/mg/ math general

/mg/ - math general - no edition edition

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Veeky
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bumping with freshly minted meme list

Aside from Stewart that list actually looks pretty good.

What to substitute it with? I feel there needs to be some kind of motivation to learn higher maths and proofs, before one can get on learning how to do proofs or just abstract algebra

unless you mean the book is shit, but it's a classic and solutions are also readily available online, etc

You should no be doing pure maths if you don't like the beauty of maths. Stewart motivation is for engineer pleb.

If you're reading my guide for plebs, you're not particularly well acquainted with math anyways. Plus, try to read any linear algebra book without seeing examples like derivatives and integrals as linear operators, or in inner products, or applications to system of ODEs. Yeah, you can't. So what should that person do, learn all of real analysis up to the point where you start to need linear algebra, just so they can understand the basic examples that aren't just R^n or C^n? Which of course, then they'd have to learn algebra without touching any of the matrix groups...

No. Stewart is more than enough. Spivak is a meme and it'll take a student more than a year to digest the amount of information, there's no need to make a calculus book this hard. Analysis on the other hand is worthy to go balls deep.

If you can't give a better reason for picking books than being "pleb", I don't think you're mature enough to be giving advice.

I'm pretty sure if someone is that autistic then they don't need his list in the first place

What's bad about Stewart?

>No talk about Discrete Math at all

So is Discrete math just a meme for Computer Scientists?

Why are sets so boring?
I'm just trying to review them by myself because I've been falling behind on my discrete math course.

There is plenty of important stuff I didn't cover at all, namely discrete, yes, elementary number theory, euclidean/non-euclidean geometry, logic, probability, physics, etc.
But they're not fundamental in the end. Discrete math doesn't have much theory, it's mostly just separate types of problems.

Anyway I apreciate your list.
I'm very very rusty on my algebra. I struggle to factor some equations and also solving most inequalities that are not linear. I can't remember any trigonometry from high-school and I never step foot inside the Analitic Geometry course (passed by being a friend with the proffessor).
So yeah, I read your list and immediately downloaded Lang's book. Looks very nice, I might even buy it.

I’m not a mathematician but if i understand correctly.
Gödels theorem states that given a set of axioms we can’t prove everything. I imagine it being like an incomplete puzzle with one piece missing.

If that’s the case, given a set of axioms A is it possible to add a specific axiom B such that all the unprovable statements are now provable but certain provable statements can not be proven.

In regards to the puzzle analogy, basically by adding B we effectively relocate the hole in the puzzle.

If so would this mean we can prove everything just not at the same time?

>given a set of axioms A is it possible to add a specific axiom B such that all the unprovable statements are now provable
no because everything that can be proven using only the axioms from A can always be proven even if you add new axioms
and also no because your new set of axioms formed from A[math]\cup[/math]B also contains unprovable statements (it's incomplete).

In other words, if you consider the set Q(A) which contains all the statements you can't prove in A and the set Q(A[math]\cup[/math]B), not only is Q(A[math]\cup[/math]B) nonempty but Q(A)[math]\cap[/math]Q(A[math]\cup[/math]B) is also nonempty. It's not a puzzle with one piece missing it's a puzzle with an infinitude of pieces missing.

Oh damn, never thought if it that way.

Thanks user

Since planar sections of a 3D conic produce circles, ellipses, parabolas and hyperbolas, is there an analogous 4D object whose 3D sections are spheroids, ellipsoids, and whatever the 3D analogue of parabolas and hyperbolas are?

Why is Lang's Algebra under 'Algebraic Geometry'?

Lang is a meme.

General opinion question related to books:
Everybody always goes nuts about posting standard curriculum lists, what's the best text for analysis, linear algebra, etc.

But what are some of you guys' favorite books that aren't "foundational" but more specific or unusual topics not required of 100% of math majors?
A personal example is graph spectra which is a really neat field that was barely covered for 2 lectures in my intro class.

Same reason he suggests some greenhorn freshman read graduate level set theory before starting linear algebra. Meme lists have meme suggestions.

