Mathematics - 2015-07-09 (page 2 of 4)

Remember me

12:00

I haven't seen artin ...... So no idea

Balarka Sen

no point in learning theory if you can't do exercises.

solving problem is the whole fun of doing mathematics!

Remember me

But then 45,46 questions are too much sometimes (though not always)

Yes I don't deny that fact

Balarka Sen

well, algebra is hard, that's why solving more exercises is important.

Remember me

Munkres has really nice questions though...

Balarka Sen

@Rememberme so, I presume you want to do galois theory?

Remember me

12:04

Not now @BalarkaSen I have many stuff to do .. Sadly

Balarka Sen

that'd be good. learn galois theory, algebraic number theory. those are fun branches.

Remember me

I need to get the basics right then only I will be doing those stuffs

Balarka Sen

right. but you don't need topology to do galois theory.

galois theory is purely algebra.

Remember me

But topology is fun to.me..... :)

Topology makes me happy for some reason....

Hippalectryon

Continuously happy ? :D

Remember me

12:06

Yes .. Haha :)

@BalarkaSen I have this professor living beside my house....
He was talking about complex tori ....
What kind of a structure is that ?

Balarka Sen

well, decide what you want to do. part of the reason i told you guys about the link is to get you to galois theory/number theory instead of fancying about algebraic topology

@Rememberme $S^1 \subset \Bbb C$ be the unit circle. give it a subspace topology. $S^1 \times S^1 \subset \Bbb C \times \Bbb C = \Bbb C^2$ is then a torus embedded in $\Bbb C^2$. this inherits a natural complex analytic structure from $\Bbb C$. this is called the complex torus.

i.e., it's a complex analytic manifold : roughly, it's a space which locally looks like $\Bbb C^n$ and is "smooth" in the complex analytic notion (the correct word is holomorphic, I think).

Remember me

Hmm.......
I will finish topology with algebra (as much as I can do, I wont do galois stuff because I remember you saying that you were deprived of the fun of altop and galloa theory), I have this craze of doing homological algebra which I yet don't know what it deals with, then I have to chose between axiomatic set theory or altop, so mostly I would do altop because I want to do Galois stuff.... , then since I have already done number theory I.might happen to go to ANT because that also fascinates me a lot sometimes more than altoo

@BalarkaSen so basically it is a torus embedded in the complex plane.....

Balarka Sen

@Rememberme "I would do altop because I want to do Galois stuff" you realize the algebraic topology have nothing to do with galois theory?

The link I told you is an analogy, not a dictionary. The branch which deals with that is algebraic geometry.

Remember me

Yes I do realize that ..... Ahh how did i leave algebraic geometry :) too many super interesting stuffs to do.. I told ya

Balarka Sen

to do algebraic geometry, there is no need to do algebraic topology at all.

you need to know heaps of algebra.

Remember me

12:19

So @BalarkaSen have you done homological algebra? I would like if you might give an insight what it is about

Also one more topic I would really love to do...
Differential topology

But all of this is future....

Balarka Sen

No, I haven't done homological algebra. You should ask Tobias.

Remember me

You like it ?

Balarka Sen

@Remember Seeing the number of things you have set up for yourself to do, that might take a couple years.

Remember me

5 years @BalarkaSen or more

Balarka Sen

@Remember How would I like something I haven't studied?

Remember me

12:22

As in you did homology right so you.might have an idea....

Balarka Sen

the flavours of homology and homological algebra are vastly different

Remember me

Okay....

Balarka Sen

like I said, many of algebraic things have relations (and/or motivated from) topological things, but the flavours are different

Galois theory has a relation with covering space theory, but you need to know absolutely no alg top to do galois theory

Remember me

Yes.... But all this always brings me to a question...

Why did we happen to require algebra to solve our topological problems?

Where did we get the idea from? And why and how

Balarka Sen

we don't need to, it's just that algebraic things are (mostly) well-understood, while topological things are not.

so having algebraic invariants for topological spaces seems fruitful

Remember me

12:27

Okay so its just a supplement which makes things easier....

Balarka Sen

we can also use topological ideas to prove algebraic things, but that's generally harder (although very very fruitful)

Remember me

Hmm..
So for example can we create a Galois group from the fundamental group?

Balarka Sen

not literally correct, but yes, we can use properties of fundamental groups to inspect the absolute galois group

that's what Grothendieck did

Remember me

Hmm....
So what is your main goal behind studying covering spaces and Galois groups?

Balarka Sen

understanding more about the absolute galois group.

