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3:00 AM
In $\mathbb{R}^3$, it should be $r\cos\theta_1$, $r\sin\theta_1 \cos\theta_2$, $r\sin\theta_1\sin\theta_2$
I might've written it wrong
 
this seems like everything's just off by $\pi/2$ and in the reverse order, no?
that works too.
hai, @Huy
 
Let me get my work out
 
yep, then your defn is basically equivalent. do you want me to explain it to you?
(if that's a problem)
 
^Does that look right? That's my work
 
at first glance, yes.
 
3:03 AM
Okay, now here's the problem I'm running into
The range of $\tan^{-1}$, obviously, is $(-\pi/2, \pi/2)$.
 
that's gonna be messy to code up for sure.
 
Yes, it's a freaking nightmare to code. But I did it
He says here that $\theta_1 \in [0, 2\pi)$ and the other $\theta_i \in [0, \pi)$.
So when you're working with the $\theta_i$, $i\neq 1$, it's pretty obvious what to do: when $\tan^{-1}$ gives you a negative value, add $\pi$
 
hmm, I think I follow
 
But how the heck do you get the $\theta_1$?
Because if I just add $\pi$ to all of the negative values I get from using the $\tan^{-1}$ formulas, there's no chance that $\theta_1 > \pi$, you know?
 
well, do you know all the other $\theta$s?
 
3:06 AM
Yes, because in order for me to find $\theta_1$, I have to know the other $\theta_i$ using the recursive relationship
 
@Clarinetist I didn't get that last sentence.
@Clarinetist then find $r$!
use that to find $\theta_1$.
assuming I'm not confused here, that should work
 
@SohamChowdhury Hmm... that's a thought. But doesn't that lead to the same problem, since $\cos^{-1} \in [0, \pi)$?
 
is this your problem?
I don't think it matters, then. your program will give you a different $\theta'_1$ instead of $\theta_1$, but they'll have the same $\cos$
hey @anon.
 
hey
 
3:13 AM
@SohamChowdhury That's what really bugs me about this assignment. It's not clear how he actually expects us to find $\theta_1 \in [0, 2\pi)$.
 
I'd quit worrying about it tbh
he probably won't care. you're in a stats program, for heavens' sake. /s
 
This is the first assignment that I've had in this class that has made me glad to have taken analysis and algebra in my undergrad. One of the previous problems required coding the Euclidean Algorithm
 
I'm around halfway through learning Clair de Lune, btw. :)
 
@SohamChowdhury Congrats, that piece is tough
I wish I had a piano and the technique to play that
In a year or so, maybe
at least the piano
 
My calluses agree, although it's hard on guitar because of the fast arpeggio-ish middle part.
That's where I'm going slowly.
 
3:16 AM
Omg, you're trying that on guitar?
 
I forgot, heh. It's been too long
Props
 
Sounds beautiful, actually. This arrangement.
The harmonics are fun!
 
listening
 
I see how I can solve it now @TedShifrin given the hint
I will just proof the hint first
and then I will mention how I did it
 
3:18 AM
Is this in the original key?
 
actually here is how to do it
so the part c mentions that
 
it's in D
half-step up.
 
Show that $R^{\omega}$ the box topology. Show that x and y lie in same component of $R^{\omega}$ iff the sequence x - y is "eventually 0".
so if we assume this is true
 
Doesn't make it much easier :P
 
consider a representative of a particular component
write that representative as follows
 
3:20 AM
we guitarists seem incapable of playing outside E, B, D, G, and A (roughly in that order)
 
$x = (x_1,x_2,.......)$
 
@SohamChowdhury Given what I know of orchestration, that's pretty much true for any string instrument player :P
 
I also learned Take Five (and how to swing, in the process).
 
consider $y = (|x_1| + 1,|x_2| + 1,|x_3| + 1,...)$
 
That is one fun piece to play.
 
3:21 AM
I am sharing this arrangement. Very neat
 
so x - y is never eventually zero
so it can't be in same component
 
Okay, I have got to start working now.
 
so since we can do this for every representative of equivalence class
 
I assume you've heard Take Five?
 
we can see that $R^{\omega}$ has infinitely many components
 
3:22 AM
Oh, of course
 
heh
This arrangement is amazing, more so because of the faces the guy makes.
I play this one, though.
It's not too hard, although I learned the solo on my own.
 
