@heather Yeah.

Right, the cross does not look like $\Bbb R$ right at the intersection point

At the crossing point, it doesn't look like a curve, it looks like two curves.

@0celo7, would 'o' be a good example of a 1D manifold?

Yes!

It's the only compact one, in fact.

What does it mean for it to be "compact"?

@heather Yes, see examples above.

~ finite

you can think of compact being "bounded"

Oh, that makes sense.

So where are manifolds useful?

now - is also a manifold if we don't include the endpoints

Oh, snap, @0celo7 I'm an idiot, aren't I? I guess technically \ isn't a manifold because of the end points, eh?

@DanielSank Yeah but that can be forgiven :)

Yeah, it's manifold everywhere except the ends.

@heather Because a lot of stuff is a manifold.

Basically all of vector calculus is just manifolds in disguise

And would these 1D manifolds be described by equations like y=x+5 or whatever?

@0celo7, vector calculus?

@0celo7 That is totally wrong and you know it. At least 50% of vector calculus is linear algebra.

@heather those are manifolds, but you can get more general

@heather He just means calculus with N dimensions.

@DanielSank Implying diff geo isn't 50% linear algebra in the end

@DanielSank, like...oh! Like spacetime in GR?

@heather Yeah, that defines a straight line, which is indeed a manifold.

@heather Yeah.

Okay, I think I'm starting to maybe kind of get this.

@heather there's a really interesting duality in manifolds. Allow me to give you an example.

In 1-dimension there are two manifolds, up to deformation: $\Bbb R$ and $S^1$ (circle)

Oh here we go...

Let us consider the circle, what @0celo7 called $S^1$.

Here's the equations way to specify it:

@Obliv People can be so hypersensitive here I don't know when that's a joke...

$$x^2 + y^2 = r^2$$

The set of pairs $(x,y)$ which satisfies that equation is indeed a circle of radius $r$. It is a manifold.

I can't stop Daniel once he goes on this rant

It's an interesting rant.

@0celo7 okay I'm going to derive the lagrange multiplier eq. without looking into the notes logically and concisely. maximize $f(x,y)$ with a constraint $g(x,y)$. We know $\nabla f(u)$ is orthogonal to the level curve $f(u) = c$ where $c$ is a constant. Thus, if we can find the largest level curve $f(x_m,y_m) = M$ (m denotes maximal) that the constraint allows, it is a solution. We need $\nabla f(x_m,y_m) = n\nabla g(x_m,y_m)$ because at this maximum level curve (a point of intersection b/t f,g)

Wait, timeout: what is a parametrization? And it is indeed an interesting rant.

@Obliv wtf is this

@DanielSank Proof :P

Their normal line must be the same. It need not be exactly the same (can vary with a scale factor) in the equation of a plane i.e $\nabla f(x) \cdot (x-u) = 0$ and $\nabla g(x) \cdot (x-u) = 0$

Reb is studying this right how for a test tomorrow

for some point $x$.

you two should form a study group

Ok, now here's the other way to specify the circle, using a parametrization:

@SirCumference are you referring to something specifically

@Obliv Nah

$$x = r \cos(\theta) \quad y = r \sin(\theta) \quad \text{for }\theta \in [0, 2\pi)$$

Honestly calling me hypersensitive triggers me

flagging

I never called you that...

@DanielSank, wait, I'm not sure I follow this part.

@0celo7 my test isn't soon I don't think. We'll be going over multiple integrals before it

@0celo7 *Hyperflagging

@heather ok, I'll pause.

What's up?

That should be a term

"Hyperflagging"

Flagging oversensitively

So, you have those two equations, but I'm not sure about the last part, for $\theta \in [0,2\pi)$

@heather Oh, I'm saying that you should think of $\theta$ as going from zero to two pi.

That's one trip around the circle.

don't think she has had trig

@0celo7 You're showing her chain rule?

Oh, okay, and then, hmm, okay, I think I get it now.

She doesn't know calc

@SirCumference I'm showing @Obliv

Oh, nvm

@heather Oh wait, do you know $\sin$ and $\cos$?

Hey @bl00

@DanielSank, yeah. With the unit circle and the x and y coordinates and everything. I do have a general idea of basic trigonometry.

@0celo7 wow. that is such an easy explanation my textbook sucks

what book is that

@bl00 The notes you gave me are actually pretty neat

Gotten over 25% through so far

@Obliv Rogawski, it's what reb is using

NO

@bl00 Wait what

STOP ASKING

@0celo7 You okay?

