I'm doing EVERY exercise in munkres' topology textbook

[Tom1992]

i think I've accelerated my learning enough, and now I'm going to start doing problems, problems, and more problems to strengthen my mathematical thinking. this thread will be devoted to munkres' well-used topology textbook. I've done all the problems in chapter 1 so far, and i haven't gotten stuck once. i know that about one third of the exercises already have solutions over the web, but i do those anyway, and then of course i do the ones not done over web. i will occasionally post some solutions to interesting problems that really intrigued me, but i don't know latek so i'll perhaps pdf my solutions. i didn't realize how much i learn by doing exercises in old topics. dis is phun!

i also want to do every question in a textbook in multivariable calculus (not single variable calculus) and a textbook in linear algebra, but only textbooks whose exercises deal mostly with proofs (not boring exercises that ask simply to compute a jacobian or an integral or a determinant, solve systems of equations, etc...) and does not hold back on topology (e.g. describes continuity in terms of open sets instead of just limits, describes the inverse function theorem by diffeomorphisms, etc...). any suggestions on such textbooks for me to practise with?

 

[mathwonk]

guillemin - pollack, spivak calculus on manifolds, adams and shifrin linear algebra, my webnotes on linear algebra (14 pages of text, lots of exercises, all proofs).

 

[tehno]

guillemin - pollack, spivak calculus on manifolds, adams and shifrin linear algebra, my webnotes on linear algebra (14 pages of text, lots of exercises, all proofs).

"calculus on manifolds ,linear algebra on 14 pages ,all proofs blabla ...and all that for a 14 year old?

I could be mistaken ,judging on his nickname (Tom1992),but where does World go these days ?

 

[Tom1992]

that's why i got to practice with some problems NOW, I've been crash reading too much, having stopped doing serious exercises after single-variable calculus.

adams and shifrin linear algebra,

this book's got a lot of bad reviews:


is there some well-established linear algebra book (at least 90% of exercises dealing only with proofs)?

as for calculus on manifolds by spivak: nice and theoretical, but aren't the problems already worked out in the web?
http://www.ms.uky.edu/~ken/ma570/

how about "analysis on manifolds" by munkres? (my dude munkres again!) is that a good book for proof exercises in multi-variable calculus? the preview at https://www.amazon.com/gp/product/0201315963/?tag=pfamazon01-20 looks good, nice and theoretical. my dad said he'll buy me any books i want.

 

[matt grime]

Why would you want to do multivariable calculus? I mean, why would you choose to use that label for what it is you want to study? It seems a bad name to use, since it implies some dull engineering nonsense. Better to stick with learning differential manifolds, if you really have to. Though I'd personally prefer to push you towards algebraic geometry rather than differential geometry. Starting with the simple book by Carter Seagal and MacDonald. It is possibly beyond where you're at now, but would be a good book to have, and is cheap.

I don't understand why yo'ure complaining that the exercises have solutions on the web for spivak. Firstly, answers to almost all exercises appear somehwere on the web, and secondly, no one is making you read them.

Jacobson is good (and expensive) for algebra. Again, I think you're making an error in wanting a book on linear algebra. Linear algebra is just the representation theory of a field, and that is a trivial subset of far more interesting subjects. Investing in Jacobson would set you up for a lot of pure maths.

Getting anything written by Serre would be useful, too.

 

[Tom1992]

ok, I've decided get my dad to order the following books for me to practice more proof exercises with linear algebra and advanced calculus:

Analysis on Manifolds - Munkres
Calculus on Manifolds - Spivak
Advanced Linear Algebra - Roman
Linear Algebra Problem Book - Halmos
Linear Algebra: Challenging Problems for Students - Zhang

can't wait to get them!

 

[Tom1992]

here is a sample question from the munkres topology book that i find interesting (and not posted in the web). I'm going to write P for the cartesian product symbol with i taken from all of I (the index set), and U for the union symbol with i taken from all of I.

let I be a non-empty index set. prove that if PAi is finite, then Ai is finite for every i in I.

not difficult, but it's only interesting if you try to prove it without using i-tuples. this is more fun!

here's my proof:
assume that I is a non-empty index set and that PAi is finite. PAi is by definition the set of all mappings x: I -> UAi such that x(i) belongs to Ai for every i. for each x, define y as the restriction of x to i for some fixed i in I. then every y (some of which are identical to each other) is a mapping into Ai , and so Ai is the set of all the y. consequently, since all the y’s are obtained by restricting all the x’s to i, then Ai cannot have more elements than PAi and hence is finite. since i was an arbitrary element in I, then Ai is finite for every i.

