serious-discussion

[deep mango]

that's the liquid trickling onto the cold hard metal plate 20 feet below

[neat frost]

FEET

Why not HANDS

[errant ridge]

oh wait nvm you need the curve to be continuously differentiable im order to do a contour integral over it

so you cant do one over the boundary of the mandelbrot set

[deep mango]

[errant ridge]

what...

[neat frost]

[deep mango]

that's a very strong condition

you just need it to be rectifiable

[neat frost]

Isn’t that what happened to Jesus?

[deep mango]

you can do integrals over a curve with a single corner

[errant ridge]

really? huh okay

[neat frost]

Wait

Rectifiable?

[deep mango]

or with many corners, if they are in the right places

[neat frost]

The fuck my ass got to do with it

[deep mango]

https://en.wikipedia.org/wiki/Rectifiable_set

this is kind of silly

one second

[errant ridge]

that's some analyst talk

[deep mango]

https://planetmath.org/rectifiablecurve

this is better

it's a curve of finite length

[errant ridge]

i see, okay i guess fractal boundaries are definitely out of the picture

[deep mango]

in particular

https://en.wikipedia.org/wiki/Fractal_curve

(see the end of the first section)

[errant ridge]

[errant ridge]

is this notion used in the theory of functions of several complex variables?

[deep mango]

I have no idea

I think then people just look at differentiable objects

[errant ridge]

[odd narwhal]

I think gmt people look at rectifiable sets

[deep mango]

Oh yeah that makes sense

[ancient flame]

I almost just exposed myself

joined vc and meant to click the unmute button

that was actually the camera button

I quickly moved out of the way LOL

[deep mango]

wow. careful!

[ancient flame]

LOL

thanks daddy ryc

well actually I've linked my youtube so it doesn't really matter

but still

[valid vigil]

#events talk where

[ancient flame]

advanced vc

[valid vigil]

O

needed #get-advanced-access to see it

[deep mango]

that was my joke

[ancient flame]

oh

smh

[deep mango]

yes, by design

[arctic grove]

u have?

u never shared with me

[ancient flame]

I'll share it w u

[bright hill]

[ancient flame]

lol

[deep mango]

All the nosey noseys are swooping in

[bright hill]

Oh, you already mentioned that

[ancient flame]

lol

[bright hill]

Lmao

[ancient flame]

LOOOOOL

[deep mango]

Darq sad he doesn't get extra insider info

[ancient flame]

ryc, this is why ur cute

[arctic grove]

this

exactly

[ancient flame]

LOOOOOL

[analog salmon]

Last time I exposed myself, I was banned from going within 50 feet of schools 😔

(for legal reasons this is a joke)

[errant ridge]

I'm banned from going within 50 million lightyears of our galaxy

don't ask what I did

[arctic grove]

u didnt study for exams!

[errant ridge]

[open aspen]

my university makes graduate students paid under federal grants (NSF) take an ethics course

even if you are a pure math student

and so now I'm going to sit around in a course while they talk about how to safely experiment on human participants

[wooden flax]

sadge

[open aspen]

for 1 hour a week mandatory physical attendance

guess that's when I'll grade papers lmao

[neat lintel]

you should file a complaint

[hollow sundial]

did you already get the anti-sexual harassment training?

[open aspen]

yes

it was a three parter

and theres a refresher next year

[hollow sundial]

something something don't date undergrads

haha refreshers now? holy shit

[open aspen]

its actually terminal

[open aspen]

I was considering it, but I dont want to cause a headache for the PI

shes already so busy all the time

if it were on zoom it wouldnt be so bad but in person is frustrating

[odd narwhal]

That's so dumb

Where do you go again

[open aspen]

especially considering they already had us take an online course/quiz series thing

UIowa

[odd narwhal]

Ah right

[open aspen]

for now, maybe lmao

[chilly hull]

when will i get my active role

[open aspen]

idk the requirement is pretty high it feels lol

its automatic though

At least... I thought it said so in #info at some point

maybe that was changed or I've lost my mind and it never existed

[pure sun]

i dont think it has ever been made explicit

[open aspen]

ive lost my dang mind

sounds about right

[odd narwhal]

The more you interact in advanced channels the faster you get it

[tall badge]

we're vague about details bc they may lead to ppl gaming the system

[open aspen]

makes sense

[hollow sundial]

ah shit there's a probability word for this

begins with an a

[pure sun]

is it a?

[hollow sundial]

arbitrage was the word i was thinking of

[zealous garden]

I prefer the phrase "Much-needed Gatekeeping"

[open aspen]

protection of the nice colors

sparsity increases value

[bright hill]

how tf is that gatekeeping? lmao

[zealous garden]

I'd say being vague about details constitutes a form of gatekeeping

[full isle]

actually??

[velvet dagger]

I mean... Calling it "gatekeeping" feels the same as calling every form of a lie gaslighting

Which to be fair I do because it's v funny

[odd narwhal]

But we do call every form of lie gaslighting

We always have

Don't you remember

[pure sun]

i cant believe he forgot that's so embarrassing

[velvet dagger]

ShiN... Gaslighting doesn't actually exist

Gas doesn't exist

[odd narwhal]

How can't it, you're full of it

[hollow sundial]

[velvet dagger]

I absorb it but then it's destroyed

[neat frost]

Good morning shin

[hollow sundial]

Sloth's full of CO

[neat frost]

Good morning dami

[velvet dagger]

I am CO

Sup slurp

[void holly]

Matter cannot be created nor destroyed

[odd narwhal]

Go to sleep slurp

[wooden flax]

bruh so now i actually have to help people in #groups-rings-fields

[neat frost]

No

[wooden flax]

and not jsut sully them

[velvet dagger]

That used to be true

[wooden flax]

to get active role

[neat frost]

I just worked out a bit

[velvet dagger]

Until I was born

[neat frost]

And now I need a shower

[odd narwhal]

You should skip the shower

Go to sleep sticky and gross

[hollow sundial]

CO King doesn't look exactly great

[neat frost]

And I need to make up some ODE material

That course is so fucking boring

[wooden flax]

if i dont shower in a morning i feel like actual piss

[velvet dagger]

Invictus you do realize that people aren't paid for having active role or anything

[neat frost]

And like it’s 6 hours straight of lecturing

[hollow sundial]

you guys aren't being paid to be helpers?