NO I'M GOING BLIND

@bl00 flagging.

@0celo7 Flagged you for flagging

I call BS

if it's only in one eye it's probably not a big deal. You can trust me I'm a PhD

How is that BS?

Flagging is very offensive to me

I take it quite personally

Prove it.

@Obliv I don't think that's how it works...

Show the flag

@Obliv He lost acuity in one eye suddenly. It could be something rather grave and frankly I think you shouldn't joke about it.

Just get a high powered laser from your lab and shine it in your eye. I think that was one of the treatment options I read on webmd

@0celo7 Here

@DanielSank It's only distance vision.

There's your flag

Particularly because he has been living with in for several days without going to the doctor, which is a very bad idea.

No time

Or here

Why isn't everything open 24/7???

That enough flags for you?

@daniel What is the worst that could happen (joking aside)?

@SirCumference Hahahahaha

@Obliv The worst? He could go blind in that eye.

@bl00 No, I'm stupid, so I go to class to appear less stupid.

Yeah that is pretty bad. Modern medicine is pretty good though so I feel like he'll be okay.

that's just my feeling

0celo, what's up with your eye?

I can't see past 4 feet

everything is totally blurry

In one eye or both?

the whiteboard looks blank it's so blurry

@SirCumference one eye

also the eye hurts from time to time

@0celo7 You need to get that checked out

Sounds like an infection

I remember on the old forum I used to use (for videogames) someone posted about how his brother got a blood clot in his leg and whether he'd be okay. They brought him to the hospital and people were really supportive about it and saying he'll be fine. Later that day he came back to post that his brother had died.

yes I have a fucking appointment with an ophthalmologist

That was a bummer.

@DanielSank, I think I get the first part of the parametrization thing.

@Obliv I want you to remember I said this for the rest of your life: If you have a sudden change in health in any way, you do not just hope it's ok. You get your ass to a doctor as soon as possible.

My father is recovering at home right now from a detached retina. Do you know what that means? It means his retina was coming off of the inside of his eyeball. If that gets too bad, you're blind in that eye. Modern medicine cannot save you.

This also happened to my brother.

In both cases, they caught it quickly and are ok.

I can live with one eye

Because they went to the doctor.

@0celo7, wrong response.

@0celo7 That is not the attitude you should have...

It's the only reasonable one

Seriously! That reduces video-game playing efficiency!

@bl00 Going around telling people they'll be ok because modern medicine is good is a terrible idea and you should stop doing that.

I don't need to be playing video games

@daniel wow how common is it? Sorry about that :(

math and physics efficiency as well!

@Obliv one of my parents has had itt

is it genetic or smthn?

@bl00 you two are the same color

@bl00 Maybe. You're both blue. It's confusing.

@Obliv Sort of. If you're myopic you're more likely to get it.

@0celo7 That's racist

Flagging

Oh no.. I am near sighted..

@SirCumference I hate blue people

@0celo7 We'll build a wall between us and the blue people

@DanielSank I'm probably just extreme near-sighted and just noticed it.

And we'll make @bl00 and @Obliv pay for it!

@SirCumference And make them pay for it

@0celo7 that wouldn't be very surprising honestly.

I mean

@0celo7 In one eye?

I do have glasses

But they basically don't work any more

@bl00 I never use them

@0celo7 They run out of battery?

@0celo7 Same

I'm 20/15 in one eye and like 20/30 in the other

It's a tiny correction

I'm 20/60 or something

But now I really cannot see at all

See a doctor

I can't read 1 feet away from me.

No pun intended

oh my god I am seeing a doctor tomorrow

can everybody just calm down

@0celo7 people care about you young ocelot.

SEE A DOCTOR NOW BECAUSE THIS IS A BIG DEAL!!!!!! (sorry, but, you know, it is a big deal.)

@0celo7 AAAHHHHHHAAHAHHA

unless of course I can't get there, which is a definite possibility

@bl00 Privileged bastard

(jk)

@0celo7 Dude, go to the ER at this point

That sounds like a blood clot in your eye

I hope it's cancer

Wtf dude

lol...

no seriously

no seriously, wtf?

people pity you so much if you have cancer

because you can die

@SirCumference oh, never thought about it like that

The cons kind of outweigh the pros

hmm, yeah

ok never mind

that was...weird

Anyways

wtf

@0celo7 XD

I remember the new Ghostbusters movie, one of the women could create a device to catch ghosts because "no one knows quantum physics better than her"

That is just so cringy...

so ACM?

