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05:00
Have you watched the MSRI video of people talking about him and his work? I own it.
because i deifned it to be?
its a system of equation
But you don't get to do that, @Faust.
i can define x-y = 3 cant i?
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
That plane likely has nothing to do with the surface.
05:00
there is an infinite line of solutions that solve that
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
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
Eric, you should definitely go to a conference where Bryant is a main speaker.
I would love to meet him he seems cool
hmm
05:02
It'd be awesome to learn stuff from him
Add him to your Stanford list, but I already told you that ages ago :P
Okay wait so I'm trying to think this through, let me see if I have the right idea. So, we want to find the area of the parallelepiped generated by $a$, $b$, and $a\times b$. This should be $\|a\times b\|\cdot \|a\|\cdot \|b\|$ because of orthogonality
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.
Ya no he's been on my list for a while
You mean the parallelogram @Daminark
Note that you're dotting $a\times b$ with $a\times b$, also, Demonark.
05:05
Oh right whoops, okay that should pick up a factor of $\sin(\theta)$
So then $|a\times b| = |a| \cdot |b| \sin(\theta)$ where $\theta$ is the angle between them
Where did that come from?
You take a parallelogram in the plane given by $a$ and $b$
Its area is $|a|\cdot |b| \sin(\theta)$
So why is that the magnitude of the cross product?
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
OK, go on.
So then $\det(x\times y, x, y) = |a\times b| \cdot |a| \cdot |b| \sin(\theta)$, but then it's also equal to $|a\times b|^2$ because of what we said above
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?
OK, except for mixing letters.
Whoops
Yeah
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
Most linear algebra classes assume you've done it in multivariable calc and don't do it, Eric.
My matrix theory class actually did this stuff
I have no idea if it works or not without working it out, @Faust. That's definitely not the approach I would take.
When Laci did it with us, geometry only came in occasionally
grumbles loudly
05:11
hi i am new on this SE site
And my multivariable education was me reading a calc book and doing every exercise cover to cover basically
grumbles more :P
hi @EmeraldRoar
i luv multivariable calculus btw
;D
Some might say I do too.
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
05:12
It was actually a horrible way to learn lol but I didn't have any guidance then
Yeah, @Faust, it works for the origin.
You didn't have me yelling at you the way DogAteMy and Balarka did, Eric.
Like, we'd learn invariant subspaces and an example would be rotation on an axis, for example
I think I ended up p decent at doing calculus tho
im going to go up and look over what you told me to do then
You're still very unusual, Eric.
05:13
i had to take a linear algebra intro class before multi-variable
@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.
Yeah no I'm a weirdo
@EmeraldRoar: Then you should have been set up to take a more rigorous multivariable course, which actually uses linear algebra.
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
@Daminark did you have to do the linear algebra? No I assume
05:14
yeah i dont know how to take the intersection of the two planes though
@Eric Huh?
I assume ha kiddos dont
Oh you mean the linear algebra class?
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 $\Bbb R^3$) ...
Yeah no we don't do that
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
hmm how do i find the normal vector to the plane in that form?
yea i also made my own 3D transform matrix before i stated to build basic projects in multi variable world
At least I don't think they are
ax+by+cd+e=0
You should have seen that in my lectures, Faust :P
$\vec n\cdot (x-x_0,y-y_0,z-z_0) = 0$, @Faust
05:16
@Daminark my SO took it and it actually seems pretty bad
It depends on who's teaching, some professors didn't reach minimal polynomials
So what are you doing these days, @EmeraldRoar?
@TedShifrin yeah i know how to do it when its in the other form i think my brain is just wreckt atm
While Boller covered quite a good bit and was a good lecturer
Wait did she take it fall or winter/spring?
Yeah, understood. You need to do reasonable numbers of hours to actually absorb anything, @Faust.
05:17
Hers did and did a lot of kind of high powered multilinear stuff
I don't remember
@Ted Shifrin now i am studying DE's
philosophically
I mean Boller is like a fantastic lecturer so
philosophically?
i do alright after i sleep on what ive done. brain seems to keep working on it while im asleep or something
OK, @Faust. We can chat a bit more about this tomorrow.
05:18
ill type it up proper when i have a chance
yea like theoretically my math teacher says you must phylosophically study any math sub before solving them
rolls a few eyes for fun
Hmm, I'm not really sure, like most of my friends took linear algebra first quarter, primarily with Boller
I like getting my hands dirty with the math personally
Some took it with Emerton and he was alright
05:19
sometimes you understand abstract things better by first doing mechanical things with them and then ...
A few did with Silberstein and regretted it
Oh ya it was him
He can't teach and ended up finishing the class with like Galois theory or something
He's Aaron right?
Which is cool I guess but also like m8 w0t?
Yeah Aaron
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
@Ted Shifrin yea i mean i am solving them while studying philosophically but i am not solving any challenging questions yet
Yeah he's really nice
I still remind you that hardly anyone is taught how to teach or mentored in grad school (other than for research).
I think he's going to Romania to sell copper pots at some point in his life
LOL, ok, @EmeraldRoar.
