Mathematics - 2018-08-27 (page 1 of 3)

Symposium

12:00 AM

I've love-hate relationship with Rudin.

Daminark

At that time I felt Rudin was rough

Ted Shifrin

I think Rudin is very rough without an inspiring teacher ... for most students.

Symposium

Rudin sometimes outlines easier more obvious parts of an argument then skips a harder less obvious bit. Pissed me off to no end!

Abdullah UYU

@TedShifrin As a student, I am also confused about the optimum level of interactiveness that a math class should have. There is also a war of philosophies behind the scene.

Ted Shifrin

@Pig: I might have tried it in my Honors multivariable math class if I hadn't retired. The videos of my lectures were finished my last year. They might have made flipping reasonable. I still can see pluses and minuses.

Daminark

12:02 AM

But in hindsight I barely got through differentiation from Spivak and was trying to read it in my own, so I was bound to not get too far. The following year when I took analysis, and since then I've liked it a lot more for metric spaces

Ted Shifrin

@Abdullah: In general, I think passive, bored students is a recipe for disaster. I recognize that some students are slower than others, shyer than others, etc., but nothing is perfect. I did my best to draw some of the quieter students in, but sometimes gave up. Some did great in office hours. Others didn't even come to office hours. Bad.

Demonark: I think students who've been through my multivariable course and done well have excelled in the Rudin course. But they had an advantage :P

Daminark

And also chapter 7 on uniform convergence is real nice. But I was sitting in on a class first year that spent a few weeks on chapter 9 and I was bored to death. The arguments felt like they had no conceptual content

It was just let h = this

Ted Shifrin

Chapter 9 is the multivariable? That chapter is an abomination.

Rudin is the antithesis of a geometer.

Symposium

@Daminark It was the same for me. I read bits of Spivak (chapter on three hard theorems I think), then went to Rudin. Even in chapter one where people usually don't encounter difficulties with (which start and largely lie in chapter 2 I think) I already ran into problem. He did some basic inequalities then used the well-ordering principle somewhere without any notice/mention (it's not even in the book).

Ted Shifrin

I had analysis out of Rudin my first year in college. I had a dull lecturer for the professor, too. But I learned enough to pass. I guess I learned most of it eventually :)

Daminark

12:07 AM

Funny thing is, the one person I know who likes chapter 9 is a physicist, says it's a nice compilation of results (though even he agrees it's not to be self-read)

Ted Shifrin

Bah. It's horrendous.

vzn

@Symposium so why looking into LSGNT? are you a student? number theory interests?

Symposium

I learned multivariable calculus from Sean Dineen's book. I can't imagine learning it from a chapter in an analysis book; it was already hard enough.

Ted Shifrin

That's why I'm proud of my book, Symposium. I do both the computational stuff and the proofs in a self-contained manner (all interwoven with linear algebra).

Symposium

@vzn I like number theory/geometry, but I'm nowhere near the levels of LSGNT. I just live near there/go to one of the universities, so naturally I was curious!

Ted Shifrin

12:13 AM

What's LSGNT?

Symposium

London School of Geometry and Number Theory.

lsgnt-cdt.ac.uk

Ted Shifrin

ahh ...

I was working from the end and saw the NT ... didn't figure out the front.

Too bad Brexit will disenfranchise the Europeans not from Britain.

vzn

@Symposium what areas of number theory? (have been poking at Collatz myself for many yrs now, even as we speak...)

@TedShifrin agreed, seems like it will have negative long-term effects on national science not unlike trumps ban.

Ted Shifrin

grrrrr ....

Daminark

Lol multi for me was a mess. Professor second quarter thought that the professor first quarter spent most of the quarter on it, as he would, first quarter guy barely did it

Ted Shifrin

12:18 AM

Yeah, Demonark, I remember. You already know my mild opinions.