Almost as bad as this

that's why I don't watch sci-fi films. @sirC

pop-culture sci-fi anyway

@Obliv The fact that they relate QM with ghosts is just so...

Dunno the word for it

@bl00 virtual particles don't count as ghosts

@SirCumference, sad? disturbing? upsetting?

theres also spectral theory

@heather screwed up is better

@SirCumference, hmm, yes.

@heather do you wanna know what a topological space is

@0celo7, sure, yeah!

@0celo7 Dude, one step at a time

ok

metric spaces first

@SirCumference, ah, going in order. Who cares. =P

good point

@bl00 will tell you what a metric space is

@0celo7, okay, sure. Anything is cool.

@heather From experience I've learned going in order is really the best

It ensures you won't miss out on much information

@SirCumference, yeah, I know. But sometimes I enjoy jumping out of order.

@bl00 practice!

good woman

the order is for chumps

Precise definition please.

@0celo7 Who needs order? Chaos is far cooler

oh crap

we didn't define functions

@bl00...um. Well. I'm not quite sure what that metric bit is.

@heather Ok we need functions first.

@0celo7, functions. Okay. @SirCumference, see, I am going in order - functions are in Algebra II and precalc and everything in the universe. =)

@heather Ok, $X$ and $Y$ are two sets.

ok

Remember what $X\times Y$ is?

hi all

I've never seen 0celo so eager to teach

When I ask him questions he laughs at my ignorance...

does LateX really not work here?

yeah, the set of coordinates (x, y) where $x \in X$ and $y \in Y$.

@ian use chatjax

@bl00 I know right.

well, I mean, the set of all coordinates. but yeah.

Never teaches me topology..

@Obliv We're unloved...

@heather coordinates?

@0celo7, er, I mean ordered pairs. wow. that was bad.

Oh, actually we need $Y\times X$

But same thing basically.

yeah

@Obliv Watch what happens when I upvote three of your answers

A relation between $X$ and $Y$ is a subset $R\subset Y\times X$.

For "$x\in X$ related to $y\in Y$" we write $yRx$.

okay

Now that probably makes zero sense.

We need an example.

what sort of relation might this be and yes^

Let's consider the relation $\le$ on $\Bbb R$

Heather is a mathematician but doesn't know it

@bl00, I am nothing, currently, but a student

^

Ok, so what is the $R$ associated to $\le$ in $\Bbb R\times\Bbb R$?

@0celo7 What'd you say the lower mass limit for a Kerr black hole was?

$e^2+a^2$

Um...I have no idea, honestly

@heather Ok let's go easier

consider the relation $=$ on $\Bbb R$

@0celo7, okay

What is $R$?

@bl00 That's unrelated

He answered my question

$\bb R$

@SirCumference dangit man I wanted to keep low rep like @bl00

I mean, however you do the set of all real numbers

What happens if you remove those upvotes? lol

@Obliv That's...not the "thank you" I expected...

@heather no

it's the line $y=x$

in the plane $\Bbb R^2$

@Obliv You still get 100 rep for every site

I don't deserve reputation as a filthy undergraduate physics student @sirC but thanks.

And for new accounts you make on sites

because if you take an $x$ and a $y$ on that line, then $yRx$

@0celo7, okay, what? I don't quite understand.

@Obliv Well you get more permissions, like upvoting and commenting, on every SE site

Not downvoting though :/

@heather here $yRx$ means $y=x$

Right

So we want the subset of $\Bbb R\times\Bbb R$ such that $x=y$ for $x\in\Bbb R$ and $y\in\Bbb R$

but that's clearly just the diagonal line

@Obliv I'm a "filthy" undergrad student and I have 4k on Astronomy

Heather's a middle schooler and has 3.7k here

oh, right! Oh! That makes sense now!

So now consider the relation $\le $

We want $y\le x$

@Obliv But did you actually know that happens when you reach 200 rep?

I know you've graphed that in algebra

@sirC not at 200 rep but I've heard that something like that happens at some point.

rekt

@Obliv Wait, you're a girl?

Thought you were a guy

what?

yes, is that a problem?

Obliv's pic

Nah, 'course not

I'm that girl in the picture.

Oh

@0celo7, yes, it sounds familiar, but I'm unsure...