 

[DeadWolfe]

I admire you greatly. I have tried, and was never able to complete ALL the problems from any textbook. Somehow I can only work on problems I find interesting.

Although, I am trying to work through every Putnam problem in history.

 

[Integral]

Tom,
Please read https://www.physicsforums.com/showthread.php?t=8997". LaTex is very easy to use, please learn to use it.

 

[Gib Z]

Yes I was going to say that too, LaTex is not as hard to use as you first think. Not to mention, it would be worthwhile because mathematicians these days are required to know some basic LaTex.

Personally I've never been bothered to do all the problems from a textbook. In the trivial exercises which I believe I am strong at, I choose some sort of a pattern, depending on how many questions, and how easy they are to me. Usually its every 2nd question. For the challenging exercises I try to do them all.

Several things have worried me about you Tom- First when I found out your level of Knowledge, I thought you must have been quite advanced from the start. Then I found out however, that you hadn't learned trig until you we're 11, which made me wonder how you advanced so far in the space of 3 years.

Now I seem to find out that you had just been reading and not doing the exercises in what you had learned, which worries me even more. I've had similar bouts where I learned the theory, I READ the entire textbook and If someone asked me anything about the theory I could do it perfect.

When it came to the questions however, It was worse than the Challenger disaster. You will find later that although you have accelerated you learning heavily, you've lost a good chunk too. Going back on subjects and re learning them from a different perspective is hard work, and would have been less if you learned it solidly from the start.

I always used to think, If I know the theory good enough, then I'll be able to apply it when I need to, no sweat. That unfortunately could not be further from the truth. Basically, the gist of it is: Learn the theory the same time you do the exercises!

O, and why do you seem to care so much if the solutions are posted on the web? No one is making you read them, do them yourself.

 

[Tom1992]

thanks for your concerns gib. you and i both started reading calculus at an early age, me at 11 (after finishing trig in a month or so), and you at around 13 perhaps. we've both done the exercises in calculus, but when i found the exercises in calculus quite easy (it's all just calculation), i felt that i did not need to spend so much time doing exercises any more and felt I could just read through an entire textbook and learn faster.

so i did this, starting with linear algebra, then over the next three years: vector calculus, differential equations (didn't like too much--too computational), groups, rings (but got bored of that), number theory, topology (which i loved, hence this thread), differential geometry/topology, and just a few days ago i decided to stop in the middle of my riemannian geometry textbook. so how much did i really learn? well, I've been diagnosed with a memory score of 150, and reading the proofs to every theorem indirectly helps one in doing proofs, and in terms of comprehension, in a few days i did all the problems in chapter 1 of munkres' topology without getting stuck, but perhaps i will get stuck later on, well see...

you're right though, i should have done the problems while learning at the same time, but i just couldn't wait to learn all the topics lying ahead of me. there is just so much mathematical treasure to be had.

 

[quasar987]

What does a "memory score of 150" represents?

Btw - 14 years old and all this math knowledge behind you already, I'm all but worried about you!

 

[dontdisturbmycircles]

14? my gosh!

I hope to learn review calculus 1 and self study calc 2 and 3 in 5 months since I only have 2 courses next semester of high school (chem and physics grade 12 lvl) but I doubt I will be able to do it... Although I will be doing most of the problems.

I think this will give me inspiration though. Every time I want to stop studying calculus 2/3 I will remind myself that there is a 14 year old that has already learned it.

 

[Tom1992]

so back to the discussion of this topology textbook, if anyone has any questions about an exercise in munkres' topology book, let me know and i will try to post my solution (once i get to that exercise, I'm going in order).

 

[Chris Hillman]

Good books and other good ideas

Hi, Tom1992,

ok, I've decided get my dad to order the following books for me to practice more proof exercises with linear algebra and advanced calculus:

Analysis on Manifolds - Munkres
Calculus on Manifolds - Spivak
Advanced Linear Algebra - Roman
Linear Algebra Problem Book - Halmos
Linear Algebra: Challenging Problems for Students - Zhang

can't wait to get them!

I know the topology text by Munkres, the cal. on manifolds book by Spivak, and Advanced Linear Algebra by Roman, and those are all good books. As someone said, they are advanced undergraduate to graduate level, but since you are already enjoying the first book, I think we can assume this level is not inappropriate for you whatever your age might be.

Hmm... one fascinating topic which can be learned by a bright person with little prerequisites (but which fits in very well indeed with learning linear algebra) is combinatorics and graph theory. There are so many wonderful books in this one area that it is hard to choose just a few!