[neat frost]

And the lecturer can’t be interesting for shit

[wooden flax]

Dami do you understand the color dynamics on the discord math server

[velvet dagger]

I'm being paid to be admin but that's different

[wooden flax]

like its a fucking hierachy

[velvet dagger]

Like that's discord level shenanigans

[odd narwhal]

I specifically am

But no one else is

[neat frost]

I am too

[wooden flax]

i cant tell if you're joking or not

honestly

[odd narwhal]

You're not even a helper slirp

[hollow sundial]

are you paid in arrows to the knee

[neat frost]

I’m paid in love and admiration

[wooden flax]

helper = permission to get pinged 10000 times every second

[velvet dagger]

Slurp you need a raise then

[odd narwhal]

No I'm paid in more current memes

[hollow sundial]

🧟‍♂️

[neat frost]

But… you’re… not… either….

[velvet dagger]

You get shat on all the time while you're here

[hollow sundial]

the burn is real

[odd narwhal]

You don't know that

Cuz my role color hides other roles

[neat frost]

It’s just because everyone’s jealous of me, I take it as a compliment

Part of my payment

[hollow sundial]

vacuously true

[velvet dagger]

Is that what we call self-gaslighting?

[odd narwhal]

Maybe one of my secret roles is actually the 'paid helper' role

Who knows

[neat frost]

So you’re saying your outer color is what defines you? That’s kinda… racist

[velvet dagger]

That's a reach my guy

[neat frost]

Shhhh

[odd narwhal]

Go shower

[neat frost]

Tbh that’s the exact opposite of what shin would be saying

[neat frost]

That’s such an effort tho

[zealous garden]

wrong

[velvet dagger]

From the "taking a shower" reference frame you're putting in a ton of effort right now

[neat frost]

[wooden flax]

its worth it ngl

[velvet dagger]

Also yeah wraithlord I just didn't wanna contest on basis of the physics because I didn't know the details

[neat frost]

How am I putting in any effort

That’s a rude thing to say

If matter can’t be created where does it all come from? Checkmate

[hollow sundial]

it's more accurate to say energy cannot be created nor destroyed

[void holly]

It came from being destroyed

[hollow sundial]

but mass = energy / c^2 sooooo maybe that's the same thing

[frozen merlin]

true

[neat frost]

Where’s shyshu

[wooden flax]

sleeping porbbaly

[neat frost]

He probably has some interesting incorrect takes on this physics shit

[wooden flax]

hes in 🇮🇳

Shyshu

jk

[neat frost]

,ti —at India

[fathom swallow]

User cancelled selection.

Aug 15, 2022, 11:25 PM

[neat frost]

Okay fine

[neat frost]

Wtf

[zealous garden]

matter is both created and destroyed constantly

the thing really being conserved is energy

[neat frost]

Is it?

[velvet dagger]

I think nowadays the way people think about it is mass-energy conservation

[zealous garden]

energy is neither created nor destroyed* (don't @ quantum mechanics on that one)

then there is a mass-energy equivalence*

[velvet dagger]

Think particle antiparticle collision

[zealous garden]

so we can turn energy into particles, and particles (matter) into energy

[void holly]

@rich citrus

• sorry for the ping

[neat frost]

Don’t ping random people lol

But good joke

[zealous garden]

IIRC in QM energy can technically be created/destroyed from nothing if it's very small amounts for short periods of time

[neat frost]

Yeah I remember reading that too

[zealous garden]

"borrowed from the vacuum" I think is the way to say it

something relating to uncertainty

[neat frost]

But isn’t the thing with quantum theory that it’s all like up to interpretation?

[zealous garden]

I don't know that math myself so I won't go on a tangent about it

[zealous garden]

A huge amount yes, for example we have the mathematical process of collapsing the wave function to get results and renormalize and all that

but no one knows what that physically corresponds to

and a lot of little interpretation quirks like that

[neat frost]

Quirky theory

[zealous garden]

I bought the Goldstein and Safko mechanics book so I could start learning classical, but everytime I open it to start reading it loses me

I'll be back for a rant whenever I do understand QM

[neat frost]

Lol

[fervent pebble]

virtual particles?

[zealous garden]

yes, and I think something with the weak force

[bright hill]

the way I understand it is that it's mostly in extremely small amounts coz anything bigger would be EXTREMELY unlikely

there is this whole theory where our universe was basically created with such an event

that event was unfathomably unlikely ofc but when you got infinite time even unfathomably unlikely events will occur

[zealous garden]

yes, so it's very small amounts for short periods of time, or very large amounts for almost instantaneous periods of time

the amount of energy you borrow is inversely proportional to the time you get to borrow it

bounded by some uncertainty principle

the weak force bosons for example are massive

which means two things, 1) unlike photons they experience time and so the weak force unlike EM has a finite range

2. the virtual bosons have to borrow energy from the vacuum because of their mass (IIRC heavier than a proton)

this makes them extremely short-lived

limiting the weak force to nucleus-scale interactions where it is responsible for radioactivity

and some other stuff I don't know

[vivid halo]

iirc virtual particles, if you do accept them as being real, violate conservation of energy and momentum

[zealous garden]

Which is why I wanted to avoid them in my original statement, but left the asterisk

[vivid halo]

so they are a kind of problematic concept, since we really do want local energy conservation

[zealous garden]

Do you think virtual particles are the metaphysical truth O great Groupoid

or do you think they're just noise along the way to a better theory

[vivid halo]

oh I mean they're definitely an artifact of using perturbative theories, and often times there are non-perturbative theories that avoid this

they're also a pretty bad way of thinking about interactions with strong coupling

it is nice that perturbation theory works so well in general though as a computational tool

[zealous garden]

I love the pictures you get from virtual particles, but I'm not a fan of their existence and I hope we settle on something that avoids them entirely in the near future

near future simply meaning within my working lifetime

[ancient flame]

whats the point of #prealg-and-algebra #geometry-and-trigonometry #precalculus if the help channels exist

[neat lintel]

lol

[ancient flame]

is that a "lol idk either" or a "lol he doesn't get it"

[neat lintel]

80% first 20% second

[ancient flame]

why 20%

[neat lintel]

maybe moreso 85 15

[ancient flame]

does it reduce clutter in the help channels or smth?

