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Self teaching Multivariable calculus

  1. Feb 6, 2005 #1
    hello, i was wanting to teach myself multivariable calculus. i am currently in calculus BC AP. that class isnt challenging enough for me. do any of you out there know any good sites or good books for multivariable calculus. any help would be greatly apprecaited.

    thanks in advance.
  2. jcsd
  3. Feb 6, 2005 #2
    Your math teacher would probably have some good resources for you. Since your AP's in May, I suggest you concentrate on BC until you take the test (just humor me, hehe), and then afterwards ask your math teacher if he or she has any books that she'd like to lend you.
  4. Feb 7, 2005 #3
    Try Stewarts "multivariable calculus", it is good for teaching yourself. Also, consider exploring differential equations.
  5. Feb 7, 2005 #4
    i was thinking about purchasing that book. i saw it on eBay. :surprised heh...
    as for concentrating on BC...i knew it before i entered the class, because i taught myself calculus during my 9th grade summer... and my school is gay so they wouldn't let me in bc in 10th grade. but thanks for the suggestions guys. any more? id appreciate all suggestions.
  6. Feb 7, 2005 #5
    If you are just studying for the AP test, any old calc text will probably do.

    If you want to learn math the right way, consider getting Spivak's Calculus. Then you should be ready for Rudin's Principles of Mathematical Analysis and Spivak's Calculus on Manifolds. Also consider looking into linear algebra. I recommend Strang's text and the video lectures available online at


    After this, you can go on to Hoffman/Kunze's Linear Algebra or Sheldon Axler's Linear Algebra Done Right. The second one is much more pedagogical and clean. The first one is the traditional Linear Algebra text and contains a bit more material.

    Anyhow, don't let your school determine (and in the process ruin) you math education. Use a combination of those books and places like this.
  7. Feb 7, 2005 #6
    well, i already know calculus. i dont have to study for the AP exam. my teacher said i should get a 5 on it...heh, she maybe a little TOO confident. i am really looking for material on multivariable calculus. well, i would really like anything that is above calculus BC. that should broaden the topic a tad.
  8. Feb 7, 2005 #7
    It depends on what you want to use your math for eventually. If you want to be a mathematician (or semi-competent physicist) start looking into rigorous calculus books like Rudin (or Bartle and Sherbert for a less steep learning curve). Otherwise, any cheap text will probably do.
  9. Feb 8, 2005 #8
    maybe start on some analysis & proving things. check out rudin's principles of mathematical analysis or pfaffenberger/johnsonbaugh's foundations of mathematical analysis
  10. Feb 8, 2005 #9


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    stewart is a mass market book for average students, but probably bettter written than many. spivak's "calculus on manifolds" is outstanding but assumes linear algebra first.

    the best classical book, and one i would recommend to you is courant's calculus volume 2. another outstanding book is apostol's volume 2. the problem is, even if you "know" calculus, where did you learn it? you could have difficulty with a high level calculus book on several variables if you learned out of a cookbook for one variable, like stewart.

    that's why you are being recommended to get spivak's one variable calculus book, or courant's or apostol's volume 1, and relearn that material right.

    you sound like someone for whom a good book would be appropriate. unfortunately the whole AP program is a disservice to students like you, who then skip getting a decent course on calculus n college from someone who actually understands the material much better than most high school teachers.

    some of them are gay though. was that a misprint, or are you a homophobe?
  11. Feb 8, 2005 #10
    Spivak also have a book called Calculus. I wonder how this compares with some of the other books mentioned.
  12. Feb 8, 2005 #11


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    I mentioned it above in this sentence:

    " that's why you are being recommended to get spivak's (one variable) calculus book, or courant's or apostol's volume 1, and relearn that material right."
  13. Feb 8, 2005 #12
    mathwonk I currently have been working from Courant's book because I am one of those people who hate "memorize this rule and solve." Do you reccommend reading the book like a story book sequentially, or skipping around? Should I do every single problem including the Appendix problems and Miscellaneous Problems? How did you fare with your classes with this book?
    Thanks :smile:
  14. Feb 8, 2005 #13