1. Bollobas, Modern Graph Theory, Springer, 1998. One of the best books I've ever read (I don't mean just math books).

2. Cameron, Combinatorics, Cambridge University Press, 1994, and Cameron, Permutation Groups, Cambridge University Press, 1999 (in between this pair you should study Herstein, Abstract Algebra, MacMillan, 1986, and you can read the first one at the same time you study Bollobas.). Wonderful stuff.

3. Wilf, Generatingfunctionology, Academic Press, 1990. I happen to prefer a different approach, based upon category theory, which fits in more nicely with the themes in Cameron above, but this book is worthwhile just for the first few chapters!

Highlights of these books include random graph theory, one of the most beautiful topics I've had some success explaining to bright persons who know no advanced mathematics at all, Moebius inversion, connections with elementary number theory, and Kirchoff circuit theory. There is considerable overlap of topics among the books I mentioned, but this a good thing, since the authors each introduce different but always fascinating aspects.

And here's another wonderful book well suited to a talented student:

4. Rudin, Principles of Mathematical Analysis, Third ed., McGraw Hill, 1976. Some students complain this is too dry, but this is a superb book, remarkable in that Rudin worked out all the (original!) proofs under the direction of Moore of "Moore method" fame.

5. Boas, Invitation to Complex Analysis, Random House, 1987.

And let me throw another element into the mix: if you have a modern personal computer, a symbolic computation system (often called a computer algebra system or CAS), such as Mathematica or Maple, can enormously multiply your power to play with examples, which is an essential part of learning mathematics! If you are registered for a class at a local college, you might be eligible for a student discount; if not, these are pricey (thousands of dollars) but worth every penny. For linear algebra, Matlab is also well worth a look. There are also many free high quality special purpose packages including Macaulay2, Singular, &c. (commutative algebra) and GAP4 (groups, group actions, and lots of other cool stuff).

If you buy Maple and/or install something Macaulay2 (which is free and which I found very easy to set up),

5. Cox, Little & O'Shea, Ideals, Varieties, and Algorithms, Springer, 1992. Many others besides myself consider this one of the best math books ever.

In particular, I feel that commutative algebra is one of the most lovely and useful topics which should be part of the standard undergraduate curriculum; it is not much less useful than differential equations in modern applications (which include, incidently, solving differential equations!).

my dad said he'll buy me any books i want.

You might want to make it clear that you are asking for books and Maple in lieu of an automobile...

You are talking to the profs in your local math department, aren't you? If not, don't hesitate to introduce yourself even if you are, as some guessed, fifteenish. Have you talked to your dad about something like "MathCamp"? This would be a wonderful experience. James Morrow (University of Washington, Mathematics) has had extraordinary success mentoring young mathematicians, several of whom have gone on to impressive research careers. There is also, I think, a fine program in Budapest (Hungarian not required; the language is English!).

Speaking of the Moore method, sounds like you might benefit from that if you can find a master. David Henderson (Cornell, Mathematics) is still teaching, but his is one name that springs to mind.

And Integral is quite correct: you should learn latex forthwith. The easiest way to do this is to click on formatted equations in this forum to see the latex code and then start marking up your own PF posts the same way. Once you get to school, you can ask fellow students for a one-hour tutorial on writing homework solutions with latex.

 

[Tom1992]

chris, nice to meet you. from your history of posts, i take it you are a relativity expert. the differential geometry and riemannian geometry textbooks I've read may be of some background for me to read about general relativity, if i should choose to explore there i may start doing all the exercises from a relativity book as well and post that thread in the relativity forum (but it will have to be a mathematical gr book).

i'm too shy to talk to the math professors, or anyone else around me for that matter. everyone there always looks over my shoulders (literally). i only have my mathematically inclined dad to help me out. I'm turning 15 in aug 14 btw.

 

[mathwonk]

as to the negative reviews of adams and shifrin by the cretins posting on amazon, one thing you need to learn is not to take the advice of students who are less intelligent than you are.

 

[Gib Z]

Actually Tom, I finished Calculus when I was 11 as well, and since then I've obviously haven't learned as broad a spectrum as you. I have focused mainly on Number Theory and finding unique proofs to everything. Maybe I should have studying a broader spectrum, seeing as Number Theory require knowledge from many fields of mathematics, but I am starting that now so I am Fine. Knowing Calculus at our age is no big deal, I have numerous friends who knew it at our age, and one who knew it when she was 8.