[neat lintel]

if it helps

i don't get it either

[ancient flame]

lmao

[chilly hull]

[pure sun]

i think the difference is like

"someone help me on these factoring homework problems" goes in the help channels and "i want to discuss factoring" goes in the topic channel

like "i'm confused on X topic" is a topics channel question but "i want help on this problem" is a help channel question

but i agree at that level it's not a really necessary distinction

[ancient flame]

yeah I don't think there are many students that want to discuss topics like factoring tbh

I can see that for the early university channels but still people ask regular questions in #calculus for example

and it is kind of borderline for some of those early university topics tbh so it's kind of confusing on where you are supposed to post it

[full isle]

bruh factoring is hard

what are the rules for 7 and 11

[ancient flame]

LOL smh

[full isle]

do you know wtf?

[ancient flame]

idk wym

[leaden torrent]

the rule for 7:

a number k is divisible by 7 if you can write k = 7n for some n

you're welcome

[storm sage]

7: Pick the last digit of the number and double it; subtract that from the rest of the number; repeat until you hit a small enough number where you can check if it's divisible by 7

11: Add together digits 1,3,5,etc, and call that m; add together digits 2,4,6,etc, and call that n; check whether m-n is divisible by 11

Exercise: prove these work

[full isle]

fuck off im not proving them lol

thanks though

[storm sage]

Lol

[full isle]

11 is a cool rule though

i should memorize rules up to 100 when i get older

[storm sage]

up to 100 o.O

[ancient flame]

I love how nami responds to this but not my questions smh

[storm sage]

Answer to your questions: Read a book

(/s)

[ancient flame]

LMAO

[errant ridge]

tbh i only ever really see calculus and below in help channels, everything above calculus i only ever see in like the early university or advanced topic channels

[storm sage]

I've seen above calculus in the help channels before

[chilly hull]

yeah but its a bit rarer

[storm sage]

some formal language theory, some multivariable calculus, some diff eqs, some linear algebra, etc (idk exactly what counts as "above calculus")

[chilly hull]

i guess any class that you take after like

calc 1/calc 2

typically

[storm sage]

yeah but

you can take a lot of classes before calc 1 if you wanted to

there's not really a set path

[chilly hull]

idk

[storm sage]

ig just any "university mathematics" is past calculus I? but calculus I is also university mathematics so idk

[chilly hull]

im kinda taking a less orthodox path so i cant say much

nvm im not

im just swapping #proofs-and-logic and #linear-algebra

[errant ridge]

yea i meant that, like you usually learn calc 1 stuff in highschool

[errant ridge]

Nice

[chilly hull]

#get-advanced-access

someday

i will reach the point where i can understand what people are saying in there

[full isle]

@chilly hull do you study math in university

[chilly hull]

self studied

still in high school

[full isle]

do you want to study math in university?

[chilly hull]

yes

[full isle]

why

[chilly hull]

its fun

do i need any other reason

[errant ridge]

nope

[chilly hull]

i like math

[full isle]

maybe?

[wild lantern]

I helped somebody with a dihedral group type problem in one of the help channels once

[full isle]

i dont have another reason

[wild lantern]

Also sometimes people post looking for help on their programming projects

[chilly hull]

,w cat curve

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Cat Bread#6889

[chilly hull]

im kinda curious

ok yeah nvm

its on a polar graph

[errant ridge]

[chilly hull]

school in like

2 weeks

[errant ridge]

your school starts in 2 week?

[chilly hull]

ya

[errant ridge]

learn all the physics in 2 weeks

[chilly hull]

no

i want to study math with the 2 weeks i have

[errant ridge]

but you wanna do both physics and math?

[chilly hull]

i want a very strong foundation in math first

[errant ridge]

lol okay calc 2 in 2 weeks

[errant ridge]

same

[zenith sand]

If you take an easy version of calc 2 with no proofs it's possible

[chilly hull]

i mean im already like

[zenith sand]

but then that isn't rlly learning calc 2

[chilly hull]

40% through calc 2

[errant ridge]

he's already half way through calc 2 i think

yea

[zenith sand]

honestly the latter half of calc 2 is easier

like convergence of infinite series

[chilly hull]

i hated integration

[errant ridge]

then finish it fast and start with calc 3

[chilly hull]

the pain

and the horror

[zenith sand]

especially if you have a discrete background

the integration stuff is harder

[errant ridge]

this is why we need to teach differential forms to high school students

[full isle]

I will disagree

[chilly hull]

i mean im going through stewart

[full isle]

because you are always given nice examples

[chilly hull]

this means i have to do DEs

[errant ridge]

DEs are awesome

[zenith sand]

Oh right, I just mean in the context of the class

In the real world I have no idea

[errant ridge]

DEs and PDEs are at the heart of physics

[full isle]

irl integration is way harder

[chilly hull]

https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus__Early_Transcendentals_(Stewart)/00%3A_Front_Matter/03%3A_Table_of_Contents

chapter 7 was

[full isle]

ig in the class it isnt as straightforward either

[chilly hull]

[zenith sand]

Most of our convergence questions were actually really straight forward

[full isle]

khanacademy is better

[zenith sand]

like you could easily memorize how to do them

[errant ridge]

both are better

[zenith sand]

you just flowcharted it

[full isle]

no

[chilly hull]

im on chapter 8 now

[zenith sand]

but then again I feel my basis in calc 1/2 wasn't rlly rigorous

like we never did delta epsilon or many proofs at all

only things like testing for continuity and such

[full isle]

yeah you shouldnt

[errant ridge]

wbu intro analysis class?

[chilly hull]

i think i see epsilon delta in my proofs book

[burnt iron]

hey was wondering whos a good youtuber to learn statistics ap, or sites to help with statistics self study

[chilly hull]

let me get it real quick

[full isle]

khanacademy

[zenith sand]

AP stats sucks so hard

[chilly hull]

yup

13.2

[errant ridge]

there's literally like a million

[chilly hull]

definition of a limit

[burnt iron]

well I dont know that idk

bruh

[zenith sand]

stats w/o calc in general was crazy confusing

made no sense whatsoever

[burnt iron]

ok what about physics

[full isle]

??