>he still doesn't formally verify his proofs
Lmfao!

mostly because 99% of the threads put up here asking for book / path recommendations, they're not asking starting from a solid undergraduate base standpoint, but instead from a "dropped-out-of-middle-school" level of education. And as it goes, one cannot produce meme lists for that 1% that are looking for niche interesting fields, which usually require a solid foundation. To be fair, I shouldn't have put Jech so far up the list, but didn't know where else to put it, and even in the text part I do mention that it is not recommended

>Everybody always goes nuts about posting standard curriculum lists, what's the best text for analysis, linear algebra, etc.
The problem is that most people posting the lists either don't know much math or haven't even read the books they list or both

I think it's in bad taste to place do Carmo's Riemannian Geometry before Lee's Smooth Manifolds. The chapter 0 in Riemannian Geometry is inadequate for actually learning the background on smooth manifolds required.

Either way, Spivak's bible should be on there instead.

>But what are some of you guys' favorite books that aren't "foundational" but more specific or unusual topics not required of 100% of math majors?
>A personal example is graph spectra which is a really neat field that was barely covered for 2 lectures in my intro class.

The wikia has stuff on it:
Veeky Forums-science.wikia.com/wiki/Mathematics#Linear_Algebraic_Graph_Theory

The problem is the losers here who would rather spend all day finding the exact perfect book to read ten years down the road than just cracking open a book and getting reading.

I really like the book selection here and the way it explains what each topic is about, so great work!
A minor piece of feedback is that the first half of analysis I works fairly well as a "primer" level book, so maybe there should be some sort of indication that one can afford to start earlier on analysis if they have an interest. On the other hand, if someone has more mathematical maturity before approach analysis (for example, by doing lots of algebra first), if might make more sense to use a more difficult book like Pugh or Rudin.

going through pic related text on analysis, would recommend

>boner for gelfand
ok freak,

>basic math
after four books on basic math??
5 books on calculus topics, really?

>james stewart
you're disgusting

>a primer
why does this come after calculus? probably because you're wasting a potential anons time with shit tier stewart

>set theory
this is already covered in How to Prove it and Book of Proof, if you had read either, you'd know. Also, odd selection as most people prefer Halmos or Enderton, and this makes me skeptical.

>LA done right
finally a decent choice, but I'd still recommend Hoffman and Kunze or Valenza over this, avoiding determinants like Axler does is silly.

>abstract algebra
finally a perfect choice

>analysis
nice user, you're really getting the hang of picking good books


Can't really criticize past that point cuz I haven't made it that far yet but overall not the worst, looks cumbersome towards the end, and could definately be leaned out quite a bit - it's non motivational to think you have to read 10 books before doing real rigorous math (LA done right in this case).

and further, as another user pointed out, no book covering any sort of discrete math?

Personally, I'd go something like this:
>How to Prove It, Velleman
>Discrete Mathematics: Elementary and Beyond, Lovasz
>Calculus, Spivak (Apostol if you're confident)
>Linear Algebra: An Introduction to Abstract Mathematics, Valenza, or the classic H&K
>Analysis I&II by Tao, Rudin if you're brave

Can't continue beyond this as I haven't rekt abstract algebra or topology yet


If you get stuck on algebra or trig, simply refer to Khan Academy, Wikipedia, Sheldon Axler's "Precalculus", Courant's "What is Mathematics" or Oakley's "Principles of Mathematics" and drill the fuck out of anything you're stuck on. The same applies to any topic that may not have been covered enough. There's an endless list of good books, the challenge is keeping it lean.

>might make more sense to use a more difficult book like Pugh or Rudin.
Rudin is a meme.

i feel like I'm 95% of the post you're talking about, since I've been doing it a lot lately

personally I post ahead of time (like when I'm halfway through a book, or know I'll have time to start another in the future, so that I waste less time searching from scratch and instead jot down potential 'leads'. I'll start making shitpost about the best book for 'x' when I'm halfway through whatever current book or so.

just saying that perhaps it's not fair to assume everyone looking for "teh best book evaaa!111!" is never going to read them, I've began a lot of the books recommend to me here. And I've even finished a couple! :^)

yeah but he's a concrete reference for someone's mathematical maturity. if you've made it through rudin (successfully), everyone knows you've made it. that being said I have not made it, but to plan to read despite there being text that I believe are better out there and i'm not excited.

hey is this

Veeky Forums-science.wikia.com/wiki/Physics_Textbook_Recommendations

any good? the "high school" section is full of second year undergrad stuff (as per usual with the braggadocios and elitist culture around here), so I'm wondering if the following:


Purcell & Morin - Electricity and Magnetism
Georgi - The Physics of Waves
Fermi - Thermodynamics (Dover Books on Physics)
Eisberg & Resnick - Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles

are really high school level or if they're gr8 undergrad material to follow with kleppner? and what about that russian guy that was in jail, Landau & Lifshitz, can I start them after kleppner? idfk about physics but wanna rocket through to grad school learnins

I also laughed at Big Jech being on there. Otherwise it's not a bad list. I'm a graduate student in set theory, so I'll list a couple of git gud grad logic books.