I know pretty much nothing compared to you or her, but I've enjoyed my time :)

 

[Tom1992]

may i suggest that if you think the reviews of adams and shifrin by the cretins posting on amazon are more reliable than advice from professors posting here, you might lose credibility as a serious student, at least with me.

prof. mathwonk, sorry i didn't mean it that way. certainly you are more reliable than those reviewers. i just like roman's advanced linear algebra more. i did go along with your suggestion with spivak's calculus on manifolds.

gib, who is your female friend who learned calculus at age 8? if you're not already interested in her, perhaps you could introduce her to me? ;)

 

[verty]

Tom, I have a quick question. In this learning of mathematics, is your learning strictly contained within the areas you have studied or have you worked them into a homogenous whole? Where does logic/set theory/category theory fit into your learning?

I am no mathematician but think you might provide me with some insight.

 

[mathwonk]

sorry for the defensive tone, please reread my edited post.

 

[Tom1992]

Tom, I have a quick question. In this learning of mathematics, is your learning strictly contained within the areas you have studied or have you worked them into a homogenous whole? Where does logic/set theory/category theory fit into your learning?

I am no mathematician but think you might provide me with some insight.

i'm no mathematician either. I've studied abstract algebra and topology. that's as far as I've gone into category theory. logic and set theory is precisely chapter 1 of munkres' topology book, from which I've done every exercise.

prof. mathwonk, your insight would be most valuable as i post various solutions of my own from munkres' topology book (and from my new linear algebra and advanced calculus books some time later) as i spot interesting problems.

 

[Chris Hillman]

chris, nice to meet you. from your history of posts, i take it you are a relativity expert.

No, if you look harder you should find some posts providing some details of my background and interests. I only wind up talking about gtr because so many people ask so many questions about this admittedly interesting topic, but I am entirely self-taught in physics. My formal training is in math.

the differential geometry and riemannian geometry textbooks I've read may be of some background for me to read about general relativity,

Yes, if you have a solid background in Riemannian geometry, including local versus global structure in manifolds and all those nifty derivatives, picking up gtr will pose no challenge, provided you first pick up Minkowski's simple and beautiful geometric interpretation of the Lorentz transformations (see Taylor and Wheeler, Spacetime Physics, FIRST edition only). Some background in Lie groups and Lie algebras will also be helpful. Don't worry about "tensor calculus"; a good Riemannian geometry course will provide all you need to know about tensors.

if i should choose to explore there i may start doing all the exercises from a relativity book as well and post that thread in the relativity forum (but it will have to be a mathematical gr book).

Someone expressed interest in doing all the exercises in Misner, Thorne, & Wheeler, Gravitation, which happens to be a classic textbook and the one from which I learned gtr, so you should look for that post and PM the participants in that project.

However, while I enthusiastically joined in the rush to recommend our favorite books, at some point I would become concerned that you are throwing your net too widely. I tend to think you might be better off focusing on math, which opens doors for all kinds of subsequent learning, including physics.

i'm too shy to talk to the math professors, or anyone else around me for that matter. everyone there always looks over my shoulders (literally). i only have my mathematically inclined dad to help me out.

I understand being shy, but you really should make the effort, since it is important for an advanced student not to be remain too isolated. Perhaps your dad can call the local math department and ask if one of the professors would be willing to have you drop by their office and introduce yourself.

i'm turning 15 in aug 14 btw.

I think it is quite likely that at least one local math professor will be happy to give you some of their time. Don't forget to ask about "Math Camp" or such opportunities. Remember, in programs like Math Camp, the others will also be 15-16.

 

[complexPHILOSOPHY]

This kid blows my mind. Do kids of this caliber, tend to approach problems differently? I feel useless now, lol. I can't even imagine opening a topology book, let alone understanding it at my level and I am a physics major.

 

[tehno]

gib, who is your female friend who learned calculus at age 8? if you're not already interested in her, perhaps you could introduce her to me? ;)

Now you're talking!
That's what I want hear from a 14 year old.
Maybe,but just "maybe" there is still hope for you...
You know,we all worry about you due to the fact you're already in a Riemann geometry stuff...

 

[hrc969]

You know,we all worry about you due to the fact you're already in a Riemann geometry stuff...

I was sondering if anyone could explain why this would cause worry about him. ... Can someone please explain?

 

[tehno]

I was sondering if anyone could explain why this would cause worry about him. ... Can someone please explain?

A nut house before he turns 18?

 

[complexPHILOSOPHY]

A nut house before he turns 18?

So the only way he won't turn into a nut house, is if he doesn't do riemann geometry and pursues girls? I spent my four years in high school, doing varsity wrestling, varsity soccer and varsity lacrosse. I was too concerned with being social and winning sports that I nearly failed high school. I had to learn (not re-learn since I never ever paid attention) mathematics from algebra up, after I graduated high school and I can definitely say, the 'experience' that I got from high school, was not worth the effort I have to invest now.