[burnt iron]

is that on khan too

[full isle]

yes

[burnt iron]

cool

[chilly hull]

you cant do physics without calculus

[zenith sand]

wdym

[burnt iron]

What

[zenith sand]

AP statistics iirc is statistics without calculus

[errant ridge]

you cant do physics without differnetial geometry

[full isle]

yes

[chilly hull]

i mean you can do like

[full isle]

its not crazy confusing

[chilly hull]

basic stuff

[zenith sand]

u just have very little basis for understanding what you're doing I felt

[chilly hull]

but without calculus its kinda unintuitive

[zenith sand]

like almost none

yeah

[errant ridge]

yes you can only do basic stuff without differential geometry

[full isle]

oh maybe lol

[errant ridge]

[chilly hull]

oh no

[full isle]

i took it at the same time

[zenith sand]

like some subjects get easier with calc imo

[full isle]

so i didn’t have the experience in isolation

[chilly hull]

another subject of math i need to add to my bucket list of studies

[zenith sand]

without calc if you do the same problems you can't really understand what you're doing

[burnt iron]

u need calc for stats?

[zenith sand]

depends what you're doing

[full isle]

rigorously yes

[zenith sand]

to understand almost any of it yea

[burnt iron]

I took alg 2 and im in ap physics and ap stats rn

[full isle]

and most times yes

[chilly hull]

ap physics 1

or c

[zenith sand]

you're learning baby stats

[burnt iron]

1

[honest veldt]

I have a proposal

[burnt iron]

oh baby stats

cool

[honest veldt]

As we all know, calculus is extraneous

[errant ridge]

also add lie theory, abstract algebra, topology and functional analysis

[honest veldt]

I propose we replace it

by what I will call

[chilly hull]

oh no

i mean

[zenith sand]

measure theory

[honest veldt]

very small algebra

[chilly hull]

half of them are on already

[errant ridge]

with differential geometry

[zenith sand]

category theory

[errant ridge]

also hello strad

[zenith sand]

look I'm throwing out subjects I haven't studied

to look smart

[honest veldt]

Aye neam how're you movin 2nite

Still slow?

[chilly hull]

my plan
#calculus -> #proofs-and-logic -> #multivariable-calculus/#linear-algebra -> #groups-rings-fields/#real-complex-analysis

idk where to put #odes-and-pdes

[zenith sand]

start w/ proofs

[chilly hull]

and #diff-geo-diff-top

[errant ridge]

actually i've sped up quite a bit

how're you movin?

[zenith sand]

I wish I had taken discrete before calc

[honest veldt]

My plan
#discussion -> #chill

[zenith sand]

it's not that important but u get more out of calc if you do discrete first I feel

[honest veldt]

That's what I love to hear, my man, excellent

[errant ridge]

wbu strad?

[chilly hull]

calc is awesome.

[errant ridge]

diff geo is like calc but on steroids times a million

[neat lintel]

pretty based of you

[errant ridge]

[chilly hull]

i might throw in #advanced-pdes for the kick of it too

[errant ridge]

yes! we're all diff geo gang here

[honest veldt]

Pretty pretty solidly, got some good velocity 2nite

[ancient flame]

yayyyy

[honest veldt]

Merci for asking

[ancient flame]

diff geo!!!

[chilly hull]

i dont know what diff geo is but

[errant ridge]

diff geo!!!!

[chilly hull]

i will get to learning it

[errant ridge]

dont worry my child, you will soon

[honest veldt]

diff geo!!! (what's a tangent vector guys)

[errant ridge]

differential operator somehow

[honest veldt]

what the crap

[ancient flame]

should I do complex analysis before diff geo

[errant ridge]

or a derivation i think

[errant ridge]

it's not really required but if u want to you can

[ancient flame]

makes sense

I feel like complex analysis is more lower level so it probably makes sense to do that first

[zenith sand]

I'm doing the stupid CS version of numerical analysis

[ancient flame]

L

[honest veldt]

what is numerical analysis

[zenith sand]

bow down

[honest veldt]

Btw

[neat lintel]

an element of a tangent space

which is the best linear approximation of your manifold at a point

[modest rune]

I think people usually recommend a book/course on riemann surfaces prior to doing full diffgeo?

[honest veldt]

You pick a point and construct a tangent space there?

[modest rune]

And complex analysis is nice to know if you're going to study riemann surfaces

[burnt iron]

can you give me 1

[errant ridge]

which is then an element of the tangent bundle?

[modest rune]

This is a rec I've seen people give several times tterra clearly I have no dog in this race lol

[neat lintel]

did you mean curve and surface theory?

[ancient flame]

oh

[modest rune]

Yeah

[neat lintel]

okay

thonk retracted

[ancient flame]

I see

[neat lintel]

"riemann surface" implies you're looking at these things as complex manifolds

[ancient flame]

yeah I'll probably do that then

[errant ridge]

:O

complex manifolds

[neat lintel]

of course, since the tangent bundle is the union of the tangent spaces

[ancient flame]

so here's my plan

complex analysis -> riemann surfaces -> diff forms / diff geo

[storm sage]

complex analysis is cool af

[errant ridge]

complex anal covers riemann surfaces no?

[ancient flame]

but first I should better learn real analysis and linear algebra but I kinda don't want to

[neat lintel]

too bad, do it

[ancient flame]

ugh

[errant ridge]

yes do LA over arbitrary fields and and real anal in R^n

[storm sage]

not necessarily, usually it covers riemann spheres tho

[errant ridge]

i see

[ancient flame]

stop abbreviation analysis as anal!!!

[neat lintel]

how do you plan on doing riemann surface theory without knowing DG?

[errant ridge]

mathematical anal

[ancient flame]

didn't max say that 😭

[neat lintel]

max meant curve and surface theory

[ancient flame]

oh ok

[neat lintel]

you do not need complex analysis for curve and surface theory

[honest veldt]

my plan is as follows

[neat lintel]

you also do not need curve and surface theory to do differential forms / differential geometry

[honest veldt]

linear algebra -> the rest

[storm sage]

true

[errant ridge]

best plan

[ancient flame]

so then what do I need for DG besides LA

[errant ridge]

either multivariable calc or anal in R^n

[neat lintel]

same thing

learn that and LA

ideally LA before mvc

[ancient flame]

oh ok

[honest veldt]

does pugh have anal in Rn

[errant ridge]

yes

[ancient flame]

I did mvc already

[neat lintel]

please don't read pugh for that

[errant ridge]

pubg does

[honest veldt]

Oh no...

Oh yeah doesn't Rudin also suck in multivar

[ancient flame]

what???? not pugh???

