Maybe Linear Algebra @TedShifrin ?

Well, one tends to teach parts in context. I have done Euler-Lagrange in undergraduate differential geometry, and I think I did some of this in the applied math class I taught 30 years ago.

Try checking MS Word @Dragon

Jasper, this is Griffiths's book.

Sorry typing error :/

@Mahmoud: I'm fond of linear algebra. But if you're going to do it, you should do a book that has proofs and proof exercises, not just mindless matrix calculations.

Aha @Jasper. I win :P

Any better suggestions @TedShifrin ?

@Dragon: I don't know what you mean by where they got the formula. Someone designed the formula for the various sizes of paper.

No, Jasper. Griffiths's book? It's from the 80s.

Mahmoud: I'm also more fond of geometry than most people are. You might also start by reading Hilbert and Cohn Vossen: "Geometry and the Imagination." It might be a little too tough; I dunno.

Yes, it is, Jasper.

I got it when it first was published.

The page I linked you to says 1983.

Does this book help building Geometric explanations for Algebraic relations as came to my mind ? @TedShifrin

Ahem, @Jasper :D

I have no idea what you're talking about, Mahmoud.

What is that book about ? @TedShifrin

"Geometry and the Imagination."

Lots of different topics, all very interesting. The Hilbert is the famous David Hilbert.

And Cohn-Vossen was a famous differential geometer, too.

@TedShifrin sorry for my bad explanation. i'm given the width of the A series sized paper. i was asked to find a general formula which would allow me to calculate the width of any given A series paper size. i have found the formula through google. i don't understand how they've woked out the formula

Okay, I guess that is a good starting point @TedShifrin Thank you :D

@Dragon: I don't know what you're looking for. Most likely, someone made up the formula first before the various sizes of paper were produced. But this is certainly not a question for MSE.

@TedShifrin ok, i apologize, thank you for your time

No need to apologize!

hey @TedShifrin

would you like to check my solution ? math.stackexchange.com/questions/1980825/…

I took your recommendation and wrote everything in terms of functions

@BalarkaSen @TedShifrin well I figured it out after all. It was a really good question I think, had me working.. (either that or I suck, both are valid options)

Good to hear!

hi chat

@Semiclassical yo

yo

are you still doing physics lately?

yup

neat

my students are starting to do circuit stuff now

which means the stuff they do in lab actually gets to be kind of fun

Hi @Semiclassical

mhm

Your solution was spectacular !

glad you liked it, heh

It took my class mates two A4 paper of computations to figure it out.

And they didn't actually understand it :P

I still like the geometry version best, heh

let me upload the picture of that

You defined $u$ and $v$ based on the equations $y=x$ and $x=-y$ Right @Semiclassical ?

here we go @Mahmoud

the black is the equation $x(x+y-1)+y(y-1)=0$, and beneath is a contour plot of $x+y$

from that it's pretty obvious that both the contour lines and the black curve are symmetric about the line $x=y$

and from there it's easy enough to conclude that the way to min-max $x+y$ is to have $x=y$

Thank you very much, it's very kind from you to give me time.

no problem

i just whipped that up in mathematica now

anyways. to get from here to the algebraic solution, note that if you rotated the coordinate axes by 45 degrees, you'd just have a standard ellipse

and that's what picking $(u,v)=(x+y,x-y)$ effectively does.

@BalarkaSen halp

that's why I was confident that those coordinates would lead to a simple solution, since they definitely make the geometry more obvious

Hi @Danu

so it's not that I pulled a solution out of nowhere. I'd have used pretty much the same logic had your equation been of the form $f(x,y)=0$ with $f(x,y)=f(y,x)$ instead

You made me realize that Geometry is a good way to solve Algebraic expressions.

I've got a pretty simple question: So if you consider some element of $\Gamma(T^kX\otimes E)$ for some bundle $E$, then Huybrechts says the following, which makes a lot of sense: Trying to define $d(\alpha\otimes s)=d(\alpha)\otimes s$ doesn't work because $\alpha\otimes s=\alpha g\otimes g^{-1}s$ for any smooth $g:X\to \Bbb C^*$ and then $d$ would give something else.

It really is. Or at least, a good way to motivate an algebraic approach.

But then he continues doing essentially just that, but for $\bar\partial$ instead of $d$. How does that avoid the exact same problem??