Enderton 'Elements of Set Theory' for standard undergrad, how math is embedded in set theory stuff.

Kunen for advanced set theory course, independence results and forcing.

Gao or Kechris for Descriptive Set Theory, I'll be working out of Gao next semester for some study in Borel Complexity theory.

Barwise 'Admissible Sets and Structures', good for a glimpse a great crossroads between model theory of infinitary logics, generalized recursion theory, and set theory.

Devlin 'Construtability.' Though I've only read the late first and second chapter, I would really like to finish the book to get some understanding of Fine Structure Theory.

Not my expertise, but I have some experience with the following.

Recursion Theory
Odifreddi 'Classical Recursion Theory.' Everything you really need to understand the field.

Model Theory
Hodges is standard. David Markers notes are better, more fleshed out.

Proof Theory
Have no idea.

One thing I'm reading through right now is Knight and Ash 'Computable Structures and Hyperarithmetical Hierarchy.' This covers recursively defined structures and classifications theorems.

>>boner for gelfand
yes, he actually has problems that make you think, covers a lot of ground in precalculus, however, it doesn't cover analytic geometry and special functions. They're all quite short and should take no time. They're all aimed at the student that has dropped out of high school and comes to Veeky Forums asking for advice.
>>basic math
it covers the rest that Gelfand doesn't cover, including analytic geometry, pure geometry, special functions and a little linear algebra. Notice none of these are calculus topics.
>>stewart
yeah it's not great but it covers all the calculus you'll ever need to know, the rest is in the realm of analysis
>>a primer
Yeah you need some sort of formal training in logic and proofs, otherwise you're not gonna know what the fuck is the contrapositive, or other standard proof techniques
>set theory
pretty funny to think that Bernstein-Schroeder is the peak of "useful" set theory. No, you need to be well versed in AoC, and equivalent definitions to even start thinking about topology and real analysis (even algebra, right inverses, maximal ideals, etc)
>>LA done right
H&F is a bit too tough for a first glance at LA

Remember this list isn't for your average undergraduate, but your average person begging Veeky Forums for starting math with no previous skill. Also, it would be extremely more discouraging if there is a selection of 5 books per subject to choose from, that person would just get lost

>Gelfand
I like Gelfand and actually poked through his trig book today, just thought the boner for him was funny, though perhaps justified.

>basic math
yeah but there's just so much overlap in coverage imo. And a lot of those topics can simply be picked up along on the way (either through in book exercises in calc books or quick youtube/wikipedia gains).

>primer, contrapositive, proof techniques
How to Prove It it starts with set theory and logic, then an outline of proof techniques. It covers, very explicitly, what you mentioned - I assume you haven't read it, otherwise you'd know..?

>set theory
I understand HtPI and BoP doesn't go in depth for set theory, but they lay a solid foundation that Tao and any LA or calc book should pick up on. whether it's enough for topology or real anal I cannot comment on, but i assume a solid text exist that is relatively self contained and introduces the essential set theory you must know.

>LA done right
Perhaps it's too tough, but there's so many videos, lecture series, etc out there, and additionally after actually having completed these books, I don't see why an aspiring Veeky Forumsentist couldn't do it.


>not for us, but for average Veeky Forums tard
then why does it go all the way into 'graduate texts in mathematics'? IDK I just personally differ in this approach and support brevity. The problem for Veeky Forumstards is not lack of resources, but more often than not being overwhelmed by too many. Perhaps many of these could be denoted as "options" for those who have more time, with a more barebones approach being denoted somehow. Maybe I'll learn how to do meme pictures and make one instead of complaining about yours :^)
I hope you know these criticisms are meant to be constructive, not just made in jest or a contrarian nature, but instead to actually continue a dialogue with purpose.

>the "high school" section is full of second year undergrad stuff

High school = Freshman level because they assume nothing. Actual high school books are a waste of time because they randomly leave out half the material causing you to have it relearn it twice. Good for high schools that don't want to create more advanced classes, bad for everyone else.