@heather these are the guys you want to avoid btw

I just always hear a guy's voice when I read posts from anonymous people

@0celo7, who?

Even Heather

@SirCumference sexist

@heather everyone in here

@bl00 exactly. and @0celo7 is a cat. And @sirC is a slice of pie

@0celo7 It's what I hear in my head

My own voice

So I project that onto other people

@SirCumference then your head is sexist

...

@heather let me draw it

PhD art skills

I am gender fluid. I sexually identify as a fluid.

@heather

@0celo7, oh yeah, duh! I haven't done that in forever.

that's $y\le x$

I hope!

Yeah, I think that's right

@bl00 oh that's what it was from. So your old old old profile picture was the same dude as well?

Right, so "relations" between sets $A$ and $B$ are just subsets of $B\times A$.

@0celo7, yeah, that makes sense now. So those points shaded are a subset of the overall set.

Can we just go, I dunno, 5 mins without discussing math?

@heather So a function $f$ between $X$ and $Y$, written $f:X\to Y$, is a relation on $Y\times X$ such that $yfx$ and $y'fx$ implies $y=y'$.

I.e. to each $x$ is only related one $y$.

@bl00 'Course it isn't

This is the "vertical line test" abstracted

It's just insane to talk only about math

Okay, so let me see if I've got this straight.

I think google is in my head. As soon as I type something it knows exactly what I want..

@0celo7, You can have a relation called a function. But this relation must "map" (that's the correct use of the term, right?) every $x$ to a unique $y$.

@heather YES

Kind of creepy. eric schmidt knows me personally :(

Victory!

@Obliv @bl00 @SirCumference sorry I have a new apprentice

Apprentice? Who?

@0celo7 You weren't exactly training us...more laughing at our ignorance...

@heather You

the crazy guy that works at google that said "If you have something that you don't want anyone to know, maybe you shouldn't be doing it in the first place;"@bl00

@0celo7, cool! I'm excited already. (Well, I mean, I was excited before hand)

Ok, not sure why were doing functions actually

I think it was because of metric spaces

Oh right!

And I didn't know what a metric was

@heather run away @0celo7 is going to turn you into another him.

@Obliv, though hopefully I'll go to the doctor when I need too =)

@0celo7, so, Master Yoda, what next?

You're gonna be a math purist

Ok, so a metric space is a set $X$ with a function $d:X\times X\to\Bbb R$ such that (i) $d(x,y)=d(y,x)$ for all $x,y\in X$, (ii) $d(x,y)\ge 0$ with $d(x,y)=0$ if and only if $x=y$, (iii) $d(x,y)\le d(x,z)+d(z,y)$ for all $x,y,z\in X$.

@SirCumference, nah. I love physics too much.

A girl version of 0celo. 0cela.

So that needs to be explained

A+ teaching @0celo7

@0celo7, whew! Okay, so $d$ is a function

$d:X\times X\to\Bbb R$ means that $d$ takes two points of $X$ at once

and assigns a real number

and it maps the set of coordinates created by X x X to R, right

$d$ is the metric or distance function

okay, so...oh, that makes sense.

@heather ordered pairs

gotta go. night folks

@0celo7, right.

Gotta remember that.

So given $x,y\in X$ the symbol $d(x,y)$ just means the distance between them

@0celo7 I'm so used to seeing $d$ followed by variables as derivatives, it threw me off

So the function follows the rules that d(x, y) = d(y, x)

right, these are all "reasonable"

The distance between $x$ and $y$ is the distance between $y$ and $x$

which makes sense because it is creating coordinates from X times itself

and stuff

and what you said

(ii) means the distance is always positive

Anyone here ever tried corn with ketchup?

Tastes darn good

unless the two points are equal, in which case the distance is 0

distance can't be negative, distance is 0 only if they are the same ordered pair

right

(iii) is called the triangle inequality

I'll draw a picture unless it's obvious

okay, I've heard of it but I don't know what that means

pictures are good

so the lines here represent the distances between the points

Oh, I get it now, the bottom can't be longer than the top, basically

the top?

that makes a lot of sense

well, the right two lines

it means taking a detour through $z$ will never shorten the path

yeah

it can be the same distance

but never shorter

so a metric space is just a way of defining distance

yes

many spaces are metric spaces

and they're the first examples of topological spaces we will encounter

now you should have a question about manifolds

one moment

okay, I'll think about this overnight

my mom is telling me I need to go to bed, sadly

but my question would be