Remember me

12:34

Ok.

Balarka Sen

probably there's no one in the world right now who knows everything about the absolute galois group

understanding it is one of the major problems in mathematics

Remember me

@BalarkaSen wow...

Balarka Sen

i should say absolute galois group of $\Bbb Q$, but whatever.

Remember me

You seem to be really enthusiastic when I take up the Galois topic @BalarkaSen

Here in chat

Balarka Sen

yes, because I want to point out that there's more mathematics than algebraic topology out there

Remember me

12:38

I know that @BalarkaSen if I had to stick to only one thing the I would have just done altop .... I have set many things for me those I feel more conceptual than real analysis ( though everything arises from it)

@BalarkaSen what do you think about differential topology

Balarka Sen

I don't know, haven't studied it.

Remember me

Hmm....
The analogy which told me still is bouncing everywhere in my head.... I have been standing out of class for not paying attention to the class and day dreaming ( though I was thinking about Galois and covering spaces)

Really super damn ......... (many more adjectives) interesting @BalarkaSen

dREaM

$\subset\overline{\underline{::::::::::::::::::::::::::::}} $ $\supset$

Soham Chowdhury

13:29

well, hi chat.

Mary Star

Hello!!

Is someone of you familiar with the following?

We have linear differential equations with polynomial coefficients depending on x.

$a_n(x)y^{(n)}+ \dots a_1(x)y^{(1)}+a_0(x)y^{(0)}=b(x)$

There are problems like if there are solutions, if the solutions are linear independent and so on and we are looking for the decidability and the complexity.

Balarka Sen

hi @Soham

evinda

13:51

Hello @SohamChowdhury

Soham Chowdhury

hello, @evinda, @Balarka.

evinda

How are you? @SohamChowdhury

Soham Chowdhury

when you first went to Mj, did he ask you to study analysis, @Balarka?

@evinda Hanging on in quiet desperation :P

I feel brilliant right now, just kidding.

evinda

@SohamChowdhury Why? What happened?

Soham Chowdhury

@evinda oh, I won first place at a quiz today :)

evinda

13:55

@SohamChowdhury Nice!!!

Lembik

what is the $n$ by $n$ $(0,1)$ matrix with the largest determinant?

Hippalectryon

@Lembik By (0,1) you mean with coefficients 0 or 1 ?

Lembik

@Hippalectryon yes exactly

is this something obvious? I mean finding a matrix with the largest determinant

Chris's sis the artist

Marvellous results flow here one after another ceaselessly ...

Lembik

@Chris'ssistheartist what is that about?

Chris's sis the artist

14:04

@Lembik Integrals.

Hippalectryon

@Lembik Well first of all that matrix most likely won't be unique

Lembik

@Hippalectryon I changed it to "a" :)

Hippalectryon

@Lembik Have you tried on a few low-dimension cases ?

Lembik

@Hippalectryon Yes but I don't see an obvious pattern.. I was wondering if the answer is obvious by considering the formula for the determinant

robjohn

@SohamChowdhury that's the English way

Hippalectryon

14:10

@Lembik What's the max determinant you find in dimension 4 ?

Balarka Sen

@SohamChowdhury yes, he did. but he says it's ok if I do it when I want to now. says I'm naturally algebraic.

I know enough analysis to get on with it :P (that is, basic point-set topology on R^n)

hello @Ted.

Ted Shifrin

@Stan: Of course I made the choice purposefully. The open covering definition belongs in a serious analysis or topology class. For $\Bbb R^n$, and for the purposes of these students, the sequential definition is superior. Note that basically the open cover definition you want appears in an exercise.

hi @Balarka

Lembik

@Hippalectryon I just found out this is a famously hard problem

Ted Shifrin

Oh oh ... Someone needs to put the LaTeX in chat announcement back on the star list.

Lembik

@Hippalectryon en.wikipedia.org/wiki/Hadamard%27s_maximal_determinant_problem

Hippalectryon

14:15

@Lembik ah ok

skill patrol

Hi Professor @TedShifrin how are you?

Ted Shifrin

hi @skull ... Doing fine, thanks ... more sorting/packing later today.

skill patrol

What day are you moving?

Chris's sis the artist

Ah, simple closed form $$\frac{\left(2-\sqrt{2}\right) \sqrt{\pi }}{2 e}$$

Ted Shifrin

On the plane two weeks from today, @skull.

skill patrol

14:18

Nice :-)

Chris's sis the artist

The annoying thing is that the problem is too easy ...

skill patrol

That's "annoying?"

