Mathematics - 2017-08-08 (page 2 of 3)

Ted Shifrin

05:00

Have you watched the MSRI video of people talking about him and his work? I own it.

Faust7

because i deifned it to be?

its a system of equation

Ted Shifrin

But you don't get to do that, @Faust.

Faust7

i can define x-y = 3 cant i?

Eric Silva

Speaking of wanting to meet people I watched that Bryant lecture on finsler geometry that's on YouTube and WAO that was interesting

@Ted actually I have watched that

Ted Shifrin

That plane likely has nothing to do with the surface.

Faust7

05:00

there is an infinite line of solutions that solve that

Ted Shifrin

There's no reason you should get the line of intersection being on the surface, no.

I told you two correct ways to do the problem. :P

Faust7

it should if i choose the original points on the surface and use the other system of equations to force the two non linear equations to act like linear equations

Ted Shifrin

Eric, you should definitely go to a conference where Bryant is a main speaker.

Eric Silva

I would love to meet him he seems cool

Faust7

hmm

Eric Silva

05:02

It'd be awesome to learn stuff from him

Ted Shifrin

Add him to your Stanford list, but I already told you that ages ago :P

Daminark

Ted Shifrin

No, Demonark. The cross product is orthogonal to the plane of $a$ and $b$, but $a$ and $b$ aren't assumed orthogonal. But you're close.

Eric Silva

Ya no he's been on my list for a while

You mean the parallelogram @Daminark

Ted Shifrin

Note that you're dotting $a\times b$ with $a\times b$, also, Demonark.

Daminark

05:05

Oh right whoops, okay that should pick up a factor of $\sin(\theta)$

Ted Shifrin

Huh?

Daminark

So then $|a\times b| = |a| \cdot |b| \sin(\theta)$ where $\theta$ is the angle between them

Ted Shifrin

Where did that come from?

Daminark

You take a parallelogram in the plane given by $a$ and $b$

Its area is $|a|\cdot |b| \sin(\theta)$

Ted Shifrin

So why is that the magnitude of the cross product?

Daminark

05:07

So the volume of the parallelepiped should be that times the height, which $|a\times b|$ because that's orthogonal to the plane

Ted Shifrin

OK, go on.

Daminark

Faust7

how this any diffrent then the case $ z= (x^2 -y^2 ) $ to $z= (x-y) (x+y)$ then set $x-y =d $ then we have $0=d (x+y)-z$ and $x-y -d=0 $ again these are both planes and there intersection should be a line, now if we choose $x_0 , y_0 , z_0 $ to be on the surface then the equation of the line should be on the surface no?

Ted Shifrin

OK, except for mixing letters.

Daminark

Whoops

Yeah

Eric Silva

05:09

I feel like this stuff should be in a good linear algebra class

Like intro linear algebra

Maybe called matrix theory or something, I took something like that in hs

Ted Shifrin

Most linear algebra classes assume you've done it in multivariable calc and don't do it, Eric.

Eric Silva

My matrix theory class actually did this stuff

Ted Shifrin

I have no idea if it works or not without working it out, @Faust. That's definitely not the approach I would take.

Daminark

When Laci did it with us, geometry only came in occasionally

Ted Shifrin

grumbles loudly

EmeraldRoar

05:11

hi i am new on this SE site

Eric Silva

And my multivariable education was me reading a calc book and doing every exercise cover to cover basically

Ted Shifrin

grumbles more :P

hi @EmeraldRoar

EmeraldRoar

i luv multivariable calculus btw

;D

Ted Shifrin

Some might say I do too.

Faust7

well i know it works in that case for the orgin cause i can draw the two planes for the origin and can see there line of intersection is exactly a straight line on the surface

Eric Silva

05:12

It was actually a horrible way to learn lol but I didn't have any guidance then

Ted Shifrin

Yeah, @Faust, it works for the origin.

You didn't have me yelling at you the way DogAteMy and Balarka did, Eric.

Daminark

Like, we'd learn invariant subspaces and an example would be rotation on an axis, for example

Eric Silva

I think I ended up p decent at doing calculus tho

Faust7

im going to go up and look over what you told me to do then

Ted Shifrin

You're still very unusual, Eric.