Purcell & Morin assumes you know special relativity so you need K&K's background for it. They are usually paired for honors physics 1&2. Sadly, there isn't a canonical set of books for physics 3+.
Georgi could be it's own standalone course on waves for physics minors. Most schools half ass their intro waves module due to time constraints but it's worthwhile to get a firm foundation on it before moving on to wave mechanics.
Fermi first four chapters (and the last) do a far better job of explaining thermodynamics than most other books. The other chapters should be just skimmed. Van Ness' book is another good short intro for beginners.
Eisberg & Resnick is a fairly standard modern physics book most schools use, it's a bit below Griffiths in difficulty but you will be ready to jump into Shankar after it.

>and what about that russian guy that was in jail, Landau & Lifshitz, can I start them after kleppner

You're should do an undergrad book that covers Lagrangian mechanics before making the jump to L&L.

>no book covering any sort of discrete math?

Because they are for brainlets.

Went to a math competition today, thought I should share some of the problems

A few I solved:
1) A crate contains several thin sticks. The length of each stick is between 1 and 100 cm. What is the minimum number of sticks the crate must contain to guarantee that a triangle can be formed from three of them?

2) In the triangle PQR, the medians from P and Q intersect at a right angle. Given that the distances PR and QR are 22 and 19 respectively, find PQ.

3) Penny and Quinny live at the opposite ends of a straight road. They both begin walking towards the others house at constant (not necessarily equal) rates. On the way across, they meet 170 meters from Penny's home. They continue, and upon reaching the other's house, turn around and continue back towards their house at the same rate. They meet again 200 meters from Quinny's house. How far apart are their houses?

And a few I didn't:
4) Let [math] P = \prod\limits_{n=1}^{100}n! [/math]. Find the least integer [math] k [/math] such that [math] P/k! [/math] is a perfect square.

5) Consider all polynomials of the form [math] x^3 + px^2 + qx +r [/math] having the property that all of their (possibly repeated) roots are of the form [math] 2^k [/math] for some integer [math] k [/math]. Find the maximum value for [math] q [/math] not exceeding 2017.

6) Five golf balls are placed into each of three boxes. On any turn, box is chosen randomly and a ball is removed from it. The ball removed is then placed into one of the other two boxes at random. After five random turns, what is the probability that all the boxes again contain five balls?

That's not all of them, just some of the more interesting ones.
Note: No calculators were allowed.

So we all know mathematics is evil and racist but which field of mathematics is the most racist and why is it functional analysis?

>which field of mathematics is the most racist
Teichmuller theory

statistics or probability obviously. Statistics isn't necessarily math tho

I take it you've never completed (or perhaps even began) any sort text that's in one of the many topics labeled as 'discrete math', that's cool though, I guess. Makes my life easier if there's wimps like u I'm competing with

I've wanted to start studying for olympiads recently just for fun. What sort of contest was this?

What does /mg/ think about Saxon?

Santa Clara University High School Math Contest
(no underage b&)

>university
>high school
a high school level math contest at a uni? that's pretty cool, did you get to attend as a uni student or a high school student?

Fuck off, are you really suggesting people should start with set theory? I've never read a set-theory text book and I don't think I'm in the minority. Boring shit

I am taking a class in abstract algebra and was noticed a pattern when you need to represent all polynomial functions in a ring that maps from Zmodn to Zmodn. What I (think) I found is pretty much why when you have some value x^n it can be replaced with x etc. This shows why there is only one way to represent all the functions withing the ring due to any x raised to a power about n-1 can be replaced with a power less than n-1. I was wondering if this sounds about right to anyone else or if I could get a counterexample thrown my way

I'm re-learning pre-algebra because I am a brainlet and I want to learn welding and you need at least pre-algebra.

Honestly I can't see how any of this shit would ever be of use to me, but god damn am I going to learn it. I found a series of YT lectures that are more concise and informative than any of the three (3) teachers that previously sought to teach me pre-algebra.

what kind of welding my dude, I tried it once and I felt it was almost an art to weld well

Are you a CS major or something?

let n=12 and x=6
x^2 = 36 = 12*3 = 0
so x^12 = (x^2)^6 = 0

x and n have to be coprime.

Yeah it will probably suck.