Chris's sis the artist

@skillpatrol Yeah.

skill patrol

Why?

Chris's sis the artist

@skillpatrol You want harder problems where you might like to put at work your creativity ...

skill patrol

14:20

I see.

Chris's sis the artist

(create new tools, new strategies to tackle the problems and so on)

skill patrol

ok, you thrive on challenges :-)

Chris's sis the artist

:D

Lembik

@Chris'ssistheartist do you just focus on integrals?

Chris's sis the artist

@Lembik Integrals, series and limits at the moment.

Lembik

14:25

I have plenty of hard problems that are not integrals!

Chris's sis the artist

@Lembik I believe you. I also have a lot of research to do in the area of integrals, series and limits.

The only ideas I have at the moment might take many years to explore them all. Well, I have greater plans to math, not now, but later.

Lembik

@Chris'ssistheartist hm... math.stackexchange.com/questions/1321242/… is missing an answer!

@Chris'ssistheartist and is a limit question under a different name

@Chris'ssistheartist any interest?

Chris's sis the artist

@Lembik That looks interesting. I also have some similar questions proposed by me.

(need to look for them)

Lembik

@Chris'ssistheartist cool. I will follow it now in case you come up with something

Chris's sis the artist

OK

@Lembik These days I focus more on the problems I'm going to add to my book. I'll come to that after a while.

Hippalectryon

14:31

@Chris'ssistheartist Btw do you have integrals with $\sqrt {\tan}$ ? (and integrals in terms of $\int\sqrt {\tan}$)

Chris's sis the artist

@Hippalectryon Yes.

Hippalectryon

:D I've never seen that integral used somewhere else other than to scare people

Chris's sis the artist

Use beta function (cleverly) $$\int_0^{\pi/2} \sqrt{\tan (x)} \, dx=\frac{\sqrt{2}}{2}\pi$$ and Q.E.D.

@Hippalectryon :D

Hippalectryon

@Chris'ssistheartist I mean, does that integral sometime arise in results (since its general closed form is quite troublesome) ?

iluso

Hi

Chris's sis the artist

14:34

@Hippalectryon well, I need to check my stuff, but as far as I remember, yes. I think I also have a tough limit involving such integrals.

That I created in the summer (toward its end) of 2013.

@Hippalectryon ^^^

Hippalectryon

:D

Chris's sis the artist

:D

iluso

Is there a standard way to turn a function which count ordered arrangements into one that count unordered arrangements ?

robjohn

$$
\begin{align}
\int_0^{\pi/2}\sqrt{\tan(x)}\,\mathrm{d}x
&=\int_0^\infty\sqrt{u}\frac{\mathrm{d}u}{1+u^2}\\
&=\frac12\int_0^\infty\frac{t^{-1/4}}{1+t}\,\mathrm{d}t\\
&=\frac12\Gamma(\tfrac14)\Gamma(\tfrac34)\\
&=\frac12\pi\csc(\pi/4)\\
&=\frac\pi{\sqrt2}
\end{align}
$$

Chris's sis the artist

@robjohn YES! :-)

Hippalectryon

14:53

@Chris'ssistheartist @robjohn But the general form is rather troublesome (as in long to get).. is it ? (is there a quick way to get it ?)

robjohn

@Hippalectryon You mean the indefinite integral?

Hippalectryon

Yes

Chris's sis the artist

@Hippalectryon What general form? You mean a system of equations for indefinite integral? :-)

Say, $I=\int \sqrt{\tan (x)} \, dx$ and $J=\int \sqrt{\cot (x)} \, dx$

(addition and subtraction -> done)

Hippalectryon

Not in terms of J though. The explicit closed form.

Chris's sis the artist

@Hippalectryon Of course you can do it, but not sure you get nicer ways.

Cristopher

14:58

@robjohn Hi. Yesterday I asked you a question about Holder's inequality (I guess you didn't see it). Can you tell me why this is true? $\displaystyle\frac{|x+1|y^4}{|x+1|^3+2|y|^3} \leq\sqrt[3]{\dfrac{16}{3125}}\left(|x|^3+2|y|^3\right)^{2/3}$

Chris's sis the artist

@Hippalectryon see here math.stackexchange.com/questions/828640/…

dREaM

15:18

-1

Q: Tricky Real Life Logic Puzzle

I thought someone here might be able to help me with this tricky logic puzzle I've recently come across. Any help would be appreciated! Let's get down to it: The puzzle consists of one box approx. 15cm long and 25cm wide and one reddish triangular-prism shaped wooden block. Box has three differ...