OMG that looks difficult
 
can you play that on the clarinet @Clarinetist
 
The thing is, when you're playing clarinet, you can only play one line at once. :P I likely could
 
3:48 AM
Hello all! :)
 
Hello
 
If I want to minimize the distance between two implicitly defined surfaces, can I just find the points at which their gradients are parallel?
 
@Simeon: It's more than that !
@Karim: Sounds like you're just assuming what you're trying to prove. I've thought about it during dinner, and Munkres's hint is exactly right. You need to look at $x$ all of whose components are bounded and $y$ whose components are unbounded.
 
But that's how you would find the critical points to start with, isn't it?
 
Critical points of what? This isn't Lagrange multipliers, unless you set it up as Lagrange multipliers.
 
3:59 AM
Oh you're right, there's no reason why they would have to be parallel.
 
No, they have to be parallel, but you need more.
Draw a picture ... when you have the closest points $x,y$ in $X$ and $Y$ respectively, what does your picture look like?
 
oh yes, opposite directions
 
Hmm, not necessarily ... you don't know which direction the gradient points. What about the line segment joining $x$ and $y$?
 
it is normal to both surfaces?
 
There you go.
 
4:04 AM
But why?
 
Think about the Pythagorean Theorem. If the chord isn't normal to both surfaces, you can move the point(s) to make the distance shorter.
 
Hmm, is there an explanation similar to the Lagrange method (if we use the chain rule for some curve to show orthogonality)?
 
Equivalently this comes back to the bit you were thinking about earlier w/ finding critical points... if you fix a pt v in one of the surfaces the closest point in the other surface to it must have w-v perp to the surfaces. why? minimize the fctn $f(w) = \|w-v\|^2$
 
Yes, @Simeon.
@Simeon: If you have learned about Lagrange multipliers with more than one constraint, you can also set this up as a Lagrange multipliers problem with $x$ satisfying $g(x)=0$ and $y$ satisfying $h(y)=0$.
Of course, if these are surfaces in $\Bbb R^3$, you'll end up with $6$ variables, $2$ constraints.
 
Oh ok, but what is the function that I have to maximize?
 
4:09 AM
minimize
 
$f(x,y) = \|x-y\|^2$.
 
yes
 
Didn't mean to step on your toes, @MikeM.
 
you're not
: - )
 
Oh that makes sense now, thinking about it as a Lagrange problem
 
4:12 AM
If you work it out, it'll tell you exactly the geometry we had: That the chord joining $x$ and $y$ should be parallel to both gradients.
 
In general, when using Lagrange multipliers, and you get only one point as a solution, how can you verify using the Hessian whether it's a max or a min?
 
The Hessian test for constrained max/min is very subtle. I doubt you're expected to do this. You can find the test in Marsden/Tromba or my Multivariable Math book.
 
Oh ok, so I should probably use another argument, like finding another point that's smaller / larger
 
Besides, the second derivative tests only prove (if you're lucky) local max/min. You need some global arguments (e.g., compactness) to know there is a global max/min.
What sort of course are you taking?
 
@Simeon: You really just need to know some things about lengths of triangles to prove that the chord is perp to the surfaces.
 
4:19 AM
Analysis II using Munkres Analysis on Manifolds
 
I did make reference to Pythagoras at the beginning, @MikeM.
 
The surfaces aren't flat... it's not quite pythagoras
 
Ah, ok, so you should make some sort of global compactness argument or something, @Simeon.
Yes, of course, @MikeM.
 
Tried to give this arg to my multivar calc students (for curves). They did not like it so much. So I just went back to the second deriv test...
 
If your surfaces are compact, then you're guaranteed a max/min.
 
4:20 AM
@TedShifrin - you're a mathematics professor, correct? :0
 
I was once, @Perplexed :)
 
Ah, okay. It's cool to know that someone of your math level chats here.
I often wonder what professors/former professors do math-wise in their spare time. xD
 
Oh my you're actually that Ted Shifrin,
I'm honoured
 
LOL @Simeon, whatever that means.
 