If I was able to do everything over again, I would have become a recluse and focused on math and physics and not concern myself with a complex social life. I have to battle with my girlfriend (whom I live with) everyday about studying, because spending to much time in books and not with your loved one, starts to have adverse effects on your relationship. However, I don't want to tell you what to do, just trying to give you some anecdotyl experience.

If he is able to understand and work through these abstract mathematics at such a young age, why would you discourage him? All of the idiots that I hung out with in high school that were 'popular' never left the town we grew up in and will never do anything beyond high school. They all go back to the varsity football games to relive the glory days.

Sorry for rambling but basically I am saying that you have your entire life to develop a strong social network with adults. Balance your social life with your work but I would definitely encourage you to use your math talents and not waste to much of your time getting caught up in the high school social hierarchy.

 

[verty]

Well I would rather see that someone who finds enjoyment in intellectual pursuits continue doing that than try to do what everyone else does. If Tom has a gift then why should he not pursue it?

Moreover, I'm pretty sure it's up to him to decide whether chasing girls is a worthy life purpose.

 

[K.J.Healey]

I, and I assume some of the people here that have read this thread, just want to say congratulations , and that I'm jealous :) Not of your intelligence; I'm fairly intelligent myself as are most people in my field. I'm really just jealous of your motivation/work ethic and your introduction to these topics at such an early age.
I would love to sit down and just start studying new areas of mathematics or physics that I haven't yet learned. Unfortunately I have so much going on in my life (Preparation for grad school, looking for work, girlfriend, etc) that I find it hard to sit down and study as a means of relaxation. I'd rather read a novel.

I'm happy you've been able to sate this interest and hope that while doing so you haven't hindered any of your creative abilities. Make sure you read imaginative works as well and help develop your creative side. Just knowing everything about a field of mathematics will not enable you to develop new ideas and theories. And just on a side note, you said Differential Equations had too much calculation. Were you speaking of algebraic/calculatory solution, or actual numeric calculations. I don't recall DiffEq having a single numeric solution when I was taught, and I think that both it and Boundary Value Problems are immeasurable tools for solving both real world and theoretical problems. It provides you with a very straight-forward analytical viewpoint on virtually any dynamic process. (Circuits, Heat, Solids, Thermodynamics/Stat Mech, Dynamics, Quantum, etc). And nearly anything that has vibrational motion uses DiffEq/BVP methods for solution.
But perhaps you're past all of that already. I do not know.

But I wish you well, and good luck!
-KJ Healey

 

[Tom1992]

please understand that i am only the youngest student in my first year classes, but probably not the smartest.

 

[verty]

please understand that i am only the youngest student in my classes, not necessarily the smartest, otherwise i might actually speak out in class.

My point is true independently of whether you are the smartest or dumbest or somewhere inbetween. And why should you need to be the smartest?

Oh, sorry, you meant you should need to not be the youngest (or be the smartest). Is age that important?

 

[Tom1992]

here is another interesting munkres problem from chapter 1. i will repeat once more that all problems i post in this thread do not already have a solution posted in the web.

prove that the set A = {a,b}x{a,b}x{a,b}x... is uncountable.

the solution is very short, but requires deep thinking.


proof:
suppose that A is countable. then there is a surjective map f:N -> A, where N is the set of positive integers. define x as the unique element in A whose nth coordinate is different from f(n) for all n in N (this construction is possible since each component has two possible values). since f is surjective, then there exists m in N such that f(m)=x. but this is impossible since by construction x differs from f(m) in the mth component. this contradiction means that A cannot be countable.

this same trick can be used to prove that the real numbers is not a countable set, using the infinite decimal repesentation of a real number. try it!

 

[verty]

If P is a countable subset of A with cardinality omega (equal to the cardinality of omega), I'm wondering what the cardinality is of the set of surjective maps from N onto P...

 

[Tom1992]

my reading has now actually slowed down. whenever i read a theorem i glance at how long the proof is. if it is short, i try to come up with the proof myself. usually such short proofs to theorems serve as valid a problem as the textbook problems. so i guess now it will take me longer to finish this book than i expected, but time is on my side. however, i don't always manage to come up with the proof: some short proofs are sooooo hard! for example:


prove that there is no surjective map f: A-> P(A), where P(A) is the set of all subsets of A.


proof: define B={a in A: a belongs to A-f(a)}. suppose B=f(c). then [c belongs to B] iff [c belongs to A-f(c)] iff [c belongs to A-B], a self-contradiction.

wow!