I'm just getting started, I have no idea but I'll be doing it for the most practical reasons. So probably contruction. Wherever the steady money is. Just got to get this pre-algebra down so I can ace the exam and skip a bunch of bullshit classes I don't need.

nah man, just havent slept enough and am pretty drunk. I noticed that shit, but the thing is if it is all funcs that map zmodn to zmodn then zmodn must be an integral domain. You cant just leave out random values of x to make sure that x and n are coprime so if you want to be an autist make sure you are 100 right

nah man, just havent slept enough and am pretty drunk. I noticed that shit, but the thing is if it is all funcs that map zmodn to zmodn then zmodn must be an integral domain. You cant just leave out random values of x to make sure that x and n are coprime so if you want to be an aut make sure you are 100 right

Pretty interesting questions desu. I think I can solve 4 and 5, haven't tried the others.

>tfw selfstudying optimisation

5) The polynomials will have the form [math] (x-2^i)(x-2^j)(x-2^k) [/math], which leads to [math] q = 2^{i+j} + 2^{j+k} + 2^{i+k} [/math]
That means that the biggest [math] q [/math] we can possibly produce with real [math] i,j,k [/math] would be [math] 1024+512+256 [/math]
A little bit of solving the linear system later we find out that for that to happen we would need [math] i = 5.5,~j =4.5,~k=3.5 [/math]
The next best [math] q [/math] would be [math] 1024+512+128 [/math], which can be produced with [math] i = 6,~j =4,~k=3 [/math]

Yea, this is basically what I did and I got k = 48 for problem 4 (This involved quite a lot of calculations so I'm not sure of the final result but I believe the method is correct).

>that garbage
>interesting ones

>primer, contrapositive
yeah i know that's the content of the book, that's why i put it there
>set theory
Munkres Topology has all the set theory you need in the first chapter for example, but he goes over a lot of the details, and leaves too much to the exercises. I don't know many other books that go into so much detail that aren't on set theory though. Rudin, for example, doesn't even cover axiom of choice, even though he uses it in proofs.
>not for us, but for the average Veeky Forumstard
People often look for a goal, and I thought that it was cutting it short if I finished with Topology, but it was a bit unfair on other subjects if I only mentioned one in particular

>t. can't solve a single problem

Yes, I don't even know a lot of the definitions in there since the problems and the related fields are uninteresting garbage.

>definitions
It's literally just high school math. Are you a biologist perchance?

so what are you brainlets thoughts on mod congruence and mod arithmetic

this vid had me space out pretty bad:

youtube.com/watch?v=kxuU8jYkA1k

>high school math
Which is uninteresting garbage, so I didn't even bother with it.
>Are you a biologist perchance?
Wouldn't that be painful?

>Wouldn't that be painful?
I don't know man you have to tell me that.

>vbm
Isn't that just needlessly obfuscated number theory?

>vbm
how about you watch it and judge the content on it's own, or do you need a label for every shit you consume? if so I don't recommend you becoming a scientist or mathematician m8

1) The condition for a triangle to be constructed is that two sticks together are longer than the third.
The worst case scenario is having the 100 cm stick in there as the first stick
Lets look at next worst case.
We would have to have two more sticks that don't add up to more than 100cm.
They should be as long as possible, but the difference between their two lengths should also be maximal, so that the next stick that needs to be added has to be as long as possible.
The worst case would be a 75 and 25 cm stick.
We continue adding sticks of ration 3 : 1 that add up to the smallest stick in our crate.
(100,75,25,18.75,6.25,4.6875,1.5625)
We cannot continue this process any further, since 1.5625/4 < 1, so we can only add one more stick without making it possible to form a triangle.
Thus the minimum number of sticks to guarantee that a triangle can be formed from three of them is 9.

For 4), notice that P contains 100 factors of 1, 99 factors of 2, 98 factors of 3, and so on. The odd factors occur an even number of times and the even factors occur an odd number of times.

If we divide P by m=2*4*6*8*...*100 = 2^50(1*2*3*...*50), the quotient P/m is a perfect square since all factors will occur an even number of times. But 2^50 is also a perfect square, so P/k! is a perfect square for k=50.

We now have to show that this is the smallest such k. Consider P/l! = P/k! *k(k-1)...(l+1). Just need to show k(k-1)...(l+1) is not a perfect square which is easy to see.

In general the upper-bound for P has to be an even perfect square and k will be half that.

Right, this makes sense. I tried counting the number the number of times each prime occurs in P using en.wikipedia.org/wiki/Legendre's_formula so it was becoming quite complicated.