EmeraldRoar

05:13

i had to take a linear algebra intro class before multi-variable

Ted Shifrin

@Faust: I will listen to you if you show me how it all works out carefully, but I have no reason to believe it will.

Eric Silva

Yeah no I'm a weirdo

Ted Shifrin

@EmeraldRoar: Then you should have been set up to take a more rigorous multivariable course, which actually uses linear algebra.

Daminark

And the linalg book that Soug gave us basically didn't have geometry at all, like it only worked with the determinant via the multilinear stuff and the sum over permutations

Eric Silva

@Daminark did you have to do the linear algebra? No I assume

Faust7

05:14

yeah i dont know how to take the intersection of the two planes though

Daminark

@Eric Huh?

Eric Silva

I assume ha kiddos dont

Daminark

Oh you mean the linear algebra class?

Ted Shifrin

You've learned how to solve systems of linear equations using matrices, Faust ... or using the cross-product of the normal vectors (since we're in $\Bbb R^3$) ...

Daminark

Yeah no we don't do that

Eric Silva

05:15

Cause they aren't offering the version of the abstract algebra course that doesn't a bunch of linear algebra second quarter anymore

Faust7

hmm how do i find the normal vector to the plane in that form?

EmeraldRoar

yea i also made my own 3D transform matrix before i stated to build basic projects in multi variable world

Eric Silva

At least I don't think they are

Faust7

ax+by+cd+e=0

Ted Shifrin

You should have seen that in my lectures, Faust :P

$\vec n\cdot (x-x_0,y-y_0,z-z_0) = 0$, @Faust

Eric Silva

05:16

@Daminark my SO took it and it actually seems pretty bad

Daminark

It depends on who's teaching, some professors didn't reach minimal polynomials

Ted Shifrin

So what are you doing these days, @EmeraldRoar?

Faust7

@TedShifrin yeah i know how to do it when its in the other form i think my brain is just wreckt atm

Daminark

While Boller covered quite a good bit and was a good lecturer

Wait did she take it fall or winter/spring?

Ted Shifrin

Yeah, understood. You need to do reasonable numbers of hours to actually absorb anything, @Faust.

Eric Silva

05:17

Hers did and did a lot of kind of high powered multilinear stuff

I don't remember

EmeraldRoar

@Ted Shifrin now i am studying DE's

philosophically

Eric Silva

I mean Boller is like a fantastic lecturer so

Ted Shifrin

philosophically?

Faust7

i do alright after i sleep on what ive done. brain seems to keep working on it while im asleep or something

Ted Shifrin

OK, @Faust. We can chat a bit more about this tomorrow.

Faust7

05:18

ill type it up proper when i have a chance

EmeraldRoar

yea like theoretically my math teacher says you must phylosophically study any math sub before solving them

Ted Shifrin

rolls a few eyes for fun

Daminark

Hmm, I'm not really sure, like most of my friends took linear algebra first quarter, primarily with Boller

Eric Silva

I like getting my hands dirty with the math personally

Daminark

Some took it with Emerton and he was alright

Ted Shifrin

05:19

sometimes you understand abstract things better by first doing mechanical things with them and then ...

Daminark

A few did with Silberstein and regretted it

Eric Silva

Oh ya it was him

Daminark

He can't teach and ended up finishing the class with like Galois theory or something

Eric Silva

He's Aaron right?

Daminark

Which is cool I guess but also like m8 w0t?

Yeah Aaron

Eric Silva

05:20

My SO like understood nothing but he did like a year long reading with her to sort it out and she learned a lot

He's like leaving academia

He's a very very nice dude

EmeraldRoar

@Ted Shifrin yea i mean i am solving them while studying philosophically but i am not solving any challenging questions yet

Daminark

Yeah he's really nice

Ted Shifrin

I still remind you that hardly anyone is taught how to teach or mentored in grad school (other than for research).

Daminark

I think he's going to Romania to sell copper pots at some point in his life

Ted Shifrin

LOL, ok, @EmeraldRoar.