[honest veldt]

What the heck do I read then

[errant ridge]

pubg

[errant ridge]

players unknown battle grounds

[ancient flame]

oof

[honest veldt]

Ooooh

[storm sage]

giving honorable mention to this cursed playlist I found https://www.youtube.com/playlist?list=PL8erL0pXF3JYm7VaTdKDaWc8Q3FuP8Sa7

[neat lintel]

read one of those

[errant ridge]

in that order!!

[neat lintel]

no

reading both is a massive waste of time, and i'm trying to make you guys not waste your time

[honest veldt]

Munkres it is, he says analysis, whole spivak says calculus

[errant ridge]

dude it's the best!!! prof ghrisr is awesome!!!

[honest veldt]

This is why munkres is a bigger badass

[neat lintel]

too much shitposting

i give up

[storm sage]

wait are you fr lol

[honest veldt]

sorry... I really do appreciate it

I'm taking the recommendation seriously I promise

[errant ridge]

I am, his derivatives, integrals, fields and forms playlist

it's amazing!

so beautiful too!

vol 2, 3 and 4

[storm sage]

i hope none of his students have epilepsy

[errant ridge]

bruh

his art work and all the animation is beautiful

and so are his explanations

[errant ridge]

spivak's calculus on manifolds and then spivak's intrkduction to differential gometey

[honest veldt]

his diff geo books have the MOST badass covers in math book history

[ancient flame]

bruh y'all scared tterra away

[errant ridge]

[honest veldt]

yeahh, I should cut down the shitpost status....

[errant ridge]

[honest veldt]

Wait this is discussion 2

What am I doing here

[errant ridge]

stard change your name to "neamesis for honorable"

[honest veldt]

OK I will

[errant ridge]

lol dont actually do it

[honest veldt]

You were saying?

[errant ridge]

Lmao

chad

[neat lintel]

ignore both and use Amann and escher

ignore those ones and read http://www.uwyo.edu/mathstats/faculty-pages/math/_files/analysis_2.pdf this
then use loring tu book

[dapper brook]

Anything worth mentioning about my solving radicals presentation other than solving radicals?

[neat lintel]

sorry?

"anything worth mention"

what do you mean

[dapper brook]

Well I have to speak for 5 minutes

I’m not sure what else to say about radicals

[neat lintel]

oh ok that what you meant

what do you have so far?

[dapper brook]

I have a in depth explanation of what a radical is, and tips and tricks for solving one, I also included a practice problem I’ll self draw and another practice problem for the class hopefully

[neat lintel]

solving radicals just boils that to either isolating it to one side of the equation, using (a+b)(a-b) = a^2 - b^2 somewhere

[storm sage]

I don't think which book you use to learn multivariable calculus will be highly influential in your future lol

Unless the book you use is complete trash

[hollow sundial]

this book i used had historical tidbits

this is one of my favorites. no bias

[neon silo]

As a physics major, is it worth learning some real analysis either thru a course or on one's own time? In the long term, would having a better theoretical understanding of calculus improve one's ability to reason about physical problems?

[vivid halo]

it depends on which area of physics you're working on. Anything involving quantum mechanics will benefit from this since mathematically rigorous treatments of quantum mechanics involve a lot of functional analysis and real analysis

[coral mason]

would physics help my conceptual understanding of calculus?

[velvet dagger]

Lots of things in calculus can be approached a few different ways

e.g. while the "linear approximation" idea of a derivative, which is the more modern way of thinking about it, can be thought of on its own terms, the rate of change idea (which is what led to the development of the idea in the first place) does take inspiration from mechanics.

Also, understanding sin and cos using the differential equation is related to simple harmonic motion/springs

[errant ridge]

why can't you do calculus on general topological spaces again?

you need a notion of distance?

[errant ridge]

physics would help your understanding of lots of different fields of math! the more advanced physics you study, the more advanced mathematics it helps to conceptually understand :D

[unborn trellis]

depends on what you mean by calculus. all you need for limits is a topology

differential calculus is a euclidean thing, since the difference quotient uses vectors/matrices

[errant ridge]

so you can do integral calculus on a topological space?

that's the bare minimum structure you need for that?

[unborn trellis]

to do integral calculus you need a measure

on a topological space then you'd want borel measures

[errant ridge]

oh wow nice, so for differential calculus what more structure do you need? like addition of the points and stuff?

[unborn trellis]

these aren't definitive answers, but my rule of thumb is differential manifolds are the most obvious class of spaces you can do calculus on

[errant ridge]

right yeah

[unborn trellis]

there are also weirder things when you take a categorical point of view

[errant ridge]

[unborn trellis]

smooth toposes, lie groupoids, diffeological spaces

but those are weird

[errant ridge]

[unborn trellis]

also, depending on the measure space you can do "differential calculus" in the sense of radon-nikodym derivatives and the fundamental theorem of calculus

[errant ridge]

huh, I've heard of the radon-nikodym theorem (no idea what that is) haven't studied measure theory yet

[unborn trellis]

you can import measure theory into smooth manifolds so that it encompasses that

[unborn trellis]

it's essentially how you take derivatives with respect to a measure

[errant ridge]

wait so you need a topological manifold to be smooth in order to do diff calculus?

[unborn trellis]

it's certainly a sufficient condition for calculus

[errant ridge]

[unborn trellis]

here's a fun post about this if you're interested

https://math.stackexchange.com/questions/2857374/lebesgue-measure-theory-vs-differential-forms

[errant ridge]

Lol I have to understand integration with measures and integration with differential forms first

but thanks

[bright hill]

Pugh is really good tho

I've read so many reviews that were practically gushing over it

Pugh is basically the anti thesis to rudin

Similar content, similar exercises (in terms of difficulty, etc), but MASSIVELY improved exposition

[fading zealot]

Next time I'm exlaining to someone how a knight moves Imma say that it moves along lines with slopes of 2, -2, 1/2 and -1/2

[vivid halo]

description unclear, moved knight 4 up 2 right

[neat frost]

Good morning DarQ!

[bright hill]

Good morning slurp!

[fading zealot]

Generalized knight

[storm sage]

yayyy

[neat lintel]

Is there a way to find for what values of k is an expression, for example 5k-4, is a perfect square?

