geometry on a bowl or sphere or saddle
leaden torrent
the exact statement is technical
which is why its explained using vague terms like "flat"
if you're interested, look into gauss' theorema egregium.
it's a good starting point.
tribal dove
like the cartesian (x-y) plane has a "flat" geometry
I don’t get it aren’t you saying that the universe is 2 dimensional
leaden torrent
no, thats a point of comparison
the universe is 4 dimensional*
- at least
but the principles translate over
humans find it easier to visualize a cartesian plane than a pseudo lorentzian manifold, believe it or not
hence why i used the x-y plane as an example.
but the principle is the same: "flat" means that the geometry is euclidean-like, in the sense of having curvature 0.
it has no relation with actual "shape"
(though the universe IS laid out remarkably in line with a plane, for reasons that date back to the big bang)
(i dont really know the details there)
(but thats an unrelated fact)
neat lintel
lolwhat
sharp mulch
To visualize the curvature, consider two beams of light that are initially traveling parallel to each other
If the universe is flat, then the distance between them will always be the same
If the universe is curved, then the distance between them will change
It can either decrease or increase
If the distance decreases, then eventually the two beams of light will cross and we're living in a "spherical" universe
If the distance increases, the two beams of light will diverge and we're living in a "hyperbolic" universe
neat lintel
Hold up
How do you mesure the distance between two beams of light on a curved plane?
good question
planes arent usually curved tho
tribal dove
I don’t understand how something can be flat or curved when it stretches infinite distance in all directions
leaden torrent
I don’t understand how something can be flat or curved when it stretches infinit...
play hyperrogue.
i dont have a better way to drill in the intuition than that honestly
pure sun
how do you know the universe isnt just like a giant sphere
leaden torrent
its an example of an infinite noneuclidean geometry
neat lintel
I don’t understand how something can be flat or curved when it stretches infinit...
no one knows if universe is infinite
fun fact
leaden torrent
now, its geometry is 2d + an independent time axis while our universe's is 4d
sharp mulch
"sphere" in some general sense
leaden torrent
but the principle is the same
also its worth noting that the question of "is the universe infinite" is like
philosophically pointless
assuming our basic understanding of relativity is correct, "the universe" is, from our perspective, literally the same thing as "the observable universe"
there is absolutely no difference between the 2 concepts
neat lintel
If our universe is 4d
leaden torrent
but it isnt unreasonable to model the universe as having a geometry that extends out infinitely
sharp mulch
*at least 4d
leaden torrent
and when we ask questions about curvature, we're asking questions about that model of our universe's geometry.
neat lintel
Do objects cast 3d shadows 
sharp mulch
3 spatial dimensions + 1 time dimension
neat lintel
not 4d like that
leaden torrent
no because light's interaction with the time axis is weird
neat lintel
yuh
sharp mulch
The time dimension is distinguished in a sense
It doesn't behave like spatial dimensions
leaden torrent
it isnt
sharp mulch
(it has a different sign in the distance formula)
leaden torrent
thats why you kind of have to learn the mathematics
to understand it
very important poll
sharp mulch
More analysis channels? Yes
leaden torrent
Important poll
1️⃣ : (-, +, +, +)
2️⃣ : (+, -, -, -)
leaden torrent
Important poll
1️⃣ : (-, +, +, +)
2️⃣ : (+, -, -, -)
leaden torrent
physicists
neat lintel
What is that
pure sun
2 
leaden torrent
the lorentzian signature of spacetime
sharp mulch
Ok so there's something called the Minkowski metric
Which is how we measure distances in spacetime
sharp mulch
"metric" in some general sense because it isn't actually a metric
mortal igloo
Is the only difference whether you want spacelike or timelike displacements to have imaginary distance?