Symposium

@TedShifrin I've consulted your book too at some point I think. To me, the idea of learning the Moore way is unthinkable. My brain just isn't wired that way. I can't even learn from a single book from chapter to chapter without finding alternative explanations . I tried to learn set theory from a Moore-style book (An Outline of Set Theory - James Henley), and it was a disaster! xd

Ted Shifrin

I sent Eric my multivariable exams, Demonark. If you'd like a copy, I can send 'em to you too. If you want some practice.

Abdullah UYU

Good night, all. @TedShifrin Thanks a lot for the advices :)

Ted Shifrin

@Symposium: One of my colleagues once taught out of Guillemin & Pollack and had the students give most of the lectures. The result? The students didn't learn anything other than what they had to struggle to prepare to lecture on.

Night, @Abdullah.

Symposium

@vzn I would like to learn modular forms, elliptic curves, diophantine geometry etc. But it's all a dream right now.

Ted Shifrin

12:21 AM

When you learn you can teach us, Symposium. I don't know that stuff.

vzn

@Symposium (math) geek dreams :)

Symposium

:)

KeJie

Does ring theory actually has any applications to the real life?

The professor couldn't provide any example.

Symposium

@TedShifrin Haha, that would be me!

Daminark

In particular, the first guy never really talked much about gradient vs derivative, so I was confused at why inner products came in the picture out of nowhere, something which took embarrassingly long to figure out when a friend of mine was like "Hey this is Riesz representation" (in R^n but we don't have a short name for the R^n result so we just call it Riesz)

Ted Shifrin

12:22 AM

Demonark: See — you needed the geometric approach!!

Daminark

And then integration in higher dimensions never happened. We knew the definitions but had no practice with parameterizing anything

Ted Shifrin

@KeJie: I don't know what real life is ... But there's a wonderful book (which I referenced in mine) called Applied Abstract Algebra, by Lidl and Pilz. All sorts of applications of rings and finite fields to coding, block designs ... It allows you to design a round-robin tennis tournament where every person plays against every other person.

Yes, Demonark, I bitched a lot about that.

KeJie

thx

Ted Shifrin

You need to have some practice (and it's not stupid to have done a few exercises on this in your life).

@KeJie: If you ever sit down and try to design such a thing, it's surprisingly hard. The theory of finite fields does it for you (with certain $n$).

anakhro

@TedShifrin do you have any exotic looking orientation-preserving homeomorphisms of S^n or B^n?

I have a fear of the unknown here.

Ted Shifrin

12:28 AM

@anakhro (you shortened your name?) ... I don't, but you should ask a topologist like @PVAL (who's disappeared) or @MikeMiller.

anakhro

I shortened it because people had difficulties with its length or spelling. :P

Ted Shifrin

well, sure.

KeJie

@TedShifrin Sometimes it motivates me to know the applications of the things im learning. Couldn't find to much apllications for convergence series etc. But since I know the example of Archilles and the tortoise I thought that convergence etc maybe habe some applications as well.

Maybe this example can be transferred to the real life somehow.

Ted Shifrin

Actually, @KeJie, I think that's an issue that we math teachers should take far more seriously. Lots of students would be more interested if they were shown such applications. The problem is the time and knowledge it takes to present such applications.

KeJie

@Ted Shifrin This might sound a bit stupid but I thought that convergence could be applied to horse racing maybe?

Ted Shifrin

12:32 AM

I really don't see that.

KeJie

To calculate which horse will finish in first place.

anakhro

@KeJie when it comes to abstract algebra you will have to take it with a grain of salt initially. The subject historically had its roots geometry, either finding the symmetries of shapes, or finding solutions to polynomial equations. It developed a lot over the course of almost 2 centuries and what you are seeing is a very well-organized, thought out presentation of ideas that were conceived many years ago.

Ted Shifrin

Convergence becomes applicable when you switch over to numerical solution schemes and want to know you're getting close to the true answer when you do an approximation.

KeJie

Are there any areas in the real life where I potentially could use convergences?

