I can link them by the fundamental theorem, but I still don't know why the two definitions are so unrelated

or seemingly unrelated

read the wiki?

it describes it well.

Turns out there's two different J.M. Lees who have diff geo books

All right, I think I got it now...

you are ready to walk through the door now.

So I'll ask question 2: doing ∫f(x)dx would find the area between f(x) and the x axis, right?

welcome to the world of university level math.

So is there any way to find the area between two functions?

Yes.

yeah.

you can figure it out yourself.

So would I call it "taking the integral of one function with respect to another?"

That's a bit much.

No.

Er...

@SirCumference If $f(x)\ge g(x)$ on some interval $I$, then the area between the two functions is simply given by $\int_I[f(x)-g(x)]\,\mathrm{d}x$.

Oh, it's that simple?

^

yeah though for some curves

Huh.

you may have to change the bounds of integration.

we talking about calculus? yay!

Like which curves?

so what are you talkmg about specfically?

like what is the question?

@TanMath Getting the area between two functions

Er, I dunno if I'm really ready for higher math though. I've only spent 30 mins learning calc. So I only know integration and differentiation.

@SirCumference that's simple...it is the integral of f(x)-g(x)dx assuming f(x) is bigger...

0celo7 already said that.

@SirCumference good job! you learnt calc in 30 mins!

Hooray?

@0537 oh...

It's a 3-4 class subject.

basically it can become complicated if one of the curves is a not a function.

I only touched upon two important things

so you will have to find intersection points and do more than one difference integral.

Okay?

@SirCumference no it's not.

it's like a 5 month subject.

calc 1&2 that is.

I meant calc 1-3/4

calc is easy!

But still, just 30 mins probably leaves me clueless on some stuff

@TanMath Well integration and differentiation were

you spent more than 30 mins dude.

everything is...ok?

though it's still little, since you used bad resources.

you going on to calc 2 then?

Not 30 mins in total. 30 mins on each. Still not very good

calc 2 is pretty neat...

vector calculus is really cool.

said no one ever

@0537 yes...

calc 3, i think is cooler...

but i know very little about it..

So my next question is kind of idiotic: why does one say "take the integral of [function] with respect to x"? Does this mean that that they want the area between the curve and the x axis?

@TanMath what year of university are you in?

but i love taylor series...

@SirCumference with respect to $dx$, because that's your differential.

@0537 2nd year...

@SirCumference yes...

@TanMath why do you have a "..." in all your sentences?

Oh, so what if I said "with respect to (something else)"

@FenderLesPaul idk...it is a habit!

What would some other options be?

fair enough

@SirCumference then it would either be a different question or not make sense.

Okay...

@SirCumference well if you said with respect to y, then the function must be in terms of y...

Oh

$\int f(x) \, d(potato)$ lol doesn't make sense.

otherwise, the integral would be equal to y, i guess...yeah guys?

@0537 You sure? Makes sense to me

lol

calc 3 is about having multiple variables to differentiate and integrate...

so an in depth explanation would be found in calc 3

no that's differential equations.

All right, last question: Leibniz's notation for derivatives doesn't exactly make sense to me. Is "dy/dx" a random way of saying Newton's "f'(x)"? Or what's the meaning?

What does "d" stand for?

@SirCumference it's a different way.

Or should I care?

the dy is a differential...same for dx...

$\frac{d}{dx}$ is the differential operator, applied to a function $f(x)$ it gives you its derivative which is $f'(x)$.

I know, but why exactly is it $d/dx$? I see $dx$ in integrals too

dy is like the infintely small change of y and same of dx...it kinda shows the definition of a derivative...

$dx$ is called a differential.

i just said that...

Excuse my ignorance, I'm still new to this

@TanMath I didn't see.

Ooooh...

A'right, makes sense now

it shows that a derivative is the slope at an individual point...

@SirCumference makes sense now?

Yep

@TanMath slope of what? lol

of the function being differentiated...

The tangent line, I assume

yeah.

Or yeah...

you didn't say that.

there can be multiple slopes at a point on a curve.

no...

this conversation is amusing

How...?

