Mathematics is hard. Walk before you run.

This is because you can look at the inclusion map $i : S' \to S$ of the submanifold; $i(S')$ is closed because it's the image of a compact set and open because $i(S')$ can be written as union of $i(U_\alpha)$ where $U_\alpha$ are the charts on $S'$. Once you know that fact I mentioned, $i(U_\alpha)$ are open so union of those are open.

By connectedness of $S$, a subset of $S$ which is both open and closed is the whole space $S$. That tells you $S' = S$.

@BalarkaSen could you kindly provide an example of such a surface?

bearing in mind that I have absolutely no knowledge of differential geometry / topology whatsoever, lol

Example of what kind of surfaces?

apart from, you know, theorema egregium aka "why you can't roll a sheet of paper into a torus"

@BalarkaSen a compact surface and a connected surface

the sphere

a sphere and a sphere?

lol

I wrote it twice and got sniped

is the torus a surface?

so you got sniped twice, I guess

yes

how can a surface be contained in another surface without them being the same?

it cannot

that is what the theorem says

but the theorem added "compact" and "connected"

oh, you need those

think of S^2 and S^2 minus north pole

@LeakyNun Small open disc in a bigger open disc

the latter is contained in the former, both are surfaces

or S^2 contained in disjoint union of two spheres S^2 \cup S^2

(or annulus for the smaller one)

hmm, this is interesting

what if I glue two spheres end to end?

not a manifold

that's not a surface because at the common point it doesn't look like $\Bbb R^2$

I apologize for my stupid questions :P

I think @LeakyNun is very smart, not stupid. =D

what if I remove the common point?

@BalarkaSen

Ah, then you get disjoint union of two "S^2 with a puncture"

disjoint union of two open disks, if you will

how do you call S^2 minus north pole?

wait, it's isomorphic to the open disc?

homeomorphic/diffeomorphic to it, yep

Since I live in Antarctica, I call the south pole home.

heh, looks like how flat earthers view the globe

Let's remove the south pole instead :P

but how do you call it?

I mean, in symbols

and are their fundamental groups isomorphic?

I mean, both have trivial fundamental groups, right

a punctured 2-sphere?

@LeakyNun yes

@Semiclassical yes

@AlessandroCodenotti thanks

hi guys

My teacher told us to do the homework examples on computer

can anyone tell me a good program to write math stuff on ?

fundamental group on 2-sphere with n punctures should be free group on n generators, if memory serves.

Hello semi !

:)

@KasmirKhaan heya

@Semiclassical n-1 generators, yeah

durr

right. disk with n punctures would have n generators

mhm

can anyone tell me a good program to write math stuff on ?

interesting

@KasmirKhaan To state the obvious, it depends what you're doing and what you have access to.

@KasmirKhaan Yes, use LaTeX to type math.

@KasmirKhaan what are you going to write?

[Chemistry] Encounter that bizarre non stationary point minima with a hessian where all eigenvalues are positive problem again. Now to check if this hessian is singular

TeXstudio fans :P

abstract algebra stuff @LeakyNun @WillHunting

If you want just symbolic typesetting, something like TexStudio or etc is fine.

woooo TexStudio

woooo abstract algebra

latex is it something i can download? or site?

there's a number of different installations.

You need a LaTeX distribution

abstract algebra is cool yes @SteamyRoot

(I recommend TexLive, but there are alternatives)

@KasmirKhaan I suggest you install TeX Live and TeXworks. The former is a TeX distribution and the latter a TeX editor. Alternatively, just use the equation editor in Microsoft Word.

@KasmirKhaan LaTeX itself is a set of programs that turn your .tex files into nice and readable files, such as .pdf

Guys thanks alot for answers but go easy on me

I prefer Kile myself, but I always tend to use weird programs nobody else likes

Overleaf lets you do it in-browser I think

am beginner on thsese stuff

It's a website that does LaTeX

In fact, there are also some online LaTeX compilers - which might be useful for somebody who just wants to try this out without having too much things to install. TeX.SE: Compiling documents online

But, again, are you wanting to typeset your calculations or to actually do abstract-algebra calculations?