Lol

iluso

@dREaM haha

Fargle

Like...why? Why would someone post this?

Balarka Sen

lol

for fun?

Fargle

I guess--but it's not like people are going to leave it up.

robjohn

@Cristopher here is what I did:

Hölder says
$$
\left(|x+1|^3\right)^{\color{#C00000}{1/5}}\,\left(\tfrac12|y|^3\right)^{\color{#C00000}{4/5}}\le\color{#C00000}{\frac15}|x+1|^3+\color{#C00000}{\frac45}\tfrac12|y|^3
$$
raise to the $5/3$ power and cancel:
$$
|x+1|\left(\tfrac12\right)^{4/3}y^4\le\left(\tfrac15\right)^{5/3}\left(|x+1|^3+2|y|^3\right)^{5/3}\\
\frac{|x+1|y^4}{|x+1|^3+2|y|^3}\le\sqrt[\large3]{\tfrac{16}{3125}}\left(|x+1|^3+2|y|^3\right)^{2/3}
$$

Cristopher

15:36

@robjohn Oh, it's clear now. So, by Holder's inequality, the limit is 0, correct? (if so, mathematica did get it right this time)

Fargle

I used to think I was good at algebraic manipulation, and then I started coming to this chat. :D

robjohn

@Cristopher yes. when the limit exists, it usually gets it right. It's when the limit doesn't exist that it sometimes messes up.

Fargle

I guess we're still better at mathematics than computers. Maybe not calculations, but definitely the mechanisms behind them.

robjohn

@Cristopher It usually messes up when the approach that shows the discontinuity is a curved path, not a straight line.

Cristopher

@robjohn Ah, I see. Thanks. Is there a general way to recognize when I should use Holder's inequality when computing a limit?

robjohn

15:42

when comparing a product to a sum, either the AM-GM or Hölder are useful

Cristopher

Thank you

Soham Chowdhury

15:56

@robjohn I am you, so . . . I'm a fan too? :)

@Fargle I never did, haha.

Fargle

@SohamChowdhury I'm from a small town--in my former worldview I was a lot more awesome than I am now.

Soham Chowdhury

@Balarka, I was half-paralysed when I went to SB. Maybe I'll tell him to make an exception next time. And isn't metric space point-set stuff essentially (a part of a basic course in) analysis?

@Fargle I think I understand. In India, you're either from a city or meant to be a farmer (with near-$1$ probability), so I'm very, very lucky.

Small towns elsewhere in the world are very different.

Fargle

@SohamChowdhury Well, it wasn't quite that small. But it was small enough where I felt bigger than the town.

Soham Chowdhury

I used to think I was good at $\LaTeX$/Markdown manipulation, and then I started coming to this chat. :P

Balarka Sen

@SohamChowdhury He won't agree, I bet.

somewhat evil grin

Soham Chowdhury

15:59

@BalarkaSen how encouraging.

Balarka Sen

no, metric space point set stuff is basic analysis

Fargle

I still live there, as a matter of fact, thanks to my most recent setback. Rather silly of me to speak of it in the past tense.

Balarka Sen

there is lot more to analysis than that

Soham Chowdhury

Look at Balarka waxing lyrical about the wonders of analysis. :P

@BalarkaSen does SB get angry?

Stan Shunpike

@TedShifrin I presumed it was intentional. I just wanted to understand your reasoning to decide how I should think about it. What is the number of that exercise?

Balarka Sen

16:00

@SohamChowdhury nah, never seen him angry.

he's very friendly.

Soham Chowdhury

@BalarkaSen ah, now it's encouraging.

I tend to be afraid of extra-calm people. They're scary when pissed.

Anyway.

Balarka Sen

prof is angry :(

Soham Chowdhury

why?

does Mj get angry?

Balarka Sen

yes, that's whom I mean by prof.

Soham Chowdhury

yes, but he doesn't seem like the angry type. what did you do?

Boni Tea

16:04

What's Mj?

Soham Chowdhury

no, it was correct before. $whom\mapsto who$

@BoniTea someone.

Boni Tea

Both who and whom are correct, I think.

Balarka Sen

@SohamChowdhury I didn't do anything. I happen to saw him scolding some phd student of his.

it was rather horrifying for the student, I guess.

Soham Chowdhury

@BoniTea I don't. "who(m) I'm referring to", but definitely "who I mean". Who knows, maybe I'm wrong.