Ted Shifrin

It's because of what I said, KeJie. You know that a finite process will actually approximate the correct answer.

You could think that adding up $1+1/2+1/3+\dots+1/10000$ would be close to the actual answer for the infinite sum ... but oops.

anakhro

12:35 AM

That being said, you find applications of some basic theorems in ring theory in cryptography as ted mentioned. In basic algebra you even get the application of RSA encryption from Fermat's little theorem. Though one might argue this is more of a number theory concept, you might have seen it in abstract algebra.

KeJie

Im legit suprised. Never imagined that ring theory has more applications then convergences.

Mike Miller

@anakhro This is to some degree the same question as "What are the exotic spheres?" Here is an operation. Take an orientation-preserving diffeomorphism $\varphi$ of $S^n$. Glue two $(n+1)$-discs together by the diffeomorphism $\varphi$ of their boundary. This produces an oriented smooth manifold. The diffeomorphism type depends only on $\varphi$ up to smooth isotopy.

Now, every orientation-preserving homeomorphism of the sphere extends to a homeomorphism of the ball: this is Alexander's trick. So this manifold we constructed is oriented homeomorphic to a sphere.

Ted Shifrin

@KeJie: What I'm saying has zillions of applications.

That's what numerical analysis is all about.

Mike Miller

So what we have is a machine that takes isotopy classes of oriented diffeomorphisms of spheres and spits out oriented exotic spheres one dimension up. It turns out that if you put the group operation of connected sum on the latter and composition on the former, this is a group homomorphism.

Ted Shifrin

Wow, @MikeM, this is cool.

KeJie

12:38 AM

@TedShifrin I did not have anything about numerical analysis yet only normal analysis.

Ted Shifrin

well, if you're interested in applications, you should take more applied courses, @KeJie.

Mike Miller

I believe it is a consequence of Cerf's pseudoisotopy theorem that this group homomorphism is an isomorphism in dimensions $n \geq 5$.

That is, diffeomorphisms of the 5-sphere.

KeJie

@TedShifrin But all courses in my university don't focus on that. Thats very unfortunate.

Symposium

@KeJie Have you learned things like Newton-Raphson method?

KeJie

@Symposium No im in my first year.

Programming courses only will start next year.

Symposium

12:45 AM

Physics surely counts as real life applications? The use of taylor series in dynamical systems etc.

Ted Shifrin

But that's very finite ... nothing about convergence. And physicists never worry about error estimates :P

Symposium

Haha

Ted Shifrin

hides head in shame

usukidoll

Meow

KeJie

But some are more obvious to me then others. The applications of matrices are more obvious to me then the applications for convergences.

Symposium

12:47 AM

When they talk about linearised equation of motion etc or in general take the first few terms of a taylor series, don't they care about the convergence of the said series?

Ted Shifrin

nope, @Symposium.

Symposium

Madness! (sorry lol).

usukidoll

I'm tired :(

KeJie

I hate sleep.

Ted Shifrin

To make things worse, I've seen calculus books that use the next term of a (supposedly) alternating series to estimate the error between the partial sum and the actual sum. But guess what ... that only works if you know the terms alternate in sign with decreasing magnitudes.

Symposium

12:50 AM

Who's writing these books. Children of congress members.

Ted Shifrin

Nope. Actually, some responsible members of mathematics society.

Physicists aren't the only ones who don't check details.

Symposium

Actually, I've a question @TedShifrin about US multivariable books. Why are some like 2000+ pages? I believe they are for high school students. The huge, huge ones!

Ted Shifrin

No, nothing like that. Seriously.

And high schools only rarely get to multivariable. They shouldn't at all.

I've never seen something like that. Maybe it's 1500 pages of pictures.

Symposium

Actually I think these might have been single variable calculus. I saw one by Ian Stewart that was enormous, for example!

usukidoll

Slept late trying to write something ugh

Ted Shifrin

12:55 AM

most books in the US are single- + multi-variable, Symposium. But I've never seen anything more than 1000 pages.