I can't really find any sources on Google nor on Wikipedia

[neat lintel]

Other than just random substitutions

[zealous garden]

A knight moves onto a space of the opposite color that is a distance of 2 away (using the king metric)

[neat lintel]

⚗️ 🧻

[mint canopy]

Let 5k-4 = n^2. Then 5k = n^2 + 4. I assume you want k to be an integer, so we now know that n^2 = 1 mod 5. This means that n = 1 or 4 mod 5. Hence k is of the form (n^2 + 4)/5 where n is any integer which is either 1 or 4 mod 5. I think this is about as good as you're gonna get.

[neat lintel]

I am a bit unsure how n = 1 or 4 (mod 5)

[mint canopy]

n^2 = 1 mod 5, do you see that?

[neat lintel]

Yes

[mint canopy]

Then you can simply check that of all the possibilities (n = 0, n=1, n=2 etc) the only one that produces n^2 = 1 mod 5 is n=1 and n=4 mod 5.

[neat lintel]

Sorry for the inconvenience I'm causing, but I'm not that well-versed with modular arithmetic

[neat lintel]

I see

Thank you

[wooden flax]

Mfw when calc teacher takes of points for not showing every tiny step

[neat lintel]

do everything formally in first order logic next time

[wooden flax]

Next time I’m gonna do everything with epsilon delta

/hj

[dire beacon]

a/b*100 = (a*100)/(b*100) is that true?

[surreal sapphire]

try some numbers

[storm sage]

what happens to your equation when a=100 and b=100?

[dire beacon]

,calc 100/100*100

[fathom swallow]

Result:

100

[dire beacon]

,calc (100*100)/(100*100)

[fathom swallow]

The following error occured while calculating:
Error: Syntax error in part "(100100)/(100100)" (char 1)

[dire beacon]

,calc (100100)/(100100)

[fathom swallow]

Result:

1

[dire beacon]

ok, its not

[honest veldt]

I probably will try it yeah, Rudin was good but now that I think back on how dry it is, I don't really feel like picking it back up...

[hollow sundial]

@thorny osprey rules don't really state exceptions

● ... We are not here to do your homework for you, but we are here to help you learn how to do your homework. See ❓

[thorny osprey]

What's the problem man? The rules don't state that we can't solve the question entirely either. Also, what matters in my opinion is whether the asker tried their way and attempted it completely before receiving a solution

[hollow sundial]

you're perpetuating people "attempting" and just coming here expecting full solutions provided by other helpers.

i'm not the one who wrote the rules, so maybe a <@&268886789983436800> can chime in

[long epoch]

Was that really worth pinging mods

[thorny osprey]

Giving away one solution is not "perpetuating" anything imo. And what other helpers do or don't do is upto them, I'm okay with others giving full or partial solutions or hints or whatever to askers

[tall badge]

it actually does include presenting full solutions

name calling doesnt fly either

[hollow sundial]

it was obvious kd won't listen to me

[thorny osprey]

I don't think the issue is worth pinging the mods over or even having a debate over

I'm not breaking any rules as per my understanding

[hollow sundial]

and i disagree, hence why i pinged.

[hollow sundial]

you're just perpetuating this further

[thorny osprey]

We can agree to disagree then

[tall badge]

im gonna say the rule does include giving full solutions

[hollow sundial]

yea that's not 100% clear

[thorny osprey]

I don't think giving a full solution to someone after they have tried their way around the problem and failed is bad or unfruitful

[hollow sundial]

i mean it's clear to 99% of helpers

[tall badge]

if the full sol is really short

[hollow sundial]

you basically handed them something they could submit as homework/exam

[tall badge]

like a one liner

then id say its ok after its clear the asker tried and failed

[thorny osprey]

And that hurts you why exactly?

[hollow sundial]

i don't see how that's relevant

[thorny osprey]

It is, as you are the only one objecting

[tall badge]

but not for a half page or smth longer

[thorny osprey]

I don't see anyone else taking it up as an issue

[hollow sundial]

i'm the only one speaking up

[thorny osprey]

Hey look man, I'm just here to do some math and enjoy solving problems and helping out others. If you have an issue with what I do, please feel free to ping whomever you want and take it up with them. Peace out ✌️

[tall badge]

im not sure if ur seeing my msgs @thorny osprey

[thorny osprey]

I am

I understand what you are saying

[hollow sundial]

pretty clear kd won't stop, but it's in your hands now mods

[tall badge]

this isnt a personal matter to us, rather a matter of academic integrity

tho i get that there may be some frustration from the asker ‘not getting it’ after many attempts

[bright hill]

B-b-b-but

Bragging rights!!!!

[analog salmon]

Everyone always talks about baby rudin, but what about daddy rudin?

[neat lintel]

people definitely talk about his real and complex analysis book

[honest veldt]

I no longer care!!!! Screw that book, I want PICTURES!!!!!

[alpine kindle]

it's usually called papa rudin

[odd narwhal]

i'll be honest

no one is really going to care that you read baby rudin

[wooden flax]

if i have a point (a,b,c) \in R^3 and then want to project this onto the plane spanned by (1,0,0) and (0,1,0) is the notation pi_{12} fine for this

Pugh on top

whats the proper notation for this

hm or ig i could say pi_1+pi_2

[deep mango]

Shin cope. Rudin is a rite of passage

Welcome to the cult my friend

[wooden flax]

ryc do you know the answer to my question

[frozen merlin]

as long as you define your notation I'm sure \pi_{12} is fine

although I would probably use \pi_{1,2}

[chilly hull]

what

im getting mixed reviews of rudin now

some people tell me to avoid it while others say that its good

[frozen merlin]

if I have some 2-variable polynomial p(x,y) and I know that p(a,b)=0, but I know that p(x,y) doesn't have a factor of either (x-a) or (y-b), what can I say about the factors of p?

maybe this is a dumb question but my brain is fried

[storm sage]

It's not good for self-study

It's good if you've already reached a certain point of mathematical maturity and you're already somewhat familiar, or if you have an instructor to guide you through it

[pure sun]

P need not factor at all

[frozen merlin]

even over the closure?