05:22
But yeah his main problem was severe disorganization and going on crazy tangents
@Ted Shifrin i am a noob (still a undergrad) xD
@EmeraldRoar: That last sentence was for Demonark and Eric, not you.
To the point where he lost track of the actual flow of the lecture
Demonark, I had more than a few colleagues at UGA who suffered from those same things all the way to retirement.
He sells copperware now
Except I heard he got into a feud with his Turkish supplier
05:23
ROFL
hope he doesn't end up in the canals
@Ted yikes
@Ted Shifrin i am from a 3rd world country tho(india) so not many people here appreciate math.
Not that many in the US do, for sure.
@TedShifrin Feels like the blind leading the blind at times.
I spent hundreds of hours mentoring grad students and postdocs in teaching, Semiclassic.
05:25
@Ted I think at Duke they have a mandatory teaching math course thing
I wonder if it's effective tho
Yeah, Eric, it takes more than one course. Most schools have something like that.
@Ted Shifrin i am a self learner can you be my mentor? :3
Ah I see
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.
IDT we do but I'm honestly not sure
05:26
I was talking about teaching, @EmeraldRoar :P
Lecturing is hard enough and that's only one part
Learning to teach is gonna be rough
Figuring out what to assign as homework is easy, just follow the Soug recipe
:P
No pls
Well, it helps to think a bit about what makes certain teachers effective and others not so, ERic.
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
05:28
Yeah that's def true
@Daminark I mean rudin has a big payoff I think
And you have to remember that some of your students may be as motivated and dedicated and talented as you, but most aren't.
Yeah when I first started grading I would put a bunch of interesting comments but I think nobody read them
So sad
I know some of my students read my comments when I graded homework, but not all.
@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
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
05:31
@EmeraldRoar: Not at this moment, but another time.
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
I'm playing bridge tomorrow night, so find me earlier in the day.
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
i am having a hard time understanding tensors.
05:34
why do you want to learn tensors?
@Daminark I did it to get paid and survive
I didn't get paid for seven months
Wait what?
i thought it might come handy in CS
The math Dept is a mess
Ah, I was paid by CS
05:35
I doubt it, @EmeraldRoar. CS uses hardly any math, as it turns out.
I think
They like lost my documentation and I will not through hell to get my paycheck
I was put under the "theory budget" which suggests compsci
I didn't get it till today
not even in graphics?
05:36
global — as opposed to local — incompetence, Eric :P
@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
graphics uses some basic linear algebra, projective geometry ideas, but not much
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
what about all that machine learning stuff?
I could like barely afford food
05:37
machine learning is mostly linear algebra, I think
just with a lot of feedback mechanisms included
Iirc machine learning uses quite a bit of linear algebra and stats
yeah, stats is becoming a big deal in large data stuff
so do DE's come handy in CS?
@Semiclassical ears perk up
I don't know enough CS to help on these questions.
05:39
lol, don't quote me on that
My guess is no but idk anything about anything
then why did i learn about all the numerical approximations in my class?
some discussion about the mathematical background needed for machine learning here: datascience.ibm.com/blog/the-mathematics-of-machine-learning
People use computers to solve engineering/science/math problems involving linear algebra, DEs, etc. But those aren't so much the computer science folks.
@Semiclassical thanks for the link
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
@TedShifrin i aspire to make simulations software's
@Eric I thought you said cohomology for a sec and I was like uh...
No she studies something something weak lensing something something
It can't hurt to know more math stuff than everyone else, @EmeraldRoar.
Idk anything about science
05:46
well, with cosmology a big deal is ... oh, which plot is it
She plots stuff a lot
She's always staring at a bunch of plots that look exactly the same
@TedShifrin yea and besides making math simulations is super joyful to me, i just love it
the multipole moments there are like Fourier coefficients (spherical harmonics instead of trig functions)
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.
@Semi I've heard all the words that are in this plot thing but I know what 0 of them mean
lol
"like Fourier coefficients" is a pretty simple gloss on it.
Oh this explains why she was asking me to explain spherical harmonics to her
lol
but there's a reason I do condensed matter and not cosmology
What is the reason
05:50
i guess i find systems that one can make in a lab more interesting than the universe
Also what is condensed matter
@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
ELI don't know the first thing about physics
The 3D-plotting that Mathematica does is some 3D projective geometry, actually. It's cool.
"A physical theory is one satisfying the Newton axioms, surprisingly not containing a long exact sequence"
05:51
Projective geometry is dope
i used the same technique but its just ironic like how similar ideas can come to all the people
@eric the opening paragraphs of the Wiki article describe it pretty well: en.wikipedia.org/wiki/Condensed_matter_physics
Lmao the first line is like "condensed matter physics is the physics of condensed matter"
That totally sucks.
Classic physicist definition
05:54
Differential geometry is the math of geometric differentials. ???
Tensors are things that transform like tensors
3
You're omitting a big part of that first line: "condensed phases of matter"
the phase aspect of it is a big deal.
Bob ward in his GR book says that christoffel symbols were tensors and it made me like actually made me angry
like phase transitions?
yikes, that's very wrong, Eric.