Symposium

Essential Calculus by Ian Stewart is 960 pages, with big dimentions (not 2000+) I suppose if it's mulvariable + single variable, it's essentially two books so it makes sense I suppose.

anakhro

@MikeMiller I don't quite see how Alexander's trick indicates that the gluing of the disks on the boundary is oriented homeomorphic to a sphere.

Can you elabourate?

I guess you get and inducded $\tilde\varphi\colon D^{n+1}\to D^{n+1}$?

Ted Shifrin

Symposium, I dunno that book of Stewart's, but it might be both single and multi already.

anakhro

But how does this apply to the gluing?

Symposium

Makes sense.

Ted Shifrin

1:02 AM

Anyhow, I agree that books are way too big ... but typical US students whine that there are never enough detailed examples. Even when there are too many.

anakhro

It's why I like Dover books. ;)

Ted Shifrin

They are of varying quality.

But, I was talking about calculus books, not serious math books :P

usukidoll

Dover is cheap and awesome

anakhro

Dover is publishing Kline's calculus book now, but that is in no way "short".

It is excellent though and worth the <$30 you pay for it from dover though!

Symposium

Early Transcendentals by Ian Stewart is even bigger: 1308 pages lol.

anakhro

1:11 AM

Yeah, it's good for flattening stuff.

Oddly enough, that's about the only good thing. :/

Ted Shifrin

It's a standard (but not great) book, Symposium. Everything's out of hand. But Stewart is dead (a number of years now).

anakhro

Is he the same Stewart who wrote the Galois theory book?

Ted Shifrin

NO

That's British Ian Stewart ... written a number of wonderful books. Lots for "general audience."

James ≠ Ian

Symposium

I've morbid fascination with these huge books, whenever I see them. Obviously they are basic, so no academic interest.

Ted Shifrin

morbid is morbid, Symposium.

Symposium

1:16 AM

Perhaps I get the itch to flatten stuff with them like @anakhro said they are good for.

anakhro

The best kind of itch is a flattening itch.

Ted Shifrin

Flattened fleas?

Daminark

Those books are probably good doorstoppers as well :P

anakhro

Much too large, you trip each time you leave or enter.

Jalapeno Nachos

@TedShifrin for a student who enjoys linear algebra, what should they do next? functional analysis or representation theory? which is a better road to fruitful research?

Ted Shifrin

1:23 AM

I can't answer that ... without knowing total background.

And "fruitful research" is a long road away ...

Jalapeno Nachos

I see

which road has more tractable problems?

Ted Shifrin

I cannot begin to answer that. Neither of those is close to my field.

(Or what used to be my field.)

Jalapeno Nachos

ah, ok

no worries, I try and post a question on the site then - thanks :)

Daminark

There aren't likely good absolute answers to that question in any event. Some people are more or less compatible with each of them, you just have to find your style

Ted Shifrin

Make things very specific and focused.

Jalapeno Nachos

1:26 AM

I see

anakhro

What did you like best about linear algebra, @JalapenoNachos, and what other fields of mathematics do you enjoy?

Ted Shifrin

that's a good constructive response.

But I still have no understanding of general knowledge/background.

Daminark

I'm pretty sure the low-hanging fruit in many subjects has been taken at this point, and I don't think either of them are likely intractable. Regarding what to learn next (ideally you'd eventually learn both), prior knowledge + preference is necessary

As Ted and anakhro have already said

Mike Miller

@anakhro Sorry I'm a little late. This is an extension of the statement that isotopic homeomorphisms induce homeomorphic gluings.

More precisely, if $\varphi$ extends to a homeomorphism $\tilde \varphi: D^{n+1} \to D^{n+1}$, then consider $D^{n+1} \cup_\varphi D^{n+1} \to D^{n+1} \cup_{\text{id}} D^{n+1}$, given by the identity on the first disc, and $\tilde \varphi$ on the second disc.