Daminark

05:22

But yeah his main problem was severe disorganization and going on crazy tangents

EmeraldRoar

@Ted Shifrin i am a noob (still a undergrad) xD

Ted Shifrin

@EmeraldRoar: That last sentence was for Demonark and Eric, not you.

Daminark

To the point where he lost track of the actual flow of the lecture

Ted Shifrin

Demonark, I had more than a few colleagues at UGA who suffered from those same things all the way to retirement.

Eric Silva

He sells copperware now

Except I heard he got into a feud with his Turkish supplier

Ted Shifrin

05:23

ROFL

hope he doesn't end up in the canals

Daminark

@Ted yikes

EmeraldRoar

@Ted Shifrin i am from a 3rd world country tho(india) so not many people here appreciate math.

Ted Shifrin

Not that many in the US do, for sure.

Semiclassical

@TedShifrin Feels like the blind leading the blind at times.

Ted Shifrin

I spent hundreds of hours mentoring grad students and postdocs in teaching, Semiclassic.

Eric Silva

05:25

@Ted I think at Duke they have a mandatory teaching math course thing

I wonder if it's effective tho

Ted Shifrin

Yeah, Eric, it takes more than one course. Most schools have something like that.

EmeraldRoar

@Ted Shifrin i am a self learner can you be my mentor? :3

Eric Silva

Ah I see

Ted Shifrin

I skipped the one at Berkeley because I already had a lot of teaching experience, but Berkeley had it, MIT had it, UGA had it.

Eric Silva

IDT we do but I'm honestly not sure

Ted Shifrin

05:26

I was talking about teaching, @EmeraldRoar :P

Eric Silva

Lecturing is hard enough and that's only one part

Learning to teach is gonna be rough

Daminark

Figuring out what to assign as homework is easy, just follow the Soug recipe

:P

Eric Silva

No pls

Ted Shifrin

Well, it helps to think a bit about what makes certain teachers effective and others not so, ERic.

Daminark

I mean tbh in hindsight I got a good bit from those psets, idk like maybe the choice of Sally was iffy, but linear algebra was quality, and Rudin was as well

Eric Silva

05:28

Yeah that's def true

@Daminark I mean rudin has a big payoff I think

Ted Shifrin

And you have to remember that some of your students may be as motivated and dedicated and talented as you, but most aren't.

Eric Silva

Yeah when I first started grading I would put a bunch of interesting comments but I think nobody read them

So sad

Ted Shifrin

I know some of my students read my comments when I graded homework, but not all.

EmeraldRoar

@Ted Shifrin can i show you my math projects they are basic linear algebra and multivariate projects? you can judge honestly ;D they are on desmos

Eric Silva

Tbf I went overboard at the beginning

Grading 10 psets took like ten hours because I was putting so much into each

It was unsustainable

Ted Shifrin

05:31

@EmeraldRoar: Not at this moment, but another time.

EmeraldRoar

k

Eric Silva

I had so many bad graders that I wanted to be a really good one but I overcompensated like waaay too much

Ok I think I'm done with Bryant for tonight

It's getting late for calculations

Ted Shifrin

I'm playing bridge tomorrow night, so find me earlier in the day.

Daminark

I graded a bit for Laci last summer, and I was moderate, though I was in a group that was mostly doing it by ear

EmeraldRoar

i am having a hard time understanding tensors.

Ted Shifrin

05:34

why do you want to learn tensors?

Eric Silva

@Daminark I did it to get paid and survive

I didn't get paid for seven months

Daminark

Wait what?

EmeraldRoar

i thought it might come handy in CS

Eric Silva

The math Dept is a mess

Daminark

Ah, I was paid by CS

Ted Shifrin

05:35

I doubt it, @EmeraldRoar. CS uses hardly any math, as it turns out.

Daminark

I think

Eric Silva

They like lost my documentation and I will not through hell to get my paycheck

Daminark

I was put under the "theory budget" which suggests compsci

Eric Silva

I didn't get it till today

EmeraldRoar

not even in graphics?