I remember watching your lectures recently, they were very helpful
 
4:22 AM
Ah ...
 
Ted, while I'm aware of the fact that you don't like Hubbard/Hubbard, would it be absolutely unworkable for me in your opinion?
Hmm, that's a thought. I could always watch your lectures.
 
I can't answer that, @Soham. I don't know you well enough, and lots of people love the book. I just find it too idiosyncratic and the exercises not systematic and helpful.
When I first created the multivariable math class back in 1998, I used Hubbard/Hubbard as the text, and just gave up in the middle and started writing my book. My students weren't finding it helpful.
 
To me, exercises are the most important thing about any math text. Of course, Jasper disagrees.
 
Well, do you know any place I can find nice exercises?
Hmm, especially books that have "guided proofs" in the exercises.
I like those a lot.
 
4:26 AM
No, most multivariable books are computations with no proofs.
 
what's your goal?
 
Soham, couldn't you wait and learn this when you get to university?
 
Knowing IFT, pretty much. I want to look at manifolds.
 
You should watch the videos, I watched the pullback one a long time ago and really liked it
 
I could, @Ted.
 
4:27 AM
The Inverse Function Theorem is sophisticated enough that most undergrad math majors in the US never learn it. I don't know why you high school people think you should be learning it.
 
But I sort of want to learn as much as possible before, because I have a terrible fear of being swamped in linear algebra homework.
 
@TedShifrin this is more a corollary of poor instruction/syllabi imo.
 
Take your time. Good grief.
No, @MikeM, it's a consequence of the fact that we have people majoring in math who don't want to be mathematicians. And I'm ok with that.
 
my undergrad had an 'advanced calc' class (mandatory for math majors) that somehow did not cover it.
 
That's why I'm not looking for a five-page ultra-concise treatment of the theorems, @Ted. I want to learn multivariable calc, properly. And that should include those theorems, from what I've heard.
That's all.
 
4:29 AM
they ousted the prof who had been teaching it for decades and rewrote the syllabus so that the people who take it now learn some calculus.
 
Well, except for the really talented ones and ones who want to go to graduate work in math, the math major has lots of other focuses and ...
 
Is it so bad that I want to learn multivariable now? I don't get it.
 
Sometimes good teachers/professors actually help you learn. All this self-study is not for everyone, even though you kids are obsessed with it.
Only a handful of my students at UGA, with my mediocre instruction, were really able to learn it deeply. The rest of them learned computations and had some vague idea about the theory. And that was typically a class with 25 students.
 
i'm baaaaack
 
That's too social of you, @Forever :)
 
4:32 AM
@TedShifrin I am tired atm I will think about it tomorrow
 
It's subtle, @Karim. Not an easy exercise.
 
well I drank 2 beers so I'm too social now
 
LOL @Forever
 
save one for me
 
4:33 AM
haha
 
or five
 
I wonder what time zone Forever is in ...
 
central time
 
ah, so getting later ...
 
lol what time zone forever is in
sounds funny
considering the nick name
 
4:35 AM
If I'd put the @ in there, @Karim, you wouldn't laugh :D
 
yeah haha
 
seems a bit late to be having your second beer
 
it was his second, @MikeM ... pay detention.
 
all the same ;)
 
when I have even one beer my ability to do math goes way down
 
4:36 AM
okay, @Ted, so what should I be doing in your opinion?
 
anyway good nights guys
 
but with coffee it is opposite, for a short time
 
cya tomorrow
 
Soham, I don't know you. It's not my business to schedule your life. I don't know your abilities or your depth of understanding of what you've learned.
Cya, @Karim.
 
Well, that's true.
 
4:37 AM
But, generally, I see a ton of people in this chatroom trying to learn stuff that's above their level to really understand and learn deeply.
I see some people in college doing that, too ...
 
Hence the exercises, @Ted. I am wary of learning half-baked stuff.
 
So, @Forever, what's the story of the moral?
 
People love biting off more than they can chew.
 