Ted Shifrin

Well, time for me to cook dinner. Bye, all.

Daminark

1:38 AM

See you

Hey Mike!

anakhro

@MikeMiller It's fine. So it is this induced map you are suggesting is a homeomorphism?

From our glued manifold to S^n?

Ah, I get it!!

I see!

Since varphi is oriented, it's isotopic to id. So that map on the gluings is a homeomorphism of spheres.

Which is probably exactly what you said, but takes me too long.

Thanks!

I find it tough to see how weird this gluing could be.

I guess I might need to work out some examples!

Mike Miller

2:03 AM

Well, exactly as bad as exotic spheres can be.

anakhro

Topologically, the spheres are not so bad, though, are they? With respect to their differentiable structure they are wacky, for sure.

I actually have never looked very far with exotic things. I have not seen many books which talk more about them.

I must go, but thanks again, Mike!

Secret

2:45 AM

Antinomy

Antinomy (Greek ἀντί, antí, "against, in opposition to", and νόμος, nómos, "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of Kant and Roberto Unger. There are many examples of antinomy. A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy because this statement is suggesting in itself to be an absolute truth, and therefore denies itself any truth in its statement. A paradox such as "this sentence is false" can also be considered to be an antinomy; for the...

Let A,B,C,D,E. Suppose A,B,C,D,E has Property P, then their union does not have property P

manooooh

4:07 AM

Hello guys

Suppose we have measured something with an uncertainty. The value is $a_0=9.36$ and the mistake made is $\Delta a=0.28$

Now I would like to find the interval of the magnitude value and the absolute uncertainty. So I write: $a=a_0\pm\Delta a=9\pm0=9$. Is it correct?

Observe that I have rounded the values, so my question is if when there is no integer part in the absolute uncertainty, if it is ok to round the first digit (which is zero)

Al t.

Hello. How to write the division symbol to represent a divided by b (but not this \frac{}{} nor /) using mathjax?

Secret

uh.... $a|b$?

Al t.

@Secret no, it's other symbol, usually used in high school or elementary school

Secret

4:22 AM

$a \div b$

Al t.

@Secret no, the symbol it's similar to the square root but without the little LHS sign -

manooooh

$\$?

Secret

17

Q: Better way to display long division?

I'm currently in the process of trying to create a worksheet for my students with long division problems for them to practice. Unfortunately, the best I've been able to come up with so far in terms of displaying long division like how they write it is: Which could work if need be, but I though...

formatting

There is no easy way

Al t.

@Secret :( ok thank you, I think i'll use |\overline{}

Secret

6:44 AM

10

Q: How is geometry defined using ZFC?

I've been trying to get a rigorous understanding for the mathematical concepts I learned in high school. I've been reading about how the real numbers can be constructed from the axioms of ZFC, but I can't find any information on geometry. I've read about axiomatic formulations of Euclidean geome...

geometry set-theory foundations

The minimalistic definition of space is irreducible entities called points and their relationships between them

Set-theoretic topology

In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). == Objects studied in set-theoretic topology == === Dowker spaces === In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M.E. Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality ...

Nick

7:09 AM

Good morning, math room!

Zee

math is so hard

Nick

9:12 AM

$5 Integral Vs. $500 Integral

►

@Zee albeit, did you know you could make money solving problems?

Vrouvrou

hello, is $C(\mathbb{R},\mathbb{R})$ with the norm sup is complete like $C([a,b],\mathbb{R})$

micsthepick

how is an integral supposed to have a price?

just checking, I thought the determinant of Matrix([
[0, 1, 0, 2],
[0, -1, 1, -1],
[1, 1, 1, 2],
[3, 0, 2, 1]]) ought to be 0

but sympy says it's 2

@Nick so the gimmick there is that the integral is done using equipment of that price range?

ok, so there is this website that says it is 2

but when I add all the products of the positive diagonals and subtract the negatives I get 0?

what is going on???

hmm, maybe the website uses sympy

oh

I am not including enough products

for 3x3 the diagonals are the only products you can do

Alessandro Codenotti

9:39 AM

@Vrouvrou What do you think

micsthepick

the thing that confused me the most was I expected it to be zero, yet I missed a minus sign

Jasper Loy

10:10 AM

I just thought of a very beautiful equation that I am not sure I have seen before: 0=-0.