Ted Shifrin

05:36

global — as opposed to local — incompetence, Eric :P

Daminark

@TedShifrin I mean, the practice of programming isn't terribly mathematical, but I see that more as software engineering, computer science has some, especially theoretical stuff

Ted Shifrin

graphics uses some basic linear algebra, projective geometry ideas, but not much

Eric Silva

The local situation was very competent in reality

The HR person who worked through this with me was a lovely person, they really saved me

EmeraldRoar

what about all that machine learning stuff?

Eric Silva

I could like barely afford food

Semiclassical

05:37

machine learning is mostly linear algebra, I think

just with a lot of feedback mechanisms included

Eric Silva

Iirc machine learning uses quite a bit of linear algebra and stats

Ted Shifrin

yeah, stats is becoming a big deal in large data stuff

EmeraldRoar

so do DE's come handy in CS?

Daminark

@Semiclassical ears perk up

Ted Shifrin

I don't know enough CS to help on these questions.

Semiclassical

05:39

lol, don't quote me on that

Eric Silva

My guess is no but idk anything about anything

EmeraldRoar

then why did i learn about all the numerical approximations in my class?

Semiclassical

some discussion about the mathematical background needed for machine learning here: datascience.ibm.com/blog/the-mathematics-of-machine-learning

Ted Shifrin

People use computers to solve engineering/science/math problems involving linear algebra, DEs, etc. But those aren't so much the computer science folks.

EmeraldRoar

@Semiclassical thanks for the link

Eric Silva

05:44

I think my SO does some data science-y machine learning sort of things to study cosmology and she uses like Fourier analysis and linear algebra and stats and stuff like all the time

EmeraldRoar

@TedShifrin i aspire to make simulations software's

Daminark

@Eric I thought you said cohomology for a sec and I was like uh...

Eric Silva

No she studies something something weak lensing something something

Ted Shifrin

It can't hurt to know more math stuff than everyone else, @EmeraldRoar.

Eric Silva

Idk anything about science

Semiclassical

05:46

well, with cosmology a big deal is ... oh, which plot is it

Eric Silva

She plots stuff a lot

Semiclassical

this one

goo.gl/images/ibcUH5

Eric Silva

She's always staring at a bunch of plots that look exactly the same

EmeraldRoar

@TedShifrin yea and besides making math simulations is super joyful to me, i just love it

Semiclassical

the multipole moments there are like Fourier coefficients (spherical harmonics instead of trig functions)

Ted Shifrin

05:48

I was very pleased to figure out exactly the mathematics that Mathematica uses to draw 3D pictures, @EmeraldRoar. I wrote a whole section on computer graphics and that for my linear algebra book.

Eric Silva

@Semi I've heard all the words that are in this plot thing but I know what 0 of them mean

Semiclassical

lol

"like Fourier coefficients" is a pretty simple gloss on it.

Eric Silva

Oh this explains why she was asking me to explain spherical harmonics to her

Semiclassical

lol

but there's a reason I do condensed matter and not cosmology

Eric Silva

What is the reason

Semiclassical

05:50

i guess i find systems that one can make in a lab more interesting than the universe

Eric Silva

Also what is condensed matter

EmeraldRoar

@TedShifrin ikr it feels like the most joyful experience in the whole world, i remember when i was learning linear algebra i figured out how to make 3D transforms by my self and also made by own 3D plot algorithm

Eric Silva

ELI don't know the first thing about physics

Ted Shifrin

The 3D-plotting that Mathematica does is some 3D projective geometry, actually. It's cool.

Daminark

"A physical theory is one satisfying the Newton axioms, surprisingly not containing a long exact sequence"

Eric Silva

05:51

Projective geometry is dope

EmeraldRoar

i used the same technique but its just ironic like how similar ideas can come to all the people

Semiclassical

@eric the opening paragraphs of the Wiki article describe it pretty well: en.wikipedia.org/wiki/Condensed_matter_physics

Eric Silva

Lmao the first line is like "condensed matter physics is the physics of condensed matter"

Ted Shifrin

That totally sucks.

Eric Silva

Classic physicist definition

Ted Shifrin

05:54

Differential geometry is the math of geometric differentials. ???

Eric Silva

Tensors are things that transform like tensors

3

Ted Shifrin

LOL

Semiclassical

You're omitting a big part of that first line: "condensed phases of matter"

the phase aspect of it is a big deal.