05:55
@Semi o true
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?
right. except that when one thinks of phase transition one usually thinks of solid ice-liquid water-gas vapor
@TedShifrin Well you're a complex geometer...
Demonark, your month isn't up yet.
Okay lol enough for now :P
05:57
in modern condensed matter physics, by contrast, you have a lot more possibilities.
Ah whoops
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
Um, you can't make a tensor out of the christoffel symbols ...
unless you're talking the curvature tensor or something
But apparently ward kind of likes being idiosyncratic or something
for instance, the phase transition from the normal state of a certain system into its superconducting phase.
05:57
that's terrible to be so inconsistent with universal terminology
@Ted I meant he like used them to define one
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
But I was skimming his book like 6months ago so idr
idiosyncratic in this case meaning wrong
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
05:59
@EmeraldRoar: Can you easily specify where the eye is and what the viewing plane is?
Whatever it was tho made me mad
@TedShifrin you mean like with x,y,z cords?
But think how mad Demonark makes you, Eric.
Right, I guess, @EmeraldRoar.
@TedShifrin you can change the sliders t1,t2,t3
He does that too but in different ways
06:01
How could I ever make someone mad?!? :O
Omg yeah no I was right he just called christoffel symbols a tensor
I see, cool, @Emerald.
I'm mad again
He ought to understand how wrong that is.
I think his definition of tensor includes things which don't transform like tensors.
06:03
Demonark: You both make people angry and drive people mad (crazy).
Then he should have his book incinerated (à la Fahrenheit 451)
And his book is usually used for the grad gr class here
covers eyes and ears
Welp
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
Oh, using structure equations for variational stuff is very cool. Have you worked out just variation of arclength using that?
06:09
Yes, I did that in chat at some point actually
Oh, I made you do it already. Cool. :)
I worked out variation of area on my own right after that too
Glad I'm semi-brainwashing you :P
@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
This reminds me of the Pushme-Pullyou from Dr. Doolittle.
06:11
Lolol, yeah, though when I defined the Riemannian metric I just did it out of vector fields I think
You didn't define it as a section of $\text{Sym}^2(T^*M)$?
This is what I stopped myself from saying
I didn't even know that definition lmao
They don't really know what the tensor bundles are
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.
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
You guys didn't talk about the cotangent bundle in that class??
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.
Ronno mentioned it for like, 2 seconds I think
I thought they talked about forms tho
You can do that without doing it all bundly.
06:15
He just did all of GP chapter 4 in like, a week
I mean yeah you don't need any of these words to do tjings
I save the bundly everything for the graduate course.
This is fair
My smooth manifolds Prof did bundle things
And then I did more with bundles on grassmannians and stuff that algebraic/complex geometers care more about than Riemannian geometers.
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 $\mathbb{R}^3$
06:16
But I had done the classical diff geo out of do Carmo by that point
And read G& P on my own
DoCarmo doesn't do any bundles.
In either book.
I mean other than tangent bundle and associated stuff.
@Ted I think the only time grassmanians have come up for me in the stuff I've learned with neves has been varifolds
@Eric are there classes that do diffgeo aside from Riemannian?
No I just read a lot of books in my spare time
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 :)
06:18
I did bundle stuff in a reading course with Benson tho
Oh wild cool
It's the end of the Riemannian geometry notes I sent you.
Right those are open on my desktop RN I think
Well, I'm calling it a night. Bye, all.
If you recall @Daminark I did diff stuff with Farb
Night Ted
Hmm, did you just go to him and be like "Yo how bout them reading courses tho?" or what?
See you @Ted!
06:20
Night, Demonark.
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
Awesome
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
See you!
06:57
Hey @Tobias!
@Daminark Hi
How's it going?
07:50
You don't happen to be at Groups St Andrews? :O
@SteamyRoot Me?
No, not sure what that even is
I'm actually not there myself (though I wish I'd been)
@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).
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
09:07
hi guys
does any of you know what book does Schuller use in his video lectures?
(just an example)
I'm interested in the very first part
of these lectures (i.e. up to the spectral theorem)
 
1 hour later…
10:20
Does this always give you the component of a vector in the direction of a unit vector?:
In an orthonormal basis, yes
It depends what you mean by component
Probably just projection
i mean like the resultant vector
in direction of unit
Because given any basis of unit vectors, you do not have $x = \sum_i\langle x, n_i\rangle n_i$
(in general)
cant see latex
10:25
See the link in the chat description in the top right
cant be bothered to set up
:(
Too bad
lol i manually inserted the render javascript into my url
that works
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?
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?
@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
@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
@VincenzoOliva I am not sure how transfinite induction is relevant here
You just need "strong" induction
If a set is well ordered, then transfinite induction is valid in it
In $\Bbb N$, it coincides with PMI
I suppose we mean the same kind of induction then
11:35
I see
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
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?
@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.
Indeed. I think one ought to stress that. In a way, this seems to me another argument to undermine the "equivalence"
@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
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)
11:53
@TobiasKildetoft Precisely. Cheers
@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
Damn, it's a pity, I'm sorry to hear that

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