@BalarkaSen eesh.

Boni Tea

My understanding is that "whom" may be used whenever it is the object of the "bit" after it.

And yea, "who" can be used wherever "whom" is allowed.

Soham Chowdhury

16:07

yes, exactly. "mean" doesn't exactly "act on" (whatever that means) "who" in that case, which is why I'm saying what I'm saying. but I couldn't be bothered to think about this now. :)

Boni Tea

And.. hopefully, I won't get my supervisor angry.

Soham Chowdhury

@BoniTea good luck with that. grad student?

@BalarkaSen what are you doing?

Boni Tea

Starting a PhD in September.

Soham Chowdhury

I definitely need to learn a little about modules.

@BoniTea oh, cool! any idea what you'll work on?

Balarka Sen

@Soham I'm writing up the analogies I have found so far.

Soham Chowdhury

16:08

@BalarkaSen nice. must feel good.

Balarka Sen

@SohamChowdhury Did you learn about group actions?

well, not writing, but LaTeXing-up.

Soham Chowdhury

@BalarkaSen will, tomorrow when I return from school.

@BalarkaSen the only things I use paper for are commutative diagrams.

Boni Tea

@SohamChowdhury Invariants and polynomials.

Soham Chowdhury

knots?

Balarka Sen

I am drawing a lot of commutation diagrams in LaTeX.

Boni Tea

16:10

Eh, I still write everything before typing them up.

Balarka Sen

arrays are good enough.

Boni Tea

@SohamChowdhury Related to knots, but not directly investigated.

Mobile version of the website is not so easy to use.

Fargle

@SohamChowdhury I'm the opposite--I only use $\LaTeX$ for school assignments, and write everything else, usually either on paper or whiteboard (or mirror, as was the case in my last apartment).

Soham Chowdhury

I'm going to do something very stupid now. Bye for now.

Boni Tea

Um.. good luck.

Balarka Sen

16:11

Have fun with EVS, @Soham :P

Boni Tea

I can't tell what EVS is.

Balarka Sen

I write up things on my doors, @Fargle. Although using chalks.

Fargle

I wish I had a good chalkboard.

Balarka Sen

I don't have one either. I had a whiteboard, but sold it a year ago because the new apartment haven't had enough room to hang it on.

iluso

Given $j\in\mathbb{N}$, does "$d^{1/(j+1)}$ is a natural number" means something for $d$ ?

Fargle

16:24

@iluso ...I mean, it means that $d$ is a perfect power of $j+1$, right?

Unless you're looking for something more nuanced.

iluso

@Fargle Oh yeah, thanks

Soham Chowdhury

16:49

@BoniTea environmental science. very exciting.

@BalarkaSen it wasn't EVS.

@BalarkaSen eh, I'm getting a whiteboard for my birthday this year. I have few wants; I'm a simple man. :P

but I prefer blackboards. (or green boards, like the ones in my school)

Balarka Sen

@SohamChowdhury oh?

Remember me

Same here...
I forcefully have to write on doors @SohamChowdhury

Hey @BalarkaSen

Balarka Sen

hi

Remember me

So you are writing all your records for the Galois question..... ?

Balarka Sen

yep

iwriteonbananas

17:57

@BalarkaSen hi

Balarka Sen

hi

Semiclassical

18:16

afternoon, chat

Chris's sis the artist

@Semiclassical Hi

robjohn

I can't believe that people would use some of the methods they claim for this question. I bet it has been a while since they have done manual computation without a calculator.

Semiclassical

@Chris'ssistheartist: hiya. i meant to point you to an double integral i saw a bit ago, which people (including me) solved by appealing to stokes' theorem and geometry. but i wondered if you'd be able to compute it directly :)

11

Q: Arnold Trivium Problem 39

We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.) Calculate the Gauss integral $$\int \int \frac{(d\vec A, d\vec B, \vec A-\vec B)}{|\vec A-\vec B|^3},$$ where $\vec A$ ru...

notation vector-analysis mathematical-physics

linked to my answer initially rather than the question itself, derp

Chris's sis the artist

@Semiclassical aaaaaaaahhhhhhh

@Semiclassical Not exactly the thing I use to play with, but it's interesting though. :-)

Semiclassical

@robjohn without looking at the answers, i think the way i'd tend to do it is via the binomial approximation, e.g. look for some way to write $2=a^2(1-t)\approx a^2 (1-t/2)^2$ for small $t$

robjohn

18:31

@Semiclassical There is little doubt that I would use the scaffold method.