@TanMath dafuq?

slope does not imply tangent.

as far as I know.

@0537 there can't!

Er, how could a point have multiple slopes?

a point by itself?

A slope is just $∆y/∆x$

@SirCumference You should think of $\mathrm{d}x$ as an infinitesimal change in $x$. Likewise, $\mathrm{d}f$ is the infinitesimal change in $f$ when $x$ changes by $\mathrm{d}x$ - think of $\mathrm{d}f = f(x+\mathrm{d}x)-f(x)$. Now writing $\mathrm{d}f/\mathrm{d}x$ for the derivative is supposed to symbolize that the derivative is the slope of the function at that point, like the slope of a linear function is $\frac{f(x+\mathrm{d}x)-f(x)}{\mathrm{d}x}$ for a real number $\mathrm{d}x$.

Ah, now it makes sense why $dy/dx$ essentially means "instantaneous slope"

Calculus is a lie.

$∆y/∆x$ is slope, but this would be an instantaneous change...so it would have "d" replace "∆", right?

kind of.

it's a limit though.

Well yeah

@ACuriousMind Is the set of $(k,l)$ tensors just $\mathrm{Hom}(V^{\ast k},V^l)$?

@SirCumference Yes, the $\mathrm{d}$ is supposed to represent that. It doesn't work rigorously like that without much further effort, though. Just accept writing $\mathrm{d}x$ as a single symbol, and don't think of the $\mathrm{d}$ as being something applied to $x$.

Okay, thanks

@0celo7 Yes.

Makes more sense now

Would anyone more familiar with the US College system be able to explain what should I read out of this?

@ACuriousMind ok, but what if $\mathrm{d}$ is the exterior derivative

@0celo7 You can't divide by an exterior derivative.

So what now? Can I go onto analysis or...?

@SirCumference why would you do that?

Interested in it

@ACuriousMind I was trying to be a smartass but it's not working...ignore me

@0celo7 k

@SirCumference then say goodbye to physics for a while.

Wait what... .-.

@SirCumference probably not

@0celo7 Don't project yourself onto others ;P

But one of the reasons I started learning calc was to learn higher physics stuff

._.

then why do you want to do analysis now?

Next time @0celo7 asked "Where's the equations?" I'd be like "Here. Ha."

Or something...

Again, it's interesting

Learning about more infinity

@BernardMeurer they send that to everyone I believe, I wouldn't read too much into it

they sent me that email when I applied

(fill disclosure I didn't get in)

@SirCumference just read the wikipedia page about infinity.

@0537 Why wouldn't I be able to learn calc and analysis

well it will slow down your progress of learning physics.

There's still more I'd like to get into

@FenderLesPaul The ammount of emails they send is just abusive (@0celo7). I always get confused on what the hell is going on

Yeah, but I can take my time with physics. No reason to go fast

Especially if I can learn some other stuff too

@BernardMeurer honestly it's better for your own mental state if you don't worry about things or constantly try to quantify whether or not you can get in

I'm only saying that because I'm the same way

and it just doesn't help at all

until you get an email from them about admissions decisions just don't read into any of their general emails

@FenderLesPaul I truly try, it's rather difficult. Specially when google inbox makes my whole house vibrate and flash when I get an email

@BernardMeurer what?

@BernardMeurer I understand it's hard. I have the same problem.

But if you don't think about it, you will be de-stressed.

@0537 Not reading emails and ponderating upon the meaning of each comma on the email while waiting for college applications to yield results

@ACuriousMind mother duck why

that hurts me

@FenderLesPaul I'm currently coding in BASH, I'll never be de-stressed I believe hahaha.

@BernardMeurer :p

The fact that an if statement is closed by a fi is enough to traumatize anyone

how ugly is that even

@0celo7 Did you have that saved for such a moment? :D

hello...

@SirCumference everything makes sense now?

@ACuriousMind maybe

@TanMath Yeah

@TanMath how was what I said wrong though?