Oskar Tegby

Any ideas? math.stackexchange.com/questions/2896001/…

famesyasd

11:00 AM

I want to translate "If exists X X = R then forall x x in X iff x in R" into FOL but I don't get it, the second appearance of X is not binded how so? Am I making some tiny mistake or does FOl simply does not capture this? I Figured out I can do what I want with "Forall X (if X = R then forall x x in X iff x in R)" but I don't get it why can't I do this with the initial form

Leaky Nun

@famesyasd you quantified both X and x

also, context

famesyasd

@LeakyNun I didn't get what you mean in the first sentence. It popped up when I occasionally replaced the meaning of "exists single x P(x)" formula with "If (exists x,y P(x) and P(y)) then (x = y)" instead of "Forall x,y if (P(x) and P(y)) then x = y"; proved a bunch of formulas and then later realized that x,y in the "x = y" part are not binded

micsthepick

11:25 AM

I'm trying to find the subset of four vectors that is a basis of their span

according to math.stackexchange.com/a/127435/227162

I should put the vectors in the columns of my matrix, as opposed to the rows

because it lets me read off the basis for the column space

so that is just the basis for all the column vectors right?

and correct me if I am wrong, but the basis for a set of linearly dependent columns in RREF where exactly one row is equal to 0 could be any combination of the columns right?

Ole Petersen

12:45 PM

Which methods is fasest to approach the right value of pi, by hand. Method 1 or 2 ?

0

Q: Compare Calculation-time for two methods

Method $1$ (Machin): $$ \pi = 16 \left(\frac{1}{5} - \frac{1}{3\cdot 5^3} + \frac{1}{5\cdot 5^5} - \frac{1}{7\cdot 5^7} + \ldots\right) -4\left(\frac{1}{239} - \frac{1}{3\cdot 239^3} + \frac{1}{5\cdot 239^5} - \frac{1}{7\cdot 239^7} + \ldots\right) $$ Method $2$: $$ \pi = 2 \left(1 + \frac{1}{...

computational-complexity computational-mathematics

micsthepick

12:59 PM

I am required to show that the inner product of two vectors is equal to 1/4 * (norm(u+v)^2-norm(u-v)^2)

isn't that RHS just equal to 0???

Leaky Nun

@micsthepick no

micsthepick

but the norms cancel after pythagoras

right?

@LeakyNun explain yourself

I could be wrong, because I am tired, but I am 99% sure that it is just equal to 0

so I found a different RHS: (||u+v||^2+||u-v||^2)/2

... no, that is wrong

but do take note of the minus sign in front of norm(u-v)^2

Leaky Nun

1:16 PM

10 mins ago, by micsthepick

but the norms cancel after pythagoras

we have no right-angle assumption here

micsthepick

ah

see what happens to my brain when it is late

so how do you expand norm(u + v)^2?

with an extra 2 u dot v

ok, everything makes sense now

Oskar Tegby

1:47 PM

Any love for my question? math.stackexchange.com/questions/2896ch001/…

Something went wrong with the URL. I try again.

https://math.stackexchange.com/questions/2896001/showing-that-a-bounded-continuous-function-is-zero-if-an-integral-with-it-is-zer?noredirect=1#comment5981177_2896001

Oskar Tegby

2:01 PM

@TedShifrin I'm having some issues with the following problems that I've posted.