But exercises without someone grading them aren't necessarily magic, @Soham.
See, @Forever, you do have a sense of humor :)
Oh, the beers do.
 
4:39 AM
That's why I come here, @Ted. Tons of people to ask.
 
Not the same, @Soham, not the same.
 
And usually it's one of "how does one even do this problem?" and "ohh, let me write it down, I see how it should be done" for me.
Well, I know a prof whom I go to very infrequently.
 
Most of my (talented) college students, even when they think they've written a good argument, get back lots of red from me.
I'm just in favor of slowing down and building really strong, deep foundations.
 
Thursdays are my day off, so usually I try to prove some things Wed. nights. But, at this point I got all the low hanging fruit. The only thing left is the big conjecture which is not going to be easy...
 
What kind of stuff do you work on, @Forever?
 
4:42 AM
connected topological spaces
 
That's a bit vague.
 
spaces that go on forever
 
i'm into those
 
@TedShifrin I don't think anyone in their right mind would disagree, @Ted.
 
it's vague, but I'm interested in lots of different types
 
4:43 AM
So you're aiming to do a thesis on point-set topology, @Forever?
 
different types of local connectedness, compact connected spaces, etc
yes that's what I'm trying to do
 
OK ... my impression is that what's left to do is very technical and very arcane.
 
@SohamChowdhury consciously, no, but there are plenty of people who claim to be in favor of that whose actions don't at all line up with it, and don't take advice from people who point this out to them
(i'm not referring to you.)
 
@MikeM is referring to Balarka in his former existence, for example.
 
I get the feeling I'm being tiresome now, so: do you have any suggestions for where someone can learn multivariable from? I want to do it slowly, as you say, but also learning the "big theorems" properly. And I've worked through a book before, but it was not at all rigorous.
Me, too, about two months back.
 
4:45 AM
yeah, what's left is very technical and mostly with non-metric spaces
 
Oh, definitely, non-metrizable spaces, @Forever.
 
is there non-technical math left?
 
Yes, @MikeM
 
maybe I should change my field...
 
In the past week I've basically cleared out most of Artin's exercises from three chapters of linear algebra. Never would've done that before.
 
4:46 AM
Number theory still has stuff that can be readily understood.
 
Which is why it attracts the cranks?
 
Not so readily solved.
Everything attracts cranks.
 
sure... by math I didn't mean the statements of hte theorems, I meant the stuff one does
:)
 
Fair enough, @MikeM.
 
by technical I mean a lot of current research in topology is tied to various weird set theoretic axioms
 
4:46 AM
but certainly I can think of stuff that's not so hard in my area i guess
 
Yup, @Forever. This is stuff that's never appealed to me, so I haven't at all kept track of where research is.
 
Well, I'll go ask on main. See ya, everyone.
 
bubye @Soham
 
it can be very abstract and semantic
like set theory
 
yup
 
4:49 AM
@SohamChowdhury: I don't think anybody is frustrated with you personally. It's hard to give personal advice when you don't know the person, and it sounds like there's not really a good place to learn all the stuff we're talking about.
Good luck with your question.
 
Well, @MikeM, Hubbard/Hubbard and my book are the two common sources these days. But mine, at least, is expensive.
And I'm not going to provide free copies to everyone in here.
 
there are still lots of open problems: carma.newcastle.edu.au/jon/Preprints/Books/…
 
open problems about open sets
 
Hi, everybody. I'm learning stochastic processes by solving the problems in Van Kampen's book. If anyone's interested, join in the work here.
 
4:59 AM
what is github?
 
@ForeverMozart It's a free graphical interface to a git repository with a lot of features which make working with other people easier.
Do you know what git is?
 
@ForeverMozart Ah.
 
I have used overleaf before
for collaborating
 
@ForeverMozart I'm pretty sure overleaf uses git under the hood.
Or maybe not.
In any case, they serve similar purposes.
 