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    i have myself never been able to read any book all the way through, except maybe a riveting spy book, like one i read once about the second world war.

    in courant i have found certain sections that i can read through and get some little nugget of wonderful stuff out.

    like the appendix in volume one where he explains about the concept of the point of accumulation. i still remember that after 45 years. and the section in the beginning where he shows the relation between decimals and points on the line.

    and a wonderful section, maybe at the end of the second volume, or first volume where he shows how path integration of 1/z gives an explanation of the multivaluedness of the logarithm, according to which route around the origin the path takes.

    or the little section at the beginning of the second volume where he explains about the formula for volume of a parallellepiped, using determinants.

    or the section in vol2 where he explains clearly the meaning of a derivative of several variables.

    just take what you want. do as many problems as you can.

    enjoy it. if you set yourself the task of reading it all in order, or not allowing yourself to enjoy the end before the beginning, you run the real risk of never going anywhere with it and giving up completely.

    of course there are exceptional people like some of my friends who just plowed right through a big book, and did all the execises. and some of them are famous now, some not.

    i have also beneifted from time to time by finally reading something basic that i should, but trying to read everything cannot be allowed to become an excuse for not reading anything.

    this book may have been too hard for most of my classes. i did not get back my evaluations for last semester yet, when we used it.

    i know that. but i would rather blow some people away than cheat the really ambitious people. also even those who are blown away now, stand to learn the stuff in future, moreso than if they never saw it done well at all.

    some students erroneously think a class is about getting a good grade now, and i think it is about exposing them to powerful ideas that will grab them and never let go until they bear fruit.

    when i took calc, i was blown away, or thought i was. then i compared notes at christmas with a friend who went to a well known engineering school. he did not know half of what i had learned. i went back encouraged to learn more, realizing they had higher standards where i was going.
    Last edited: Feb 8, 2005
  15. Feb 8, 2005 #14

    So you did, so you did... I have an excuse but I'll spare you :biggrin:
  16. Feb 8, 2005 #15
    thanks a lot. Frankly if I did every single problem in the book, it would take years for me to finish. What I am doing is skipping to other sections, really understanding them, and doing a couple of problems. Is this discontinous, or would you say that this is allowable? Because Schwarz inequality and all of the stuff in the beginning are not important for me right now. I first want to glean information about the important topics, morever enjoy it. Do you think this is a reasonable way to study Courant?

    Thanks a lot :smile:
  17. Feb 8, 2005 #16


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    sure read whatever you want. i always skipped that horrible section on cauchy schwartz myself.

    if you want to learn calculus though eventually you need to read about continuity, pages 46-56, and integration 76-86, and differentiation, and their connection, on up through page 120.

    as for cauchy shwartz, it follows from the law of cosines, which itself follows from the pythagorean theorem, that if v and w are vectors then v.w = |v| |w| cos(t), where t is the angle between them, and v.w is their dot product. hence since |cos(t)| is never greater than 1, we get |v.w| <= |v| |w|. thats the c-s inequality.

    apparently we are talking about vol 1 here.
  18. Feb 8, 2005 #17
    yes. I plan to finish vol 1 within a week and go to vol 2. I am reading about the important topics.