I need one program please

One of those is Overleaf - mentioned by Akiva.

@TobiasKildetoft Kile has too many buttons, and I prefer no buttons at all, lol.

@KasmirKhaan LaTeX is fairly easy to get started with once you find a template. Then you just need to look up the codes for all the symbols, which can be done at detexify

okay thanks alot all :)

overleaf and sharelatex are getting merged eventually, anyway :O

huh

@WillHunting I don't really use any of the buttons, as I have keyboard shortcuts for them all

Wolfram Alpha?

so I google latex and see what happnens

Wolfram Alpha is very hit or miss, unfortunately

If you want to do math you could use like Mathematica or something

though it's paid

codecogs.com/latex/eqneditor.php

cough sage cough

this what I got

If you've got a Mathematica license through your uni, it's great

@MathematicsAminPhysics ?

Did you guys mean that?

but if you don't....

@Semiclassical I do

@KasmirKhaan That one is not really for typesetting documents. You can only write single equation in this.

Ah. Then that's good for some stuff. (Though it's not a word processing/ presentation software.)

@KasmirKhaan Are you on Windows or Linux or Mac?

Damn it >< well guys let me explain , we have homework problems, we gonna do them and then send them on pdf format

With regards to abstract algebra calculations, I'm not sure what's out there

@TobiasKildetoft windows

I think the LaTeX distributions are called different things depending on that. It also changes which editors are available

TexLive + TexStudio is available on all platforms :3

@KasmirKhaan Ahh. So you just need to be able to do word processing with symbolic math?

@TobiasKildetoft TeX Live and TeXworks works on all 3 platforms, lol.

I never thought this will be this complicated :D it is harder than the homework problems

@KasmirKhaan Then Overleaf is good if you don't want to download anything

@Semiclassical Hmm did not know that word can do that

Didn't mean it was.

overleaf ill google that @AkivaWeinberger

You guys are the best :)

@WillHunting Ahh, I see. This may have changed since last time I installed it, or maybe I just never really bothered to find out

@KasmirKhaan microsoft word also offers symbolic math typesetting

I meant word processing in the sense of document creation.

@MathematicsAminPhysics I believe so

Assuming everything's without boundary

@LeakyNun Right. I know that Word's equation editor is better nowadays, but I"m not sure how much better

Just in general if a manifold is contained in a connected manifold of the same dimension, they have to be equal I think

Compactness is not required EDIT: Yes it is

@AkivaWeinberger he has already asked the same problem an hour ago. Balarka claimed the proof needs an understanding of algebraic topology or manifolds, and he claimed to know neither of those, so there's no point.

Right, guys?

Well, you've basically done the equivalent of asking a car enthusiast what their favorite road is :)

@AkivaWeinberger Huh?

@AkivaWeinberger that isn't right

S^2 sans the north pole is contained in S^2

Going to get a lot of answers :)

@AkivaWeinberger compactness is really required, as per all our earlier examples

@Semiclassical well tell me what it cannot do

@TobiasKildetoft On Mac, there is MacTeX, which is a modified version of TeX Live, and on Windows, there is proTeXt, which is a modified version of MiKTeX.

@MathematicsAminPhysics Oh OK compactness is required derp

If I knew, I'd tell you. That's my point :/

Sorry

@WillHunting I think I used to use MikTeX, but I have not opened my Windows in a long time

I use TeXworks for my editor, but I should probably switch to TeXstudio or something

TeXworks is very bare bones.