5:02 AM
multiple people can edit the latex file
 
@ForeverMozart The idea with git is that everyone has their own version controlled repository of all files. I can change my files without you knowing anything about it. Then, we can merge our respective changes together when we choose to do so.
Github provides some nice user-interface stuff to make that merging process easier.
In fact, can can use github without knowing anything at all about git. You can do your editing online. Github doesn't render TeX like overleaf does though.
It's mostly for programming code.
Wait, what the heck? @ForeverMozart overleaf uses a "secret link" to keep your work private?
You can't set user access permissions?
 
yes
you share it with people you want
 
Oh I see, it supports both. There's a "secret" (LOL) link and user access permissions. That's good.
Looks nice.
 
yeah it is very easy to use. we used in in a graduate class. the professor listed theorems and assigned them, and we all typed in our proofs
 
Looks great. One thing I like about github is the ability to file "issues" which are resolved by subsequent work. Does overleaf have something like that? Some kind of "todo" support?
The idea of "issues" is to collect discussion and work focused on resolving a single problem or adding a single new feature.
 
5:12 AM
i don't know too much about that. You can create a project and have multiple tex files in it
I doubt I have used all their features
 
ok, thanks.
 
prenex normal form If someone could help me with this simple question, I would appreciate it. It is regarding free variables when converting to prenex normal form. Thanks
 
6:00 AM
Lol this is truly nasty.
2
Q: Why is this sum equal to $0$?

JHMWhile solving a differential equation problem involving power series, I stumbled upon a sum (below) that seemed to be always equal to $0$, for any non-negative integer $s$. $$ \sum_{k=0}^s \left( \frac{ \prod_{r=1}^k (-4r^2+10r-3) \prod_{r=1}^{s-k} (-4r^2+6r+1)}{2^s (2k)! (2s-2k+1)!} \times (2s-...

 
@DanielSank nope, AFAIK
 
I really wish there was a way to depict 4-dimensional objects. >:(
We could embed them onto a 3 dimensional space much the way we can embed 3 dimensional objects onto a 2 dimensional space, couldn't we?
 
6:45 AM
@SohamChowdhury That's too bad. The built-in issue tracker on github is one of it's best features.
@PerplexedGuest When you represent an n+1 dimensional thing in n dimensions, you're not "embedding" it, you're "projecting" it.
And sure, you can project 4D objects in 3D.
 
 
2 hours later…
8:23 AM
@robjohn would you mind having a look at this question please?
 
man @skillpatrol you are up so late! I caught you on your desktop! yay!
 
@TanMath Hi pal :-)
yep, laptop actually :P
I think desktops are going to go the way of the dinosaurs.
I only use mine because of the printer...
 
Could someone help checking if there is a fallacy in this exercise? i.imgur.com/0gmPCEP.jpg My TA said he don't understand this proof.
Could someone help checking if there is a fallacy in this exercise? i.imgur.com/0gmPCEP.jpg My TA said he don't understand the notation $N_{\epsilon}(a)$ in this proof.
 
 
2 hours later…
10:30 AM
Would this question posted on SO be on-topic on math.se?
 
10:45 AM
OK, I'll flag for a mod to get it migrated then, tnx
 
@DanielSank yep, I know
 
 
1 hour later…
12:13 PM
You know when you do a } under something, how do you do that?
So you can say "this part of the expression is" and put a } under the part to show you mean this part.
 
@AlecTeal This thing?
 
Did you have to edit that 3 times? Can't you just write it out?
 
Bad typos, sorry. Does each edit send a ping?
 
It does. Also it isn't that helpful.
It would have been faster if you'd just said \underline thanks for trying though
 
I didn't know that. I wouldn't have done it if I did. Apologies.
 
12:18 PM
hi
has anyone knowledge of complex numbers?
 
Thanks @Boni for trying, I meant that.
 
user174558
You can turn off pings.
 
user174558
Also, if one is too scared of pings, one can just not enter the chatroom.
 
user174558
@SohamChowdhury and @Huy look cute.
 
well, that's a nice thing to know.
 
Huy
12:28 PM
I'm cuter
 
.....pedo and a Justin Beiber fan. @Jasper thanks for the life advice!
 
Our man Mr. Huy has been busy these past couple of days.
 
user174558
@AlecTeal Bieber.
 
