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A book After Spivak: Calculus

  1. Mar 8, 2013 #1
    I'm finishing up Spivak, after a break from "pure" math. I'm looking for a good follow up on multivariate analysis. I've heard that "Calculus on Manifolds" uses a lot of linear algebra (which I know very little of), so I'm on the look out for another suggestion.

    I'm looking for a book on the analysis side, rather than computational, in the spirit of Spivak (which is rather challenging, yet rewarding!).

  2. jcsd
  3. Mar 8, 2013 #2


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    Courant volume II( Calculus) might be want you are looking for. I think you'll find a lot of similiarity between the way Spivak and Courant write and introduction material.

    However, I do suggest you hold off on proof focused multivariate book, until you have a decent foot hold in linear algebra. I think, in many ways, the concepts form a more clear and concise picture when you can look at the subject from the lense of linear algebra.
  4. Mar 8, 2013 #3


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    I doubt there exists a multivariate analysis book that doesn't heavily use LA. On the most basic level, arguably the most important theorem in multivariate analysis, the inverse function theorem, is itself one that relies on linear algebra.
  5. Mar 8, 2013 #4
    Alright, so, I think I'll use Stewart for multivariate (computational).

    Can you recommend a LA book? And after this, I can move on to a proof based text, correct?

    By the way, I like the smell of Spivak's book. o_O
    Last edited: Mar 8, 2013
  6. Mar 8, 2013 #5
    I really like Axler, Linear Algebra Done Right. You can go to the math subtextbook forum to read reviews and opinions of it.
  7. Mar 8, 2013 #6
    Last edited by a moderator: May 6, 2017
  8. Mar 8, 2013 #7
    x2. I think it will satisfy both your computational and theoretical needs. Beautiful book. Everything is well motivated and there are even many interesting applications. This book gets me incredibly excited about math every time I read it. If you include the appendix, you will be busy for a while.
    Last edited by a moderator: May 6, 2017
  9. Mar 8, 2013 #8
    I'll check this one out, thanks.
    This is pretty expensive, is the 2nd edition really better than the first? I'd like to save some money, if possible.
  10. Mar 8, 2013 #9


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    There is also the much more gentle but unfortunately very computational "Analysis on Manifolds" - Munkres that you can take a look at.
  11. Mar 8, 2013 #10
    I'm really looking for a theoretical pure math approach to it. I've got a book (two in fact) for computational, which is good enough for physics, I suppose, I just happen to enjoy pure math.

    I'm thinking of two different approaches. 1. Buy the all in one book, or 2. buy Linear Algebra Done Right, and I'll buy Spivak after.

    Has anyone taken a look at Calculus on Manifolds from Spivak?
  12. Mar 8, 2013 #11


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    Yes I am rather well acquainted with the book. The exercises are much better than those in Munkres but it isn't the best multivariate analysis book out there; it is rather unmotivated and is not as good as his amazing single variable calculus book. Still, it's better than Munkres in my opinion. There aren't many epsilon delta proofs in it which is a rather large disappointment since they are so fun, even in higher dimensions, but he has a lot of important elementary results regarding differential forms (and more importantly integration of forms which is so important in physics that you just HAVE to look at it from a pure math perspective). Micro's suggestion of Hubbard is probably the best at the level you are in considering Spivak makes use of a lot of linear algebra that he assumes the reader already knows.
  13. Mar 8, 2013 #12


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  14. Mar 8, 2013 #13


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    Even if you get another book, Advanced Calculus by Lynn H. Loomis and Shlomo Sternberg is very good at a great value of 0$.
    It has self contained basic linear algebra.

    Many books are good.
    Vector Analysis by H. B. Phillips is good, but out of print. General Vector and Dyadic Analysis by Chen-To Tai is worth looking at, particularly for its coverage of common errors people make, but way too expensive to buy. Vectors, Tensors and the Basic Equations of Fluid Mechanics by Rutherford Aris is cheap and worth having even though it is a bit old fashioned, the first three chapters cover vectors, later chapters unsurprising cover tensors and the basic equations of fluid mechanics.
    Last edited: Mar 8, 2013
  15. Mar 8, 2013 #14
  16. Mar 8, 2013 #15
    You can buy the current edition straight from the publisher (Matrix Editions) for approximately $70. I don't know why everyone charges so much for this book (even used on Amazon). This book is very worth the $70 (I usually won't pay more than $15-20 for a book). But I would jump on that $25 older edition.
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