Semiclassical

don't know what that is off the top of my head

robjohn

@Semiclassical It's described in the link from my answer and I work out a bit of $\sqrt2$

Semiclassical

mmkay.

i'll take a look after i think a bit more about how i'd approach it

though it's sort of funny how much easier life becomes when you have something even as simple as a hand calculator with only basic operations (i.e. no square root)

in which case i'd probably do newton's method and iterate

Ted Shifrin

@StanShunpike #12 in the first section.

Stan Shunpike

Excellent! Will look at it this afternoon :)

Semiclassical

18:40

afternoon @ted

Ted Shifrin

Hi @Semiclassic

Stan Shunpike

@TedShifrin Hows moving going?

Ted Shifrin

Well, @Stan, I have a week left to be ready. I'll be glad when the next month is over :P

So, you learning it all?

Stan Shunpike

Well, I am learning a lot.

I really felt yesterday like I have never understood length before.

Since I tried to understand compactness and got confused with countably finite, countably infinite, uncountably infinite.

I thought uncountably just meant you couldn't because it went on forever

not that even if you counted forever that you would still miss some numbers

Ted Shifrin

length?

Stan Shunpike

18:44

4

A: Why can a closed, bounded interval be uncountable?

I suspect you're conflating two meanings of "finite". Some sets are finite, meaning they have only finitely many elements. An interval like $[0,1]$ is not such a set. On the other hand, $[0,1]$ has finite length, which is a quite different matter. As the other answers have explained, finite le...

see the top answer

Semiclassical

measure != cardinality

Ted Shifrin

Hmm, you have a lot of basics to iron out, @Stan :)

Stan Shunpike

Yes, I know that. :)

Ted Shifrin

For easy but interesting reading, you might check out "How to Think Like a Mathematician," by Kevin Houston.

Stan Shunpike

Hmmm, I will look it up. I wonder if our library has that.

Ted Shifrin

18:46

It's actually quite affordable, but check the library.

Did you sort out the intuitive explanation I gave for Lagrange Multipliers?

Stan Shunpike

In the PDF? I skimmed the whole thing. It looked very similar to your lectures, which I did follow I think. I now understand the multplier as a scalar needed to make $\nabla g(\mathbf{x})$ equal to $\nabla f(\mathbf{x})$. One thing I am confused about is what info the Hessian gives me vs the first derivative

The first derivative of $L$ being zero tells me I am at a critical point right?

Ted Shifrin

That's the definition of a critical point.

Stan Shunpike

Okay. So is it the second derivative that tells me whether I am at a max or min?

Ted Shifrin

First go back and review single-variable calculus and think about what the second derivative tells you at a critical point.

Stan Shunpike

Well, I did and I know that

I am just checking/verifying.

Ted Shifrin

18:50

OK, so it's just generalizing that. Remember that you can have $f''(a)=0$ when $a$ is a critical point and then you don't know. Same thing happens in the multivariable setting, but much more complicated.

Stan Shunpike

right, you went through all the cases in your video and how that relates to the determinant iirc

Ted Shifrin

The best way to understand — both single and multi — is in terms of the second-order Taylor polynomial.

Stan Shunpike

right with the 1/2 term and the quadratic form with the Hessian

Ted Shifrin

not just determinant ... it's the signs of the $D$ entries in $LDL^\top$ or the signs of the eigenvalues (which we get to at the end).

Stan Shunpike

Yes, exactly.

Ted Shifrin

18:52

So for your exercise, when you look at $F=f\circ\Phi$, you're then back to a standard multivariable function (with no constraints).

Stan Shunpike

Is this why you said I am not working with the full Hessian? iirc u said something like that

Ted Shifrin

Right, because you're on this $(n-1)$-dimensional hypersurface $g(x)=c$, and the full Hessian takes into account all directions in $\Bbb R^{n}$.

user1667423

Is there any significance to the integral of the Hessian determinant of a function over space?

Does it tell us something about the behavior of the function?

Stan Shunpike

@TedShifrin So parameterizing gives us a standard multivariable function? Why is this advantageous? Because we can now work on the hypersurface?

Ted Shifrin

Because then you're doing regular, unconstrained calculus, @Stan.

user1667423

18:57

e.g. the integral of the squared magnitude of the gradient of the function tells us how variable it is (the Dirichlet energy)

Ted Shifrin

@user1667423: Right, I don't know a comparable interpretation. For one thing, although the Dirichlet energy is not coordinate-system dependent, your Hessian will change when you change coordinates.

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