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I'm doing EVERY exercise in munkres' topology textbook

  1. Jan 21, 2007 #1
    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?
    Last edited: Jan 21, 2007
  2. jcsd
  3. Jan 21, 2007 #2


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    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).
  4. Jan 21, 2007 #3
    "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 ?
  5. Jan 21, 2007 #4
    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.

    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?

    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 http://www.amazon.com/gp/reader/0201315963/ref=sib_dp_pt/105-7684738-0871604#reader-link looks good, nice and theoretical. my dad said he'll buy me any books i want.
    Last edited: Jan 21, 2007
  6. Jan 21, 2007 #5

    matt grime

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    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.
  7. Jan 21, 2007 #6
    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!
    Last edited: Jan 21, 2007
  8. Jan 21, 2007 #7
    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.
    Last edited: Jan 21, 2007
  9. Jan 21, 2007 #8
    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.
  10. Jan 21, 2007 #9


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    Please read this thread. LaTex is very easy to use, please learn to use it.
  11. Jan 21, 2007 #10

    Gib Z

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    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.
  12. Jan 21, 2007 #11
    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.
    Last edited: Jan 21, 2007
  13. Jan 21, 2007 #12


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    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!
  14. Jan 21, 2007 #13
    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. :biggrin:
  15. Jan 21, 2007 #14
    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).
    Last edited: Jan 21, 2007
  16. Jan 21, 2007 #15

    Chris Hillman

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    Good books and other good ideas

    Hi, Tom1992,

    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!).

    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.
    Last edited: Jan 21, 2007
  17. Jan 21, 2007 #16
    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 any one 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.
    Last edited: Jan 21, 2007
  18. Jan 21, 2007 #17


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    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.
    Last edited: Jan 22, 2007
  19. Jan 21, 2007 #18

    Gib Z

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    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 Im starting that now so im 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 :)
  20. Jan 21, 2007 #19
    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? ;)
    Last edited: Jan 21, 2007
  21. Jan 22, 2007 #20


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    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.
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