I am not really saying there's no point in helping a person understanding a proof. But I don't have the willingness to do this while getting interrupted by "I do not know this FULLSTOP" at every sentence I write without any engagement or effort from the opposite side for understanding what I am trying to say

@MathematicsAminPhysics Differential? It's true in the topological category as well, I think

yeah just significantly harder

@BalarkaSen I missed the earlier conversation where this happened

@BalarkaSen Topological category usually is

true

@TobiasKildetoft I use latexmk to compile everything with one click. It comes with TeX Live. You don't need to click multiple times to build the output of a complicated file. On MiKTeX, there is a latexify for this purpose.

@BalarkaSen is this true in point-set topology?

(To be clear, "differential category" means we care about smooth maps and diffeomorphisms, while "topological category" means we care about all maps and homeomorphisms. In particular, the latter contains fractals and such.)

@MathematicsAminPhysics Smooth surfaces?

@BalarkaSen There is no point learning Fermat's last theorem if you can't even solve a quadratic equation, in other words.

@BalarkaSen So there's an easier proof for smooth surfaces?

@LeakyNun The theorem statement is about manifolds. What do you mean by point-set topology?

I don't know differential topology so I wouldn't know @BalarkaSen

whether it holds for topological manifolds rather than differentiable manifolds, maybe?

@BalarkaSen is a surface a set contained in R^3 where every point has a neighbourhood homeomorphic to R^2?

@SteamyRoot Personally, I don't like to use the word differentiable for C infinity, lol. I prefer smooth, but almost everyone uses differentiable, lol.

@WillHunting doesn't differentiability imply smoothness?

@AkivaW Yeah you end up trying to prove that for an injective smooth map $f : U \to \Bbb R^n$, $f(U)$ is open for any open subset $U \subset \Bbb R^n$. That should be significantly easier.

Most people also include Hausdorff in the definition of surfaces/manifolds.

@SteamyRoot That's not "many people," that's just legit part of the definition

Otherwise you get the line with two origins and stuff

$df$ is an injective linear map $\Bbb R^n \to \Bbb R^n$, isn't it? So that's an isomorphism

@BalarkaSen The phrase 'due diligence' often comes to mind when it comes to questions here or on site. If the asker isn't willing to do their due diligence, why should I bother to answer?

@AkivaWeinberger Not historically.

so you can cover $U$ by small open balls $B_\alpha$ such that $f$ is a diffeomorphism restricted to $B_\alpha$ (inverse function theorem); that means $f(B_\alpha)$'s are all open

$f(U) = \cup_\alpha f(B_\alpha)$ is open

That should do it

Ah so the point is that $df$ isn't zero anywhere near the centers of these balls, so it's a diffeomorphism?

Or singular, I guess

Determinant zero

on that note, something I'm forgetting

this is so cool lol

Suppose we want to distinguish between matrices with and without full rank. That's easy enough: The former has determinant zero, the latter doesn't.

@AkivaWeinberger yeah.

@Semiclassical the matrices must be square

True, but then the word 'rank' is ambiguous

@Semiclassical why?

Column rank or row rank?

@Semiclassical they're the same: that's a theorem

Yeah ^

Even for nonsquare matrices

...huh. am I being forgetful? I knew that was true in the square case

how can one utter the specific terms "column rank" and "row rank" without knowing that they are the same

I mean it's not apparent from the definitions @LeakyNun

I don't know. How can one manage to fit so much disdain into one sentence?

Because you usually define them separately and only later prove they're the same?

Is there a diagonal rank, lol.

@AkivaWeinberger I mean, he knows those two terms.

Condescendingness over 9000 though.

@Semiclassical hmm, I apologize

Okay.

I think @Semiclassical said that to @LeakyNun so that he can get 5 stars in chat.

wut

Some users of this chat needs to stop throwing snide piles of mud at other users

Nah, I said it because I was feeling a bit salty :/

@BalarkaSen did you mean some user?

Whatever, time to move on I guess.

AAAANYWAYS

Square matrices have nullity zero iff nonzero determinant

true

iff lots of other things...

ya I once saw a list with like 10 items

What does one do if one wants to characterize the set of square matrices with nullity exactly one?