[pure sun]

Yeah

Take your favorite irreducible 2-variable polynomial

Say, y-x^2

Pick any a in your field

[frozen merlin]

that is a pretty good one

[pure sun]

Then (a, a^2) is a zero of that poly

[frozen merlin]

oh yes

so it was a dumb question

[pure sun]

All you can say from p(a,b) = 0 is that p is a sum of a multiple of (x-a) and a multiple of (y-b)

(Not uniquely)

[frozen merlin]

how would you find those multiples, say for y-x^2 and (a,a^2)

[pure sun]

You can do it algebraically, basically you want to rewrite your polynomial in terms of (x-a) and (y-a^2)

So like, let’s see

[tender tulip]

oh god

[neat lintel]

no

[tender tulip]

i don’t want to touch multi-variate poly rings yet

[pure sun]

y - x^2 = (y-a^2) - (x-a)^2 + 2a(x-a)

So in this case the reresentation is unique

In general you can compute some partial derivatives

[tender tulip]

I decided to consider orbits of automorphisms and shit and I fell down a rabit hole but now galois extensions make a bit more sense

[pure sun]

Basically youre doing a multivariable taylor expansion

And the coefficients are just various partial derivatives

[frozen merlin]

oh yes of course

makes sense

cheers

[pure sun]

And since p(a,b) = 0 the constant term is 0

And all of the rest of the terms are multiples of (x-a) or (y-b) or both

Np

[pure sun]

… then dont

The conversation was not about you

[tender tulip]

i was half joking half replying to Ttera

[neat lintel]

what about "no" did you miss? 🙂

[pure sun]

But interrupting a conversation about math to let everyone else know that you dont like the math being discussed

Is not okay

[tender tulip]

alr

[pure sun]

Literally nobody cares

[tender tulip]

That’s pretty fuckin cool ngl

[pure sun]

That is a more appropriate thing to say

[frozen merlin]

I think it's not so bad over text bc it's easy to ignore, but yeah this is annoying irl

[tender tulip]

Is there an application of multivariate taylor expansions for multivariate polys

beyond this specific case

[open aspen]

welcome to the mathematics community

[pure sun]

Pick your favorite application of single-variable taylor polynomials

[open aspen]

just say "Hatcher's Topology" or "Lang's Algebra" and immediately spark a debate

[pure sun]

Now you can do the same thing for multivariable functions

[tender tulip]

this fuckin shit, what

[unborn trellis]

It's surprisingly clear once you work it out

[open aspen]

yeah its not so bad

[velvet dagger]

Did someone say Hatch... jk

[neat lintel]

hatcher is fucking terrible

[unborn trellis]

FUCK HATCHER

[true quartz]

Hello

[open aspen]

its a hatcher sucks hour it seems

try again in a few

[unborn trellis]

average hatcher lover:

[dapper badge]

Pictures make me go

[true quartz]

Wth is that

[unborn trellis]

just kidding that's a transphobe I found on twitter

[dapper badge]

Proofs make me go

[open aspen]

average algebraic topologist

[true quartz]

His head look like grenade

[tender tulip]

h o w

[open aspen]

just draw some diagrams

[pure sun]

Im sorry mizalign but like

[neat lintel]

this is the algebraic geometry definition of the cotangent space. the idea is, very roughly, to single out the "first order" behaviour of functions near x

[pure sun]

How do you not understand

[unborn trellis]

it's the map f -> df_p for f in the germ at p

[pure sun]

That some math becomes easy to understand

When uou have more background

Maybe if that definition seems like nonsense to you

It means that you dont have the background to understand it yet

[true quartz]

Math is easy when you take it in parts and learn the logic from it

[unborn trellis]

The I² part corresponds to the product rule basically

[neat lintel]

wdym buncho bananas it's simple it's just the M coseparator of blah blah blah

[pure sun]

Hahaha

[tender tulip]

this is more understandable

[dapper badge]

today nG helped me relearn real/quaternionic representations and frobenius schur indicators!

[hollow sundial]

great job @vivid halo

[full isle]

lol

this is bullying

[dapper badge]

there are still some things I should consider with regards to analogies with Casimir invariants, but I think I've forgotten too much about Lie algebras to look into that right away

Or I guess I could look into higher Frobenius Schur indicators since I've heard/seen them occasionally but never really "got" them

[devout sage]

A museum opens with a collection of 250 items. During the first week after
the opening, an additional fifteen items are donated to the museum’s
collection. In each subsequent week, eight more items are donated to the
museum’s collection than were donated in the previous week. Work out how
many items are in the museum’s collection after sixteen weeks.

So let $a_0$ be the items already in the meseum and $a_i$ be the items that is added in the ith week after opening.

a_0 = 250, a_1 = 250 + 15, a_2 = 250 + 15 + 8, a_3 = 250 + 15 + 82, a_4 = 250 + 15 + 83, ...., a_16 = 250 + 15 + 8*15
is this correct?

[fathom swallow]

WhyPerson
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[dire root]

Can someone please explain why Euler's formula doesn't imply that 2pi * i = 1 ? Since it implies e^(2pi i)=e^(0 i) = 1 ?

Sorry if this has been done before lmao

[zealous garden]

how would it?

[storm sage]

first can you explain why (-3)^2 = 3^2 doesn't imply -3 = 3?

[dire root]

Oh right, since the process of squaring isn't one to one?

[storm sage]

yep

[zealous garden]

wtf that's not even the same form but it worked

[storm sage]

?

what's not the same form

[dire root]

So it would be the same with that formula, how would you describe that restriction?

[storm sage]

what restriction?

[zealous garden]

Your equation is a different inference

the question is "why doesn't a^x = 1 mean x = 1?"

[storm sage]

no...?

wait

[dire root]

I assume since it relies on sinusoid that there would need to be a domain restriction for the inverse

[storm sage]

oh I read it as 2pi i = 0 i

not 2pi i * i = 1

Lol

where did you get 2pi i * i = 1 from?

[dire root]

oops sorry i meant 2 pi i = 0

[storm sage]

okay that's what I figured haha

[dire root]

yeah from the e ^ 2pi i = 1 lmao

So just when we use that formula and try to take the inverse, we have to restrict the domain then?

[storm sage]

Well

[zealous garden]

it's more we have to restrict the range innit

[storm sage]

Yeah I guess that's one way to put it

Are you familiar with "branch cuts"?

[dire root]

Not that term no

[storm sage]

Okay

[dire root]

Right that's what i meant lmao, words

[zealous garden]

It has multiple possible answers so we have to choose one to go with

[dire root]

right, similar to the inverse sinusoids correct?