https://math.stackexchange.com/questions/2896001/showing-that-a-bounded-continuous-function-is-zero-if-an-integral-with-it-is-zer?noredirect=1#comment5981177_2896001

https://math.stackexchange.com/questions/2896198/computing-the-derivative-of-a-system-of-equations-in-the-neighborhood-of-a-point?noredirect=1#comment5981606_2896198

and I still wonder how close to a given point where we can get if we have $M_n=constant$ and $x_n\to0$ as $n\to\infty$. I know that I'm asking for a lot, but I'd appreciate your hel…

jesterII

@XanderHenderson Thanks for reviewing my edit at math.stackexchange.com/review/suggested-edits/1058631. Could you please clarify why the edit was rejected? The posted reason "does not make the post even a little bit easier to read" does not seem sensible. The edit significantly improves readability of the post by using MathJax and fixing formatting. (The other user Key Flex who rejected the edit has been suspended from the site.)

Jasper Loy

4:03 PM

Hi @Semiclassical I hope you are well.

Semiclassical

hi

Ninja hatori

0

Q: about isometry classes

This is Theorem 2.3.8 from Scharlau's book Quadratic and Hermitian Forms I know that $\mathbb{F}_q$/$\mathbb{F}_q^2$ has index 2 so it has only two elements ; since 1 is also square in $\mathbb{F}_q$ why it is one of element of $\mathbb{F}_q$/$\mathbb{F}_q^2$. Also what is meaning form $\langl...

linear-algebra group-theory field-theory finite-fields bilinear-form

Ted Shifrin

@Oskar: With regard to the uniform convergence thing, under those circumstances you know convergence will NOT be uniform on $\Bbb R$ (indeed, on any interval containing $0$). But on $[a,\infty)$ (with $a>0$), provided $f_n$ are decreasing on $[x_n,\infty)$ and $f_n(a)\to 0$, you'll be fine. Work it out. ...

Re the other two, in part you're misreading the iff. You get to assume $f$ is bounded, continuous. Then you want to prove $f=0$ iff the various integrals vanish. I would make a change of variables to turn it into $[0,1]$ and use Weierstrass. Re implicit fn thm, I haven't checked your work, but $f(z)=(x(z),y(z))$ and $f'(z)=(x'(z),y'(z))$, so why are you worried? You might want to check out some of my videos on IFT ...

Aaron Hall

4:19 PM

anyone mind if I make a link to the main site in the room description?

Jasper Loy

Nobody will mind.

But one can navigate to the main site by clicking the icon at the bottom of this page.

Aaron Hall

oh, nice, I missed that, then nevermind...

Jasper Loy

That's the trick I always use when I am in the chat and want to see the main quickly.

Aaron Hall

ok, so... the fundamental theorem of calculus substitutes h for delta x - is it just more convenient? Is there a deeper reason we take the limit as h -> 0 instead of delta x -> 0?

Is it just a historic reason? Does h stand for or mean something significant? If so, is that meaning different from delta x?

Jasper Loy

Well, delta x is just the change in x, and you can call it anything you want, like h, for example.

h does not stand for anything special that I know of.

Aaron Hall

4:30 PM

The substitution seems to add an unnecessary step in the derivation.

Ted Shifrin

hi @Jasper.

@Aaron: $h$ is simpler to write and type than $\Delta x$. No difference otherwise.

Abdullah UYU

@AaronHall I'm not sure, but possibly you mean two slightly different limit definitions of derivative?

Ted Shifrin

Typically, $h$ is used in more advanced math treatments, and $\Delta x$ is used more in engineering-style calculus textbooks.

Jasper Loy

@TedShifrin Hello. Are you thinking of writing more books in your retirement?

Ted Shifrin

Hell, no!

Abdullah UYU

4:33 PM

Ah, ok. I think I misinterpreted. @AaronHall

Ted Shifrin

@Jasper: At one point I thought I'd undertake the project of TeXing up notes for graduate differential geometry, but I decided I wasn't interested enough and there wouldn't be that much interest.