Eric Silva

Bob ward in his GR book says that christoffel symbols were tensors and it made me like actually made me angry

Ted Shifrin

like phase transitions?

yikes, that's very wrong, Eric.

Eric Silva

05:55

@Semi o true

Ted Shifrin

sorta like a book I reviewed that had all sorts of diff geo actually wrong

why didn't they get a real geometer to criticize it before it was published?

Semiclassical

right. except that when one thinks of phase transition one usually thinks of solid ice-liquid water-gas vapor

Daminark

@TedShifrin Well you're a complex geometer...

Ted Shifrin

Demonark, your month isn't up yet.

Daminark

Okay lol enough for now :P

Semiclassical

05:57

in modern condensed matter physics, by contrast, you have a lot more possibilities.

Daminark

Ah whoops

Eric Silva

I think it mightve been that he like made a tensor out of christoffel symbols and then just called it the christoffel symbols

In any event it was like offensive to my sensibilities

Ted Shifrin

Um, you can't make a tensor out of the christoffel symbols ...

unless you're talking the curvature tensor or something

Eric Silva

But apparently ward kind of likes being idiosyncratic or something

Semiclassical

for instance, the phase transition from the normal state of a certain system into its superconducting phase.

Ted Shifrin

05:57

that's terrible to be so inconsistent with universal terminology

Eric Silva

@Ted I meant he like used them to define one

EmeraldRoar

here i have done something similar to how 3D Mathematica working just that in my algorithm projections can be changed in real time desmos.com/calculator/rjybyp9rlc

Eric Silva

But I was skimming his book like 6months ago so idr

Semiclassical

idiosyncratic in this case meaning wrong

Eric Silva

So he could've just been calling christoffel symbols a tensor

Which is like so horrible

Cause like, you can just check that they're not without much work even

Ted Shifrin

05:59

@EmeraldRoar: Can you easily specify where the eye is and what the viewing plane is?

Eric Silva

Whatever it was tho made me mad

EmeraldRoar

@TedShifrin you mean like with x,y,z cords?

Ted Shifrin

But think how mad Demonark makes you, Eric.

Right, I guess, @EmeraldRoar.

EmeraldRoar

@TedShifrin you can change the sliders t1,t2,t3

Eric Silva

He does that too but in different ways

Daminark

06:01

How could I ever make someone mad?!? :O

Eric Silva

Omg yeah no I was right he just called christoffel symbols a tensor

Ted Shifrin

I see, cool, @Emerald.

Eric Silva

I'm mad again

Ted Shifrin

He ought to understand how wrong that is.

Eric Silva

I think his definition of tensor includes things which don't transform like tensors.

Ted Shifrin

06:03

Demonark: You both make people angry and drive people mad (crazy).

Then he should have his book incinerated (à la Fahrenheit 451)

Eric Silva

And his book is usually used for the grad gr class here

Ted Shifrin

covers eyes and ears

Daminark

Welp

Eric Silva

Bryant's derivation of the EL equation for willmore is so much better than what I did when Andre gave me that as an assignment man

Working through that paper has actually been really great

Ted Shifrin

Oh, using structure equations for variational stuff is very cool. Have you worked out just variation of arclength using that?

Eric Silva

06:09

Yes, I did that in chat at some point actually

Ted Shifrin

Oh, I made you do it already. Cool. :)

Eric Silva

I worked out variation of area on my own right after that too

Ted Shifrin

Glad I'm semi-brainwashing you :P

Eric Silva

@Daminark did you hear me almost give that abstract nonsense definitions of a Riemannian metric in lecture

I had to stop myself from pulling a you

Ted Shifrin

This reminds me of the Pushme-Pullyou from Dr. Doolittle.

Daminark

06:11

Lolol, yeah, though when I defined the Riemannian metric I just did it out of vector fields I think

Ted Shifrin

You didn't define it as a section of $\text{Sym}^2(T^*M)$?

Eric Silva

This is what I stopped myself from saying

Daminark

I didn't even know that definition lmao

Eric Silva

They don't really know what the tensor bundles are

Ted Shifrin

If they're at that fanciness level, my notes aren't a reasonable choice.