[zealous garden]

yes, this is the basics of the idea of a branch cut I think

[storm sage]

Basically, in complex analysis, the logarithm is only a uniquely specified function if you exclude part of the domain ("a branch cut") and specify a point

[zealous garden]

I haven't encountered them [branch cuts] formally yet

[zealous garden]

yes

in fact

[storm sage]

For example, you could consider the logarithm which is defined on everywhere in C except for the non-positive real line

[zealous garden]

you'd be taking the inverse of sinusoids

[storm sage]

And then let log(1) = 0

This would be one "branch" of the log function

[zealous garden]

in taking the inverse of euler's formula for e^it

[dire root]

wdym by "uniquely specified function"?

[zealous garden]

one output

[dire root]

yeah that's what i meant

[storm sage]

For example let's say you want to find the inverse to y = x^2, you have two possibilities which are inverses, either y = sqrt(x) or y = -sqrt(x)

you specify one by saying that sqrt(4) = 2 instead of -2

for example

or you just say the "principal square root"

[dire root]

right, we cut off one of the sides to make the function one to one then?

[storm sage]

similarly in complex analysis, we have the "principal branch of the logarithm"

[storm sage]

yeah, of the x^2 function

similarly, the principal branch of the logarithm function restricts the domain of the e^z function to be -pi < im z < pi

[zealous garden]

yesu yesu

[vivid halo]

a good way to think of the logarithm example is that log(x) is the integral from 1 to x of 1/z dz in C-{0}, but along which path? The branch of log depends on how many times you let this path wrap around 0

[storm sage]

wait the branch of log depends on the starting point

you always restrict the domain so that the path doesn't wrap around 0

but yeah this is why you have to restrict the domain

[zealous garden]

that tastes like hot sweet honey to my mind, I like these words magic man

[bright hill]

I know lol, I'm mainly shitposting copium

[dire root]

what does "wrapping around" exactly mean in this context?

[storm sage]

do you know what a path integral is?

if not then ignore that lol

I suggest Welch Lab's series on complex numbers on youtube :)

[dire root]

Sorry I'm only familiar w just the calculus integral for now lol

[storm sage]

it is a neat intro to complex analysis

you're all good haha

[odd narwhal]

Line integral

Path integral is the physics thing

[zealous garden]

What

those are two different things

[odd narwhal]

If you try looking up path integral you'll get qft results

Yes that's what I'm saying

[storm sage]

?

[odd narwhal]

Hmm

[zealous garden]

yes that's the path integral

[odd narwhal]

Well path integral is not a very commonly used term

[zealous garden]

the line integral is "directed" so-to-speak

[storm sage]

I thought it was just called the path integral formulation because it uses path integrals

[odd narwhal]

So is the complex line integral

[storm sage]

I always thought line integral sounded weird because it's not a line (well I guess it is in the input space after you parameterize it)

[dire root]

I mean I will have to do more looking into this specifically tysm you guys

[zealous garden]

a true directed integral

[odd narwhal]

A path integral is an integral on the space of paths

[storm sage]

no problem!

[storm sage]

ohh okay

that makes sense

after some looking up, I think they are synonymous terms in this context

[odd narwhal]

Sure

I just haven't really heard it be referred to as a path integral

In part to not be confused with the physics path integral

[storm sage]

I'll just call it a line integral then lol

[storm sage]

?

line integrals over the complex plane are directed

if you reverse the path orientation, the sign flips

[zealous garden]

I know

that's my point

I'm complimenting them

[storm sage]

okay lol

[zealous garden]

they collect flow and flux at once as well

the question to my determination is whether ds is a scalar or vector (pseudoscalar in this context)

if it's scalar then path, vector then line

how do you make the distinction?

[storm sage]

I just use them synonymously, I guess I'll use line integral from now on though

[zealous garden]

like you'd take a path integral to find the length of a curve, but a line integral (over the same integrand) would give you a vector pointing from the beginning of the curve to the end

[modest rune]

I think ive heard path integral a lot

in complex analysis

since line integral is like an objectively bad name i think its reasonable

[odd narwhal]

Fair I guess, I've never heard it

[vivid halo]

path/line/contour integral

whatever you choose to call it

integrating along a path means choosing some parameterization of your path (so like writing down equations for the x and y coordinates of that curve in some "time" variable t in [a,b] say) and then turning this line integral into an ordinary integral from a to b by change of variables

[storm sage]

What do you call a line integral over a closed curve

I've always called that a contour integral

but now I'm realizing people only really use that for complex analysis

[vivid halo]

contour usually

err

only in the complex setting yeah

[storm sage]

what do you call it in the real setting then

[vivid halo]

I'd be hesitant to call this a contour integral outside the complex analytic setting

[storm sage]

seems odd that the terminology would be different

[vivid halo]

I mean I tend to just say "integral along path" or "integral along loop"

since there's usually a particular path I'm referring to

[storm sage]

man

[deep mango]

Uhhh

[storm sage]

I hate math terminology

[deep mango]

Path integral also has a very very different meaning

Im not sure if thats been brought up

[vivid halo]

yeah

it has

[storm sage]

yeah it has

[deep mango]

Ok good

[storm sage]

ryc what would you call an integral over a closed curve? (in a real setting)

[deep mango]

This just confuses a lot of people so I much much prefer calling the other kind line integral or contour integral

[deep mango]

An integral over a closed curve?

[storm sage]

okay lol

[deep mango]

I would not give that a name

Or a closed contour integral

For a curve in R^2

[storm sage]

wait aren't all contour integrals closed?

[modest rune]

that is probably a definition written down somewhere

[storm sage]

wait so what do you call this symbol then $\oint$

[fathom swallow]

Eric Tao (he/him)

[modest rune]

i never use it

[deep mango]

I would not assume that a contour integral is closed

[modest rune]

I just use $\int$

[fathom swallow]

MaxJ

[deep mango]

I would not call it anything

I would call it oh-int, idk

[modest rune]

$\int_\gamma f dz$

[fathom swallow]

MaxJ

[modest rune]

where \gamma is a path

and i'd just call it "the integral"

[deep mango]

Yeah i would absolutely never write a circle on an integral

[modest rune]

or "along gamma" if i need to specify

[deep mango]

That would get way too confusing

[storm sage]

okay lol

[modest rune]

I just think its like

[storm sage]

just gonna call everything an integral then, screw it

[modest rune]

useless

[zealous garden]

"Integral along the closed .... "

[modest rune]

I would say let gamma be a closed path

if I wanted to take an integral over a closed path

but for some reason did not want to specify which one

[zealous garden]

because you can put higher dimensional things than paths on the bottom

but yes "Integral along the closed path C" is $\oint_C$