Jasper Loy

@TedShifrin I see. I am thinking of buying a MacBook. Do you know if Pages is good for typing equations? =)

Ted Shifrin

I have been a Mac user since 1988, but I've never used Pages. It's just their free version of Word, and I HATE Microsoft Word. I have no idea if there's an equation editor. I guess I could look. ... I use TeXShop for everything, including letters.

Jasper Loy

@TedShifrin Wow, I use TeXworks on Windows, and it was inspired by Richard Koch's TeXShop.

Ted Shifrin

It looks like they've built in connectivity with a spreadsheet, but I don't see an equation editor.

Oh, they do have it.

You can include mathematical expressions and equations in your document. To add an equation, you compose it in the Pages equation dialog using LaTeX commands or MathML elements, then insert it into your document.

You can add equations inline with body text or as non-inline objects. An inline object is embedded in the text flow and moves with the text. A non-inline object is fixed to a particular position on a page. You can drag non-inline objects anywhere on a page, wrap text around them, or layer text or other objects over or behind them.

Jasper Loy

4:38 PM

Well, I do know it can type equations, but I don't know whether it works well, like whether it can be used easily, just like Word. =)

Ted Shifrin

Since I hate Word, I haven't tried.

Jasper Loy

lol

Ted Shifrin

For certain documents for the administration at UGA I had to type things up in Word. It raised my blood pressure 20 points every time.

5

@Jasper: It seems to work just fine. With a keystroke or menu click you get the equation editor box and you just type the TeX into it, then hit "insert."

And you can edit it by double-clicking to bring the equation editor back up.

Jasper Loy

OK. Yeah maybe I will get the MacBook Air, the cheapest apple computer.

Aaron Hall

I switched to Linux from Windows a decade ago while working on my graduate business degree, and completed the remaining 2/3 of the degree with Linux (joint projects were edited on Word in the computer lab...).

Ted Shifrin

4:43 PM

Other than teaching in classrooms, I've managed to escape Windows my entire life.

Jasper Loy

@AaronHall Same but I went back to Windows cos things work so much better on Windows. =)

Abdullah UYU

A star from me to hatred to Word :D

Aaron Hall

@JasperLoy things like viruses and malware?

Ted Shifrin

I've been a Mac devotee for 30 years now.

LOL @Abdullah

@Jasper: You saw what I said? May I close Pages up now? :)

Aaron Hall

I switched to Linux for the same reason. :P

Jasper Loy

4:45 PM

@TedShifrin Yes, thanks. =)

Ted Shifrin

Whew ...

:)

Since it's native to Macs, I bet it's a whole lot less frustrating than Word imported. Maybe I'll try it out sometime.

Jasper Loy

@AaronHall Maybe because I like things to be more or less perfect, but the software in Linux, even though developers do fix it, often introduce new problems or inconveniences.

Leaky Nun

@TedShifrin I just learnt today that the pivot columns form a basis for the column space lol

Abdullah UYU

@JasperLoy There are no differences, in others you don't hear of them, in Linux it's open to all. At least the problems are clear.

Ted Shifrin

Congratulations, Leaky. You could have found that proved in my books :)

Leaky Nun

4:50 PM

well I don't exactly have your books

Ted Shifrin

"exactly"

Jasper Loy

Some Ted books are available currently on Russian servers.

Ted Shifrin

It's not difficult, @Leaky. The key idea is that if you have an element of the nullspace (kernel), it gives you linear combinations of the columns that add up to $0$. So once you have a basis for the nullspace, each vector has a $1$ in one of the non-pivot positions, and that allows you to solve for the non-pivot column as a linear combination of the pivot columns.

Jasper Loy

However, other Ted books have never been available there.

Ted Shifrin

GRR @ anything Russian.

Leaky Nun

4:53 PM

ok

Ted Shifrin

Despite my Russian heritage, I'm not too thrilled that the Russians are manipulating everything, including our voting.

Transcript for

$Mathematics$ Mathematics