I wrote multivariable calculus notes for MIT students when I was an undergrad, but I never wrote things other than seminar lectures as a grad student. Somehow, though, I had decent exposition skills — or so I've believed.
I think it's a shame that new mathematics keeps exponentially growing but the exposition is done so poorly (people want to write quickly and move on to newer projects, etc) that it is nearly impossible to understand for someone outside a narrow field of expertise.
I genuinely believe that it would be untenable to do math in a few years if people don't step back and try to communicate more than proving newer stuff
Well, it's sorta like promotions value research over teaching at almost every institution of higher learning. Similarly, journals prefer short articles to ones that have exposition that less expert readers can learn from. I fought to keep a reasonable amount of exposition in my published thesis.
couldnt agree more. its a sad state. also it seems like those who dont have access to profs that are "in the know" have no hope of learning by themselves as there is too much info that is not explicitly stated but implicitly "understood" by the experts.
anyway thats my 2 cents. total trigger topic for me ha. im off! have a nice weekend everyone
I actually typed a lot of my multivariable math book as I went. Of course, revising is a lot easier in the computer age than it was in the old typewriter days.
Take a monic integer polynomial. Now take the polynomial whose roots (with multiplicity) are squares of the roots of the polynomial we started with (all over C). This polynomial has integer coefficients again. is there a direct way of seeing this?
The only argument I have is that the polynomial is the characteristic polynomial of some integer matrix and the polynomial with the roots squared is the characteristic polynomial of the square of that matrix.
I mean, if $\alpha$ is a root of the original polynomial, $\Bbb Z[\alpha]$ is integral over $\Bbb Z$, so in particular $\alpha^2$ is also integral over $\Bbb Z$, and the Galois conjugates are $\beta^2$ where $\beta$ is a conjugate of $\alpha$.
Because Galois automorphism are field automorphisms, it commutes with multiplication
there was no further punchline, just whether you know an explicit description of the intersection form (where by explicit I mean something like giving a concrete basis of H_2 by explicit generators which realizes an isomorphism to E8+E8+3H)
@Koro: First of all, for the first direction, can't you prove quite generally that if you take $f=(f_1,f_2)$, then $f$ is continuous if $f_1,f_2$ are both continuous?
I was told that it might not actually be well-known this explicitly, though I wouldn't know. The reference I've looked at so far deduced the isomorphism type only from an abstract classification of unimodular lattices, which reduces the problem to computing rank, signature and whether it's even or odd.
I went back to Rudin. I think he's considering only real-valued functions, so the answer to my question is $Y=\Bbb R$. It's true in total generality here.
I didn't remember that as an issue with Rudin, but it really is. OK, I want to think of the graph as a subspace of $E\times Y$. No, he's giving a specific example. You have that backwards.
Well, I want to take a closed set $C$ in $Y$ and look at its preimage. By definition, we're taking the preimage of $f(E)\cap C$, and that is closed in $f(E)$.
No, it's natural to have $Y$ or else I have no idea where $f(E)$ even lives.
Rudin is being truly sloppy (and very old-fashioned, but the book is old).
At any rate, how do I see $f^{-1}(C)$ from the graph?
assume $\alpha\in\mathbb{C}$ is an algebraic integer of absolute value. then $\mathbb{Z}[\alpha]$ is a lattice and it contains all powers $\alpha^n$, which all lie on the unit circle. so these should only be finitely many values, so $\alpha^n=\alpha^m$ for some $n\neq m$ and cancelling shows that $\alpha$ is a root of unity.
however, it's not true that every norm 1 algebraic integer is a root of unity, so where am I being stupid
@Koro: I'll give you a battle-plan, so you can work it out for yourself. I want to turn $f^{-1}(C)$ into the projection of some compact subset of the graph (onto the first coordinate).
rudin teaches you ways of looking at stuff, but it's not that citable as a reference. i ran into that several times, not quite finding the thing i wanted to use as a textbook theorem.
I hadn't paid that much attention, never having actually taught out of it (as a student, it seemed fine to me), but talking with Koro, I realize he's totally vague about defining a function with domain and range, etc.
i think sometimes rudin falls into the trap of not wanting to explain too much, or give people citable things they can just plug into other things. i don't know why. there's sometimes an impression that this sort of treatment dumbs math down. i should ask my friend who took classes from him whether he was like that in real life.
ahlfors is similarly weak on the kinds of fundamental things you would expect a modern book to do better.
complex analysis is tough. sometimes you don't want to define a series as only a function on an open disc, because it converges for lot of the boundary of the disc too and maybe continues beyond that in some sense. so maybe you leave out exactly where you're talking about it being a function.
Ahlfors didn't bother me, though, even though it's dated, because his mathematical taste was good and those were mature students studying from it. At least one year more mature.
i had an 'intro to proof' book that spent what i thought to be far too much time on a function being a tuple of domain, codomain, and then a subset of ordered pairs of domain x codomain. precisely so you could say f(x) = x^2 was surjective R to [0, infty) but not surjective R to R and that those were different functions.
it was hyper formalistic.
they're not different as sets of ordered pairs. i didn't understand the need for the attention. the sharp students could appreciate the distinction and the rest were completely confused by it.
there's something about 'the gift of gab' which is true. the culture encourages non substantive storytelling. nothing revelatory of emotions or intellect, just shaggy dog stories that take forever and go nowhere.
