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Book recommendation for studying multi-dimensional analysis?

  1. Mar 14, 2012 #1
    I'm in an introductory analysis course, and next quarter we're doing multi-dimensional analysis. So far, we've done single variables and I've done well with Spivak's Calculus. However, next quarter, we're using this book:

    [1] https://www.amazon.com/Advanced-Calculus-Gerald-B-Folland/dp/0130652652

    Based on the Amazon review, this looks like it sucks.

    Is there any book you would recommend I could supplement studying it? I tend to learn very well when the rigor in stuff is very clear - with consistent definitions - and based on the reviews, unfortunately, this book sucks at this...

    Also, I've had no multi-variate calculus before, if that influences your recommendation...
  2. jcsd
  3. Mar 14, 2012 #2
  4. Mar 14, 2012 #3
    Considering crat has already successfully studied from Spivak's Calculus, Stewart's book would be highly inappropriate. He obviously isn't looking for a cookbook calculus text.

    That being said, Spivak's Calculus on Manifolds is certainly an option. But if you haven't had multivariable calculus before, this may be rough going: it is extremely concise. However, Spivak (as you probably already know) is very clear and insightful. Ideally, you could check it out from a library and see if it suits you.

    A similar, but much less condensed account is Edwards' Advanced Calculus of Several Variables (a Dover, so it's cheap too), which I also recommend highly. It is rather gentler on the reader, starting with a review of the requisite linear algebra and topology. Speaking of which, have you taken linear algebra yet? Because linear algebra is hugely important in rigorous multivariable calculus...
  5. Mar 14, 2012 #4
    Many members have said Apostol (same rigor as Spivak) book II is good.
  6. Mar 14, 2012 #5
    And as I understand crat is learning new material, neither does he want a needless mix of topology and quantum math like analysis texts with no goal at all but distract the user from the title. Crat WANTS a first-course multidimensional text, and these texts I mentioned are the standard, not the cheapest though. Dover's Advanced Calculus is okay for advanced calculus topics but it does not cover topics deeply. A beginner needs a text with nice illustrations especially in subjects like multiple integration and vector calculus. If the advanced thing is a must then I would rather suggest this:


    To learn math, or even anything, start simple, learn the basics, don't push yourself into "rigorous" proofs and explanations because simply no one (majority at least) will be able to grasp it right from the beginning. The more complicated stuff comes after you master the basics, and when your brain is mature enough to do rigorous math you will do so in a more advanced specialized courses. BTW I still consider analysis is an outdated title for calculus. Better get rid of antiquated terms.
  7. Mar 14, 2012 #6
    Just because the material is new to him doesn't mean that he shouldn't or can't learn it properly; the fact that he has worked through Spivak's Calculus suggests that he has the mathematical maturity to translate between rigor and intuition.

    Dijkarte, why do you say that Edwards doesn't cover topics deeply? It sure covers them more deeply than, say, Stewart. And for the record, there are plenty of illustrations as well.
  8. Mar 14, 2012 #7
    By the way, Apostol is also a very good book. It depends on what you are looking for, as Apostol sticks to a very concrete, classical approach, and consequently doesn't even mention differential forms. (Rather, you'll see a lot of div, grad, curl, and all that.)
  9. Mar 14, 2012 #8
    Also, if you need to learn linear algebra, I suggest either Hoffman and Kunze's Linear Algebra or the relevant chapters in Artin's Algebra.
  10. Mar 14, 2012 #9
    Not Edwards in particular, but generally speaking I found that Advanced Calculus texts don't go into the details of the basics, not many examples, not many illustration.

    For someone new to the topic, they need to get it right, regardless of how skilled or gifted they. And I'm not underestimating cart's abilities at all :)
  11. Mar 14, 2012 #10
    Have you seen Edwards? There are plenty of illustrations, and the level is certainly appropriate for someone coming off of Spivak, even without any prior experience with multivariable calculus. It can definitely be used as an introduction, as long as you have experience with proofs and linear algebra.
  12. Mar 14, 2012 #11
    For linear algebra I highly recommend Elementary Linear Algebra by Bernard Kolman or Howard's. Both are pretty good introductory texts. Don't do it all one course. I mean linear algebra is not a sophomore course. Unfortunately many linear algebra courses are taught as a single course for second year students screwing up the beauty of the topic.
    Take one introductory which should focus on linear equations, matrix calculations and basic vector algebra in R^3. This should be taken before heading to more advanced topics like vector spaces and generalized vectors.
  13. Mar 15, 2012 #12
    Since the author has already gone through spivak's calculus, he is not "pushing" himself into rigorous proofs. After going through spivak's calculus, I do not see why anybody would want to settle for Stewart's book as the next step. Also, the thread title is multi-dimensional *ANALYSIS* -- this is a specific term, and it usually means calculus "done right".
    And topology is CERTAINLY not "needless" in the mix. Knowing basic topology is absolutely important at this point in time ( for example, integration theory and the inverse function theorem ). I'm sorry, but I think you are giving bad advice, especially if the author is actually looking to learn strict mathematics.

    Calculus on manifolds (Spivak ) and Analysis on Manifolds are good books to go through next

    P.S. beginning topology is hardly "quantum mathematics", whatever that means. It's pretty basic, if you understand delta-epsilon ( which I assume the author does ) then you'll get through it like nothing.
  14. Mar 15, 2012 #13
    With sufficient mathematical maturity, you can start with a book like Axler's "Linear Algebra Done Right". This is actually very reader friendly, and you can learn a lot from it.
  15. Mar 15, 2012 #14
    I'm not going to argue my advice but simply look it up yourself what the best texts are on calculus, and I don't think there's something called calculus "done right" and calculus "done wrong."

    Delta-epsilon is explained in the texts I referenced, it's not rocket-science. :)

    Could you please check what some good universities require for undergraduate calculus? Cheap books like Dover's are good but they lack of the teaching strategy.
  16. Mar 16, 2012 #15
    I would go for Apostol Volume II Part 2 (Nonlinear Analysis).

    Spivak's Calculus on Manifolds is, imho, too dense and brief. It's a book for someone who already "knows" the topic, not for a first approach.

    However, go and ask your teacher. He/she will give you the best possible advice.
  17. Mar 16, 2012 #16
    I agree with everything A. Bahat and wisvuze said... Spivak's Calculus on Manifolds and Edwards is the way to go..
  18. Mar 17, 2012 #17
    Ha! We used Folland in my Advanced Calc class. It is not a bad book, just not very friendly. There are gems in it, but for the most part it is better as a reference than as a learning book.

    If you are familiar with basic analysis, the first quarter of the book shouldn't be too bad. The rest of it is ok for learning the nuts & bolts of (fairly rigorous) applied multi-dimensional analysis, if your teacher is solid.

    For the person recommending Stewart's Calc III book, it is a completely different beast. You should already know that stuff before doing Folland.
  19. Mar 18, 2012 #18


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    Are you serious??? The OP finished Spivak and you suggest him to read Stewart??? Really??
    I don't think you should give advice if you don't know what you're talking about...

    The OP should probably go for Spivaks calculus on manifolds book, or Edwards.
    And Lang for linear algebra is also not bad. Axler or Hoffman/Kunze are good as well.
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