I know it's something simple but I'm forgetting.

@Semiclassical rank-nullity theorem?

You need an (n-1)x(n-1) minor to have nonzero det, right?

uh, maybe not

@BalarkaSen Yeah, I think that's what I had in mind.

Anyone under 18 is a minor.

I'm sure you could look up algorithms from CS that compute rank very efficiently

"The rank of A is the largest order of any non-zero minor in A."

There we go.

@Semiclassical Ok, cool

we're good

@AkivaWeinberger gaussian elimination?

Oh duh yeah that's probably how they do it

And that's stated in a way that doesn't require square vs. not square, so I'm happy

I'm not interested in it now, but

@Semiclassical How is minor defined, again?

@Semiclassical Ooooh, now that you mention it... I use this a lot somewhere in the projective geometry I teach.

Bit of matrix minus a row and a column?

Can't remember where though, but it's useful

@SteamyRoot Yeah, p sure you're right

singular locus or some such

I have used that to figure out an exercise in G&P I think

Yeah, that sounds about right.

figuring out the manifold of matrices with rank at most something

Crap, I really have to read up on that course again by the time the academic year starts

A while back I was interested in algebraic curves of the form $\det(A-\text{diag}(x,y,z))=0$ where $A$ is a 3-by-3 matrix

Don't only square matrices have minors? @Semiclassical

I do need to relearn this sort of linear algebra at some point

@SteamyRoot You teach projective geometry? Hmm, most places I know about don't even have such a course...

Okay@Semiclassical I got my mathematica programm working, how to turn what i write into pfd?

"In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns." And non-square matrices still have square submatrices.

headdesk

headdesk was for me ?@Semiclassical

@KasmirKhaan You misunderstood what I was saying. What I said was that, if you wanted to do actual calculations, then Mathematica can be handy.

facepalm

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

:(

But if you want to do symbolic math typesetting, then some implementation of TeX is what you need.

@WillHunting It's a part of the Geometry II course I teach. It's basically 3 mini-courses; the first in projective geometry, the second algebraic geometry, the third euclidean geometry

can you please just give me a site?

@SteamyRoot Wow, I would have thought the first one would be differential geometry. =D

Like something i click on and its there , ill worry about the notation needed alone :D

@WillHunting Geometry I is all differential geometry, though.

Took me like 20 mins now to download and get activation key for that matematica

facepalm indeed

@KasmirKhaan As suggested before, you could try Overleaf.

@MartinSleziak Thank you sir

@KasmirKhaan If you will use LaTeX a lot, you might as well download and install a full version.

And you can find links to some similar websites/tools here: Compiling documents online

And if you don't, you might as well use Microsoft Word for now, lol.

@KasmirKhaan Please, do not call me sir.

@WillHunting can you give me a site ? ill download it because ill be using it for a good while =p

@MartinSleziak my bad , I use that as a form of respect

@KasmirKhaan I suggest you install the full TeX Live, which on Windows, also comes with TeXworks, an editor.

@KasmirKhaan tug.org/texlive

hi

My daily religious prayer to the MSE chat: Ravioli ravioli give me the dramaoli.

@WillHunting they said they are many versions of latex , can they all do the job that I need? like write stuff ( abstract algebra ) and turn it into pdf?

I know that it might be respectful in some parts of the world, but still it feels strange. Where I live, I only hear "sir" in movies, and always when addressing much more senior people.

Haha okay =p I understand your point

@KasmirKhaan The TeX world is very complicated. You will have many questions, and there will be many complicated answers. But for now, if you do what I suggested, yes, you can write anything and turn it into PDF.

@WillHunting Thank you ! :)

@MathematicsAminPhysics why do you assume downvotes have anything to do with your MSE University idea?