But they don't know diddly about multivariable calc, so toned down is better.

Daminark

06:14

The only bundle anyone in class knows is the tangent bundle, unless they did Neves' class and learned the normal one as well

Eric Silva

You guys didn't talk about the cotangent bundle in that class??

Ted Shifrin

You don't need all this fancy s**t to do curves and surfaces. And trying to do all the abstract stuff without having seen concrete examples/motivation sucks.

Nope, Eric. I never did it when I taught it.

Daminark

Ronno mentioned it for like, 2 seconds I think

Eric Silva

I thought they talked about forms tho

Ted Shifrin

You can do that without doing it all bundly.

Daminark

06:15

He just did all of GP chapter 4 in like, a week

Eric Silva

I mean yeah you don't need any of these words to do tjings

Ted Shifrin

I save the bundly everything for the graduate course.

Eric Silva

This is fair

My smooth manifolds Prof did bundle things

Ted Shifrin

And then I did more with bundles on grassmannians and stuff that algebraic/complex geometers care more about than Riemannian geometers.

Daminark

And the pset that week didn't really deal with any of the technical stuff, it was a few problems on degree/forms and a few on getting Stokes' in the plane and in $\mathbb{R}^3$

Eric Silva

06:16

But I had done the classical diff geo out of do Carmo by that point

And read G& P on my own

Ted Shifrin

DoCarmo doesn't do any bundles.

In either book.

I mean other than tangent bundle and associated stuff.

Eric Silva

@Ted I think the only time grassmanians have come up for me in the stuff I've learned with neves has been varifolds

Daminark

@Eric are there classes that do diffgeo aside from Riemannian?

Eric Silva

No I just read a lot of books in my spare time

Ted Shifrin

You should read my proof of Gauss-Bonnet, Eric. It uses Schubert cycles on Grassmannians and all sorts of moving frames on Grassmannians and universal bundles :)

Eric Silva

06:18

I did bundle stuff in a reading course with Benson tho

Oh wild cool

Ted Shifrin

It's the end of the Riemannian geometry notes I sent you.

Eric Silva

Right those are open on my desktop RN I think

Ted Shifrin

Well, I'm calling it a night. Bye, all.

Eric Silva

If you recall @Daminark I did diff stuff with Farb

Night Ted

Daminark

Hmm, did you just go to him and be like "Yo how bout them reading courses tho?" or what?

See you @Ted!

Ted Shifrin

06:20

Night, Demonark.

Eric Silva

Literally I was just like "I wanna learn geometry and topology what do"

And he was like ok let's do a course

And I'll give you a solid foundation

Daminark

Awesome

Eric Silva

Basically we did the contents of the first two quarters grad top stuff plus a bit more stuff

O shit it's late I should go to bed

I'm out

Daminark

See you!

Daminark

06:57

Hey @Tobias!

Tobias Kildetoft

@Daminark Hi

Daminark

How's it going?

Tobias Kildetoft

Good

SteamyRoot

07:50

You don't happen to be at Groups St Andrews? :O

Tobias Kildetoft

@SteamyRoot Me?

SteamyRoot

Yup

Tobias Kildetoft

No, not sure what that even is

SteamyRoot

groupsstandrews.org/2017

I'm actually not there myself (though I wish I'd been)

Tobias Kildetoft

@SteamyRoot Hmm, not that many of the talks are particularly interesting to me. It would mainly be the one by Donna Testerman, but it looks fairly similar to one she gave last year in York anyway (though probably updated somewhat now with new results).

Tobi

08:15

Can this vector algorithm be simplified?:

var dn = ob1.coords.copy().subtract(ob2.coords).unit(),
    dnn = dn.copy().multiply(-1),
    n1d = dnn.multiply(2 * ob1.velocity.copy().dot(dnn)),
    n2 = dn.multiply(ob2.velocity.copy().dot(dn)),
    v2 = n1d.copy().multiply(ob1.mass).add(n2.copy().multiply(ob2.mass - ob1.mass)).divide(ob2.mass + ob1.mass);

ob1.velocity.add(v2).add(n2).subtract(n1d);
ob2.velocity.add(v2).subtract(n2);

Let me put it in mathematical notation

user8469759

09:07

hi guys

does any of you know what book does Schuller use in his video lectures?