    Thanks a lot :smile:
  19. Feb 9, 2005 #18
    mathwonk, how does, in your opinion, Courant compare to Apostol?
  20. Feb 9, 2005 #19


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    well i like em both very well. apostol as i recall is more precise, having been written 30 years later, in the era of careful mathematical writing, the 1960's, whereas courant was a pioneering text from the 30's and is by a guy who is also a writer on mathematical physics. so it has more applications. but i think apostol has some of those too. apostol is more meticulous and dry, courant has more charm.

    so courant is not by modern standards as "rigorous" as apostol, but the essentials are there.

    apostol is incredibly scholarly and careful, and clear too. both of them start with integration instead of differentiation, the correct historical ordering.

    gosh I cannot pick one over the other, but i'm leaning towards courant.

    to be honest though, as a young student, courant was hard for me to get far in.

    i did not realize i should read it in palatable pieces, and got discouraged by not making it through the dense parts.

    apostol can do that to you too.

    as a teacher though, i was amazed at how carefully apostol covered all the bases.

    i myself have seldom learned anything straight out of one book, at one session. everything has to go around and around for me, and settle out, in ways that are partly psychological.

    so I use more than one book.

    young people really like spivak though. he works hard to make it appeal to the very bright, but also naive young student.

    i actually learned much of calculus for the first time out of spivak, while grading the course as a grad student.

    then years later as a teacher i realized that the same stuff was in courant, and that spivak had just cleaned it up, and repackaged it.

    then more years later, i read apostol and was again impressed that i was still learning a lot i still did not know.

    you cannot go wrong with any of those books. but don't feel left out if they do not suit at first encounter.

    shoot, my best math buddy at harvard, taking the elite honors calc class said his favorite book was silvanus p thompson's calculus made easy, and i like it too.

    it took me years to realize that the down home funky stuff thompson says is actually right though, because he didn't prove anything. i like proofs.

    stewart can be useful too, and thomas, or edwards and penney. i sniff at some of these as cookbopoks, but I am willing to learn anywhere i can.

    i get a little out of each source, whatever that source does well.

    there is no oine source in general that does everything best.

    (I might make a one or two exceptions in my own very specialized field of research, as there are a couple of experts who have written some great texts on things they excel everyone at.)

    but the perfect calculus book does not exist.

    I like Spivak, because it spoke clearly to me, and allowed me to make the amterial my own, so that I no longer need to refer to the book. When someone gets beyond a book, and begins to pooh the very book he used to value, that isa big compliment to that book. i.e. the book gave him all he needed and allowed to amke it his own, so that he no longer needs the book. then the book falls away and becomes superfluous. that is a good book.

    spivak did that for me.

    but i got some ideas from courant and some from apostol.

    So it is one thing to say how scholarly a book appears to an old person, and another to say which one a young person can best learn from. that is a personal matter. One of my friends however who is a professor of topology learned very well from apostol as a student at MIT. Spivak was used at Chicago recently, and Courant was used in the 60's at Harvard.

    I myself own them all, as well as Joseph Kitchen's fine book, and also G.H Hardy's classic, Pure Mathematics, and also Jean Dieudonne's Foundations of Modern Analysis, and also Lynn Loomis' Advanced Calculus, and also Serge Lang's Analysis I and II. I especially like Lang's books. They are so clear, and to the point.

    I also own Calculus Made Easy, Stewart, Edwards Penney (6 different editions), Lipman Bers book, Thomas (several editions), and am recently trying to incorporate the ideas of lebesgue integration into my grasp of calculus, from books like Rudin's Real and Complex Analysis, and Riesz Nagy's Functional Analysis.
    Last edited: Feb 9, 2005
  21. Feb 10, 2005 #20
    Thanks for the insightful response. I really appreciate it.

    I'm currently using Apostol and I'm find the first few chapters a bit dry. I've already studied the topics of the first 10 chapters in school, but I intend to go through them quickly to learn the basics in a more rigorous manner before proceeding to the chapters I know very little about. Hopefully it won't be too difficult.

    I'm finding the proofs of seemingly 'trivial' things tedious and hard -- like, for example, proving that the ordinate set of a nonnegative bounded function is measurable and its area equals its integral. I realise that this is an important result, but I'm really not enjoy proving this like this analytically, if you know what I mean.

    I'll just trudge along in the meanwhile and wait till I graduate and go to college, where hopefully I'll have access to several books and get a wider point of view. :)
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