@KasmirKhaan If you have problems with TeX, the best SE site is TeX SE, not math SE. =D

Haha okay got it! :D

Good mornin Typhon

@TimTheEnchanter Hello

@MathematicsAminPhysics I don't know who you are and I got nothing agaisnt you, but if you stop posting that comment , I think alot more ppl will like you :D

@MathematicsAminPhysics then why do you take so much offense to downvotes? People do downvote because they actually read yer post and feel it is ill-written/ill-researched!

it's normal

@LeakyNun Are you here?

this is the internet

it doesnt matter whether they actually have a grudge against you

some people will just take advantage of seeing the post and use it as an excuse to cause trouble

fair enough

okay, too much drama for me to stick around

there's a mod chat

ask there

@Semiclassical My favourite drama series is Hindsight (2015), you should watch it too. =D

Nevemind

Okay guys this is my last question about latex things :D

I got the latex code ready for first problem of my list

How do i put it on pdf format?

@KasmirKhaan Have you installed what I sggested?

@WillHunting I did not want to bother you with that again , but could not do it , was super comfusing, this what I got github.com/TeXworks/texworks/releases

If so, read the tutorial A Not So SHort Intro to LaTeX which comes with it, and you will know what to do. =D

@KasmirKhaan No, no. That is nooooo use alone.

:(

I did nto download yet , but saved that link untill i could figure it out

Maybe just use Word for now, and install later?

That could be the best idea >< but we gonna have 6 homework problems so I really need to get these stuff

I got deadline on 6th september

the first problems are easy but writing them on pdf using latex is the hard stuff-.-

Equation editor works very well in Word, very very well.

I just reformated my pc so I dont think I have word atm

But hmm I got this site

OMG

hostmath.com

I could type stuff there and they came cool and all but have to figure out how to put that on pdf =p

I know this is the first time you see someone like this =p but I got lots of problems learning these stuff , Ilike doing math on paper

I don't know how those sites work, so I can't help you with that!

Its fine Ill figure it out =p thanks alot @WillHunting

@KasmirKhaan Again, this is not the type of the site you want. This can be used when you want to type single equation. If you want to use something more complex (document/pdf file which contains both text and equations) you need something else.

Several people here recommended overleaf (if you want and online tool), why not try that one?

I did but that is not what I was looking for

@MartinSleziak Am trying to do homework problems on (abstract algebra ) not write a papper

Yes, but you are trying to make pdf file which contains both some text and some mathematical expressions, right?

@MartinSleziak Our teacher dont want hand-written solutions, he wants all homework to be on pdf format

@MartinSleziak Yyes

Well, then overleaf is one of possible tools to do that.

Hmm

I just noticed that I can delete the stuff that was written there

@MartinSleziak I should try it out some day too. =D

If you prefer sites like hostmath - which you linked above - because there you have editor where you can create formulas by clicking; that site might still help you - you can copy LaTeX code from that to Overleaf (or whatever editor you are using).

All right :D thanks alot ! :)

Am very new at this sorry for that =p

@KasmirKhaan IIRC in overleaf you have option to share your document. (Either read-only or even allowing the person you share it with to edit the document.)

So this might be the way how people here in chat can even help you get started, since they will see your code.

I found the botton that turns it into pdf but IIRC did not find ><

You can than edit it further.

Ahhhh :D I got it thanks :D

I saw that text from the first page ,I assumed it was used to create a "pro" pdf text about some subject, did not know I could type anything =p

thought it was used by teachers to make texts and stuff

If you have problems with your first document in Overleaf, you can ping me in my chat room and I can try to assist you abit.

@MartinSleziak much appriciated thanks! :)

If something is used by teachers, that does not mean it cannot be used by students.

haha =p I meant that the format : abstract, discussion , ect , I thought it was meant for something more professional

You can simply delete the abstract and it's no longer there.

Yepp that was the part I did not get untill now =p

@AkivaWeinberger Hey Akiva, sorry I didn't respond, I was doing an assignment so didn't see your ping.

@Mr.Xcoder I'm here and I leak everywhere

Solved it. Sorry for pinging