That solution is already wrong. If you take Fibonacci sticks
(1,1,2,3,5,8,13,21,44,65)
then you can't form a triangle and those are already 10.

you're right. I should have started from the smallest stick instead of the largest and I might have realized this.

I think it's a wacky meme, getting into that shit but with polynomials

Let f(x, y) be a functon from [a,b]x[c,d] to R^2
f(x0, y) is continious for all x0 in [a,b]
f(x, y0) is continious for all y0 in [c,d]

Does if follow that f is continious on [a,b]x[c,d]?

Of course not.

Hey guys quick question, Im having trouble understanding this, the problem is binomial coefficients b =3, n=4, and k=2 and the result of this is 16. I thought this problem was binomial(n+k-1, b) so the answer would be 10 right? What am I missing here, how are they getting 16?

Binomial coefficients only need two parameters and you gave 3 in your post, so I don't know what you're getting at.

H I N T: this problem is binom(n,b) when k=1 and binom(n+b−1,b)when k=b. Note the solution increases as k move from 1 to b.

Thats right from the paper I got the problem from

Has anyone here read Xavier Gourdon's books ?

Pic related. Problem 6

Proof?

> matlab

Proof: think

It's octave you plebeian

:thinking:

Actually I wasn't the original user who responded to you and now that I've read the question it seems that f actually is continuous.

My uni uses almost only Griffiths for the theory based classes.
Other than that, I've been recommending Feynman lectures left and right, still think they are very worth it for an introduction.

You don't even bother posting the problem and you expect others to bother helping you?

Anyone know any dark magic on how to prove that a polynomial doesn't have a specific root? E.g. you have some integer cff. polynomial and you want to prove that (x-a) is not a root, where a is rational.

I know that Strasman's theorem + Newton Polygons rules out a lot of roots. Then Cohn's irreducibility criterion does likewise, but its hard to apply in many cases. Rule of signs gives some indication. Plus various well-known rough bounds on the size of roots.

>primer, contrapositive
I'm not sayign HtPi shouldn't be there, but that it should be before calculus so you can look at a more rigorous textbook in calc.

>munkres
if munkres covers so much set theory, why have seperate text on it? this is the kind of redunandancy and overlap I'm trying to address.

>Veeky Forumstard
fair enough, if it were me I'd have three seperate images for the three recurring anons: absolute middle school level noobs, ambititous undergrads looking for non-stewart text, and then upper level grad school crap

plug the root into the polynomial dumbass

Thanks Sherlock, sometimes you have polynomial forms where you don't know the coefficients because they are generated by another process like with equidistributed sequences, recurrences, partial sums of generating functions, etc.

>No link to a pdf repo of all of these
Someone really needs to upload them and fix that.

[math] f(x,y) = x^y [/math] is not continuous in [math] (0,0) [/math] despite being continuous in in one dimension for fixed x or y.
Proof:
Let [math] x^1_n = 0,y^1_n = \frac{1}{n} [/math] and [math] x^2_n = \frac{1}{n},y^2_n = 0 [/math]
Then we have
[eqn] \lim_{n \to \infty} ( x^1_n,y^1_n) = \lim_{n \to \infty} ( x^2_n,y^2_n) =(0,0) [/eqn]
but
[eqn] \lim_{n \to \infty} f( x^1_n,y^1_n) = \lim_{n \to \infty} 0^ \frac{1}{n} = 0 \neq 1 = \lim_{n \to \infty} \frac{1}{n}^0 = \lim_{n \to \infty} f( x^2_n,y^2_n) [/eqn]

That function is not even defined at (0,0). It can't be continuous on it.

Thanks.
I found a bit cleaner explanation here if someone interested.
calculus.subwiki.org/wiki/Separately_continuous_not_implies_continuous

It is fucking depressing that my lecturer uses the opposite.

You can define it at this point as 1 explicitly.

meh i didnt think about that
problem is, that f(0,y) isn't continuous then

this is better

In those cases wtf do you have then?
>No coefficients
>no roots
so you only know the operations?

You can make a triangle with those sticks, it's just that you can only make flat triangles.

Yeah you might have a recurrence relation between coefficients with a few of the first coefficients explicitly. Or you might have an infinite product and a power series which are equal to one another, so you do generating function manipulations to get the values of the coefficients of one in terms of the other. Sometimes its useful to get partial sums for various calculations, in which case you're essentially working with polynomials where you only know some general attributes of the coefficients.