Self adjoint and essentially self-adjoint operators - Lec 07 - Frederic Schuller

►

(just an example)

I'm interested in the very first part

of these lectures (i.e. up to the spectral theorem)

Tobi

10:20

Does this always give you the component of a vector in the direction of a unit vector?:

Astyx

In an orthonormal basis, yes

It depends what you mean by component

Balarka Sen

Probably just projection

Tobi

i mean like the resultant vector

in direction of unit

Astyx

Because given any basis of unit vectors, you do not have $x = \sum_i\langle x, n_i\rangle n_i$

(in general)

Tobi

cant see latex

Astyx

10:25

See the link in the chat description in the top right

Tobi

cant be bothered to set up

:(

Astyx

Too bad

Tobi

lol i manually inserted the render javascript into my url

that works

Rumplestillskin

Does anyone have a reference that works out how to obtain a line element in Cartesian coordinates starting from a line element in polar spherical coordinates?

Vincenzo Oliva

11:24

What's the point in saying the Well Ordering Principle is equivalent to the Principle of Mathematical Induction, when the latter is a direct consequence of how $\Bbb N$ is defined, i.e. as the intersection of all the inductive sets? I mean, if the PMI is a priori true, isn't it inevitable that the WOP implies it?

Tobias Kildetoft

@VincenzoOliva It all depends on context

Also, I find it a shame that these are always proposed as equivalent, when well-ordering of a set does not actually allow one to do the "usual" induction

It just happens to work on the natural numbers

Vincenzo Oliva

@TobiasKildetoft That's precisely what bothers me. It's somewhat vacuous that the WOP implies the PMI because the latter has to be true in the naturals, and whenever you get in another set, the implication is no longer true and you have to settle for transfinite induction

Tobias Kildetoft

@VincenzoOliva I am not sure how transfinite induction is relevant here

You just need "strong" induction

Vincenzo Oliva

If a set is well ordered, then transfinite induction is valid in it

In $\Bbb N$, it coincides with PMI

Tobias Kildetoft

I suppose we mean the same kind of induction then

Vincenzo Oliva

11:35

I see

Tobias Kildetoft

I think the main place the PMI makes sense is in the context of Peano axioms, where well-ordering does not make sense

in ZF it is more of a theorem (or practically a definition) instead

Vincenzo Oliva

11:47

@TobiasKildetoft Yes, my professor did illustrate it in that context. However, in the chapter on orderings, this equivalence is shown. Freiwald goes so far as to say "it is the well-ordering property that lets us do mathematical induction in $\Bbb N$". Do you think it is a plain mistake to put it like that?

Tobias Kildetoft

@VincenzoOliva Not entirely. I mean it is mostly the fact that the naturals are well-ordered that allows induction. It just also needs the fact that there is a unique element with no immediate predecessor

Note that an "intermediate" version of induction requires a base case for each such element, and this version also works in any well-ordered set.

Lucas Henrique

hey.

Vincenzo Oliva

Indeed. I think one ought to stress that. In a way, this seems to me another argument to undermine the "equivalence"

Tobias Kildetoft

@VincenzoOliva Indeed. It is already a bit silly to speak of things being equivalent when really they are just true statements. It is like claiming that PMI is equivalent to $1$ being a natural number

mathiu_lady

Why [\mathbb{Q}(\omega, 2^(1/3)): \mathbb Q]= 6 ? From Galois theory it will be equal to the degree of the minimal polynomial containing \omega and 2^(1/3).

We can take a polynomial as (x^2+x+1)(x^3-2)

Vincenzo Oliva

11:53

@TobiasKildetoft Precisely. Cheers

Tobias Kildetoft

@VincenzoOliva I actually wrote a blog post here on the MSE blog some time ago about induction and how it has nothing to do with natural numbers. But unfortunately, the blog is no more

Vincenzo Oliva

Damn, it's a pity, I'm sorry to hear that

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