Thank you @Semiclassical

Why is $\bar\partial g$ always zero?

glad you liked it :)

@Danu I sincerely doubt he does precisely that.

@MikeMiller He has $\bar\partial_E\alpha:=\sum \bar\partial(\alpha_i)\otimes s_i$.

Am I missing something stupid?

I will focus on applying that in my Exam tomorrow @Semiclassical

Good luck!

All tough it won't be the same problem

I'm skeptical. He needs a derivative on the second factor.

Neither I'll have access to Mathematica @Semiclassical

This is consistent with other sources I've checked.

There is no derivative on $s_i$.

nope. though the mathematica was just a tool to get a nice graphic. I was able to visualize it fairly simply

at least to the extent of it being a tilted ellipse

@Semiclassical I will have four problems of the same difficulty each on 5 points which makes for a total of 20/20

What kind of problems do you expect?

The teacher said that if digest the Course and work out all 24 problems we did, then do a bit of searching.

We can have a good mark.

also e.g. page 70 of Griffiths & Harris has the same thing.

that's a bit vague

Everybody keeps saying it doesn't depend on frame (because transition functions of $E$ are holomorphic), but not commenting on the other thing.

I asked : How much exactly ?

Teacher : six or seven, or maybe 11 if you lucky enough !

Ew.

I don't know should I cry or laugh ?

yeah, that's

pretty brutal :/

hi @ted

Hi @TedShifrin. It's time you smack me again!

@Danu: Because $s$ is a holomorphic section of $E$.

@Mahmoud: I'm also a huge fan of using vector algebra and vector geometry to prove neat Euclidean geometry things. Lots of such exercises in my linear algebra text :P

@TedShifrin Ah, wait, $s$ must be holo so if we multiply by some function $g$ it must also be holo?

Right.

Okay, that's as simple as I'd hoped it would be.

Thanks.

Sorry for not pointing that out @MikeMiller

So the transition functions disappear when you apply $\bar\partial$.

@Danu That's crucial. The s-derivative does not vanish if it's not holomorphic.

@Semiclassical What do you recommend me to do ?

@TedShifrin That's clear.

I meant the problem with $d$ not being there now.

But yeah, it's the same reason, because we need $g^{-1}s$ to still be holomorphic

Not sure, tbh. That's really really open-ended

I prefer to look at it even for general sections but point out that $\bar\partial$ is well-defined because $\bar\partial \phi_{ij} = 0$.

maybe take the problems you've done and try to figure out ways to make them harder in interesting ways? idk

@TedShifrin Now you see why I was confused!

@TedShifrin That says it's independent of frame. Fine. But I'm talking about this:

This $g g^{-1}$ business

Any tips on not freaking out in the 2 hours exam ?

No, it says you get a well-defined section of $\Lambda^{k+1}T^*X\otimes E$ when you apply $\bar\partial$.

probably the thing to remember is that your classmates will be in the same situation

@TedShifrin Sorry, what is $\phi_{ij}$ supposed to be then? The $g$ from my notation?

I assumed you mean a change of frame

No, transition functions if you work in two trivializations.

I usually give up after trying one single Idea and blame myself for not solving it @Semiclassical

I can't understand what Huybrechts is doing. How could you not try to differentiate $s$?

Right. I was never worried about that issue. I didn't understand why the argument that with $g g^{-1}$ doesn't apply.

I mean, what he has is right, but it's stooopid.

@TedShifrin Wait, whaa?

My mind gets blocked and I stop reasoning.

Griffiths & Harris have exactly the same thing

Huh? Where?

well, don't do that :/

Page 70.

How ? that's my question :/

Oh, they take a local holomorphic frame.

I have a hard time giving advice on this, though, because by and large I never had test-anxiety

"Set $\bar\partial\sigma=\sum \bar\partial \omega_i\otimes s_i$.

It's not an arbitrary section. So it's what I said when I descended into the room.

But Huybrechts does too, @TedShifrin.

I have writing-anxiety to my wits end, but tests always were fairly straightforward for me.

@TedShifrin Yeah, okay.

so I don't have good advice for that

What about you @TedShifrin ?

Thanks @Semiclassical :)

What would you do if the trivialization wasn't holomorphic?

I think others here could give some advice, though (nudge nudge)

Then you need the product rule, @Danu.

Just add $\alpha\otimes \bar\partial s$ terms?

Yup.

With a sign for moving $d$ across a differential form, as usual.