Please #cut4beiber
 
Can anybody help me with a question about complex numbers?
 
12:29 PM
Ah. Jasper scratches my itch these days.
sup, @Huy?
 
@GniruT what does it say for the topic here?
 
ups
 
just tell us. go on.
 
how do you draw in Complex plane the imaginary part of "iz+1"
 
you don't. you draw the whole complex number in the plane. as a point.
 
12:30 PM
Write z as a+bj and work out what iz+1 is
 
filthy engineer
$i > j$
(/s of course)
 
user174558
Too much strong language in this chat. First pedo, then filthy.
 
the alphabet must be different in wherever you come from @SohamChowdhury because the natural ordering on letters gives $i<j$
 
user174558
I suspect the flags will come in soon.
 
Also given we use i as the vector for the x axis.... j just makes sense for the complex part.
 
Huy
12:33 PM
correcting some geometry exercises from students
and then I have to attend a seminar on Regularity Structures
what u up to today @Soham?
 
user174558
Some places have no alphabet.
 
@SohamChowdhury I have to draw the set ç
{iz+4:z of C, Im(z)>0}
 
user174558
I think people who have too much difficulty in elementary math should just drop the subject. It is not needed in daily life. We only need simple arithmetic to survive.
 
Eh. I couldn't do group theory for a long time.
 
Just dropping in to let you guys know that someone is flagging an awful lot of messages in this chat. I've been invalidating them because nothing offensive has came up, but you might want to watch out.
 
12:40 PM
kk thanks
 
user174558
See what I said?
 
user174558
I know these flaggers.
 
This place is almost professional compared to Lounge<C++> over at StackOverflow.
 
Maybe you're the flagger @Jasper
@Boni there are a lot of circlejerks on this site. It's not a very nice place.
 
@AlecTeal You can't flag your own messages.
 
12:43 PM
@DrRDizzle I can't see what was flagged either. So that's inconclusive.
 
@AlecTeal putting aside the fact that his messages are being flagged, I don't think throwing around accusations is a very constructive thing to do
 
@Doorknob clearly you've never played Cludo. Nor have I but I know the gist.
 
hah, "it was @AlecTeal in the chatroom with the keyboard!" :P
 
@Boni I also hate the hot questions list, because some dumb-ass is like "why is a negative number times a negative number positive" and it becomes hot and gets so much attention but a good maths question, not accessible to the audiences OF THE OTHER SITES gets nothing.
Just skews rep further.
 
user174558
LOL, someone flagged me? LOL
 
user174558
12:45 PM
This is too much.
 
user174558
Also, better not use circlejerks and dumbass either @alec.
 
user174558
But I wonder who flagged me and for what.
 
@AlecTeal There were questions like that...? I am around mostly for the chat rather than the SE site itself. So I just meant that this chat itself is very mature in general.
 
Yes @Boni - as a rule of thumb, the less accessible the topic the more mature it is. So here you can expect undergrads and above, as you know SE has a much broader set of users.
 
user174558
I am sick of these flaggers. They like to ban me from the chat.
 
user174558
12:51 PM
I did not flag anyone. If I want to flag, I will tell so openly.
 
How to draw the set ${iz+4: z \epsilon C, Im(z)>0}$
 
Huy
mark one point belonging to that set. then mark the other points belonging to that set
 
@Jasper Can you? I didn't know that. Not that I wanted to turn them off anyway, it's just interesting to know.
 
user174558
@TRiG Look for the loudspeaker beside "all rooms" on the top left.
 
@Jasper Ah. Nice.
 
Huy
12:56 PM
that's clearly a lamp
 
user174558
ROFLMAO
 
@Huy started working on Hubbard and Hubbard
waiting for prof to tell me what next to do in algebra.
did some linear recently.
got interested in Lie stuff (which you probably know tons about). do you know about Naive Lie Theory?
 
user174558
I have only heard of naive set theory, lol
 
user174558
What is naive lie theory?
 
yeah, the title is inspired from there
it's like Naive Set Theory, but for Lie theory
 
user174558
12:59 PM
Very enlightening answer, lol
 
so your question reduces to "what is Lie theory?"
 

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