@0537 that a point can have multiple slopes?

that is so wrong...

there can be multiple slopes at a point on a curve.

not possible

@ACuriousMind Is the bundle of tensors equal to the tensor product of vector bundles?

Or is that true by definition and I'm blöd

@TanMath yeah, just rotate the tangent line.

wtf are you people talking about

a point does not have a slope

if I'm being retarded.

^

@BernardMeurer You know how case is closed in bash? Yeah...

@0537 no! then it isn't tangent!

Rotate the tangent line? It wouldn't be tangent to the function

yeah.

so?

it's still a slope at a point.

@0celo7 How do you define the bundle of tensors if not by that?

it isn't a tangent line!

I didn't say it was.

@0537 wat

you didn't imply tangent line.

@0537 oh... so by that logic, a point can have infinite number of slopes

yeah.

Okay, so you can change the tangent line's slope. But what good would a "non-tangenty" line do for you

@TanMath that's what i'm saying.

@BernardMeurer When I need a refresher I usually read the python docs.

@SirCumference nothing.

.-.

@SirCumference No -y needed. "tangent" and "non-tangent" are already adjectives

@alarge Thanks man, thanks a lot for reminding me something that will keep me from sleeping tonight...

@ACuriousMind Oh yeah... ._.

@TanMath when you said it shows that a derivative is the slope at an individual point...

you weren't being precise.

@DanielSank Huh, didn't even know the Python docs covered Regex, great addition! Thanks a lot!

because a point can have infinite slopes.

Now you're just arguing semantics

@ACuriousMind Easily: the disjoint union of tensor products of tangent spaces

Crap

@ACuriousMind also what's the TeX for a disjoint union

that's all i was saying.

@0celo7 And your definition of "tensor product of vector bundles"?

@0celo7 \sqcup

"semitics" Thanks autocorrect...

@0537 you are making a big deal...i bet @SirCumference knows what i mean...

hey @DanielSank

@TanMath be more precise. lol

big deal...

big deal for me.

Bar fight starting

@SirCumference did you understand what i meant?

Yeah

@SirCumference lol

::breaks bottle::

XD

@SirCumference so that is all that matters...

@0537 bye...

@ACuriousMind hopefully not on our heads. lol

@TanMath what?

@ACuriousMind disjoint union of tensor products of fi....oh

@0537 Nah, on a table. Broken bottle is the traditional bar fight weapon, right?

@0celo7 Exactly :)

@ACuriousMind boot knife in the glorious south

@0537 i do not want to have a bar fight...so I a, leaving before anything happens!

@TanMath dude the discussion was over...

and i'm a calm person.

watch out for that ayy lmao @ACuriousMind! He'll shank you!

@0537 He says as he takes out a knife

@SirCumference D:

you can be calm and still murder people

@0celo7 I waited for you say that ;)

That's the most screwed up type of murder

When the murderer doesn't care

You dream about me?

@ACuriousMind awk

lol, coward

@ACuriousMind NO...

I had a dream where 0celo7 was teaching me relativity.

remember?

what is going on?

you're just obsessed with me @ACuriousMind and want me to dream about you

@0537 seriously?

@0537 If I remembered your dream that would be pretty messed up...

@0537 how would I remember your dreams

@TanMath 0celo7 is dreaming about ACurious

Jesus no

Do with that as you will

At least not intentionally

@ACuriousMind I implied that I shared that with you guys before.

You can dream intentionally?

^

@0537 I read that wrong

I thought you said "0celo7 was touching me relatively"

I was dreaming about Mordin Solus obviously

that makes more sense

^

and it carried over

@FenderLesPaul go away lol

@SirCumference uh, yeah

@FenderLesPaul How does one touch relatively

you can touch relativistically.

^

@ACuriousMind it's a tumblr term

you wouldn't know!

wat

just for full disclosure

I had a dream that @0celo7 was touching me relatively

So then would @0celo7 be your relative?

Great job, the star wall finally returns to its usual non-sensical content!

@JohnDuffield

@0537 what

@0537 he is following your footsteps...

wat

@ACuriousMind Honestly, I do not remember dreaming about you