sharp mulch
No it's not the minkowski inequality
tribal dove
no one knows if universe is infinite
I just thought that in order for something to be “mostly flat” it had to have a finite height or something
leaden torrent
and i've said multiple times that that's not what we're talking about
tribal dove
But apparently not? Idk it’s beyond my understanding rn
leaden torrent
when we use the word "flat"
words can have multiple meanings
we're not talking about a literal shape
pale orchid
i think you're still not grasping that the "flat" used here is not the flat you're thinking of, the one used in day to day life
neat lintel
I just thought that in order for something to be “mostly flat” it had to have a ...
the earth is mostly flat in our perspective
but thats because its big as fuck
we have no idea the shape of the universe
because its big as fuck
clever knot
I like how dog is phrasing this
leaden torrent
(the earth metaphor is a bit misleading though since the earth is NOT flat from an observer's perspective, it's very easy to see that it's actually at least somewhat curved)
neat lintel
smh
leaden torrent
(for example, look over a flat horizon)
neat lintel
if u want buzzwords
leaden torrent
(but its good enough)
neat lintel
look up extrinsic and intrinsic curvature
mortal igloo
The earth is flat in my room
leaden torrent
i just bring it up to emphasize why the answer to the question isnt easy for physicists
mortal igloo
I need to parallel transport a vector around my room 100000 times to see the curvature
leaden torrent
for an earth scientist its easy to confirm the curvature of earth in your own backyard (+ a mile or so)
neat lintel
lol what
leaden torrent
for a theoretical physicist it actually takes a lot of work and data to test the curvature of the universe
neat lintel
for anyone
just be exposed to sunlight
On the other hand what if the universe is not infinite, what will you find at the end of it?
and use shadow calcs
clever knot
On the other hand what if the universe is not infinite, what will you find at th...
what if it loops around
neat lintel
but then that questions our model of sun
leaden torrent
again questions about the "end of the universe" are moot unless you can clarify what existence means to you metaphysically
neat lintel
and idk why we know the sun doesnt move
tribal dove
i think you're still not grasping that the "flat" used here is not the flat you'...
Wait what’s the other meaning of the word flat
leaden torrent
assuming our understanding of relativity is correct, for literally ALL intents and purposes, the universe is finite
and it ends at the boundary of observation
its possible to introduce some notion of metaphysical "existence" beyond that
but any claims you make "past the boundary" will be, by their nature, unfalsifiable
mortal igloo
The universe could be compact
leaden torrent
in the same way that claims about alternate dimensions are unfalsifiable
leaden torrent
Wait what’s the other meaning of the word flat
curvature 0.
sharp mulch
Curvature in some technical sense
pale orchid
Wait what’s the other meaning of the word flat
https://en.wikipedia.org/wiki/Gaussian_curvature, as nami told you several times already
sharp mulch
That tries to capture our intuition about curved 2d surfaces
neat lintel
This seems like it needs you to understand a lot of things in order to get it so I'll shut up for now
mortal igloo
Curvature tensor in the case of 4d manifolds
neat lintel
This seems like it needs you to understand a lot of things in order to get it so...
nah not rly
leaden torrent
the geometry of hyperrogue is not flat
neat lintel
linear alg and some imagination
I am studying it rn
if you have good intutitive understanding of dot products and projections
then you can read wikipedia article on curvature and be fine
leaden torrent
if you "zoom in to" a hyperrogue game (into a distance of like, 2 tiles), it "looks almost like" a standard 2d hex plane
but start actually moving and that falls apart
neat lintel
I still haven't got there
leaden torrent
or "zoom out" and it falls apart
since its not flat
ie the geometry isnt flat
the game itself is "flat" in that it takes place on a 2d computer screen lmao
neat lintel
np friend
leaden torrent
its really the best way there is to explain this since you can play around and "feel" the difference
like
travel in a straight line
then stray off course 1 tile
neat lintel
also if u want video i got u
neat lintel
also if u want video i got u
Sure
leaden torrent
but it might take 40000x as long
leaden torrent
yeah lmao
neat lintel
just think its a good talk
leaden torrent
the holy grail is a funny example
neat lintel
ig there is a small section