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Spivak's level vector calculus book

  1. Dec 15, 2011 #1
    Besides Apostol 2, is there any good, rigorous and suited for self study book on this subject? Thanks
  2. jcsd
  3. Dec 16, 2011 #2
    I assume that the title of your post means that you're aware of Spivak's Calculus on Manifolds, but are looking for other texts. Here are three you might consider:

    1. Analysis On Manifolds by Munkres. Covers topics similar to Spivak, but in a more leisurely fashion.
    2. Advanced Calculus: A Differential Forms Approach by Edwards. Again, similar to Spivak, except introduces differential forms right at the beginning.
    3. Functions of Several Variables by Fleming. More advanced than the other texts (uses Lebesgue integrals), but is still intended for undergrads.

    You can search inside all three texts at Amazon to get a better idea. IMHO, some of the reviews on Amazon are not very helpful. If I had to choose one for self-study, I'd go with Munkres.
  4. Dec 16, 2011 #3


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    Courant and John - 2 volume Introduction to Calculus and Analysis is quite good. Years ago I followed Spivak Calculus with Volume 2 of Courant and John at university so I have some fondness for it.
  5. Dec 16, 2011 #4
    Loomis and Sternberg is a very nice free book. Its quite challenging though.
  6. Dec 16, 2011 #5
    I would recommend taking linear algebra and maybe even some real analysis before attempting Spivak, Munkres, or Edwards.

    Unfortunately, I don't think that there is a multivariable calculus textbook in the same style as Spivak's Calculus. You might like Vector Calculus by Marsden and Tromba. It's in a style that is more rigorous than Stewart Calculus, but not as rigorous as Spivak's Calculus. There will be a new version sometime this month that is supposed to contain the typical Definition, Theorem, Corollary layout.
  7. Dec 17, 2011 #6
    I thought the OP was referring to Spivak's Calculus on Manifolds, but it's more likely he meant Spivak's Calculus. If so, then my suggestions are inappropriate. Fellow posters: Spivak, Apostol, Rudin, et. al. authored more than one book. You'll get better replies if you're more specific.
  8. Dec 17, 2011 #7
    Thanks for all who helped, and sorry about my bad question. I'll explain it more specifically:

    I've finished a first course on single variable calculus, and I intend now to review some parts using Spivak's Calculus (not calculus on manifolds), which I found to be a very nice read and far better than the books I used.

    As in the next semester I'll be starting multi-variable calculus (calculus B/2), I'm looking for a good textbook, not so easy as stewart's, but not so hard that I take hours on each page.

    Petek: I'm aware of Spivak's Calculus on Manifolds, but does it cover a standart Calculus B course? It seems so small to fit the whole subject of most other books i've seen, like stewart's 2, or marsden vector calculus.

    Sorry if it isn't very clear again.

    [Edit]By the way I know basic linear algebra: maps, inner product spaces, adjoint, orthonormal operators, self-adjoints, basically what's covered in Lang's linear algebra (though not as deep as it's taught there because I used an easier book for a start), and I have nothing on analysis.
    Last edited: Dec 17, 2011
  9. Dec 17, 2011 #8
    Spivak is a little light on computations unless you do most of the exercises. Calc on mainfold is good if you are willing to do the problems. The other books in this thread are also good choices.

    One thing I have learned is that larger books often contain less information than shorter ones. Certainly Spivak has more content then a book like Stewart.
  10. Dec 17, 2011 #9
    Spivak's Calculus on Manifolds is at the level of an upper division math course, probably best used after taking a course in analysis of one-variable real functions. The level is about the same as the other books in my first post. So, it's more advanced than a typical course in multivariable calculus.
  11. Dec 17, 2011 #10
    Functions of Several Variables - Wendell Fleming is my favorite...

    I also like Advanced Calculus of Several Variables by C. H. Edwards
  12. Jan 2, 2012 #11
    Two books:

    Advanced Calculus - Nickerson, Spencer, Steenrod.

    Review by C.B. Allendoerfer
    Am dying to get this book.

    Calculus of Several Variables - Goffman
    This book is almost identical to Spivak's book, except for the fact that the explanations
    are far clearer but I would definitely read both Spivak & Goffman. If anybody reads either
    of these books (Steenrod/Goffman) please post your experience with them, would be nice to read.
  13. Jan 2, 2012 #12


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    Opinions of course differ, but if you don't already have it, I suggest you don't bother with Nickerson, Spencer and Steenrod. I have never met or heard of anyone learning the subject from this book. Loomis and Sternberg is only a little less useless. These books are mainly for showing off, by the authors that is. Sort of "look how abstract and difficult I can make an important subject look!"

    A book that many people in the past considered one of the best is the vector calculus book by Williamson, Crowell and Trotter.

    That link about Spencer et al....from Hugo Rossi's introduction to his own calculus book, which seems never to have caught on, actually tells you how unsuitable the Nickerson book is. I.e. Hugo says there that he taught from it to a class of "exceptionally brilliant" students at Princeton and found it necessary to essentially write another different book to fill out the treatment in Nickerson et al.

    I myself have a copy of Spencer et al, since Lynn Loomis required it for his course at Harvard that I took in 1964. But he never, ever, used even a page of it and I have never learned anything out of it, in spite of opening it several times over the years. His course resembled more the famous book by Dieudonne, Foundations of modern analysis, which I do like, although very abstract.

    Years after I took Loomis's course I realized I had learned almost nothing there except for the very basic and important fact that a derivative should be thought of as a linear map, and the almost trivial fact that in infinite dimensional Banach spaces, not all linear maps are continuous.

    What is the point of teaching differential calculus in Banach space if you never show anyone even one example of the derivative of an interesting function? E.g. since the space of paths has infinite dimension, a natural choice might be the derivative criterion for a smooth path between two points to have minimal length, confirming that a straight line is the shortest path between them. This example is a non trivial topic in its own right, the first one in the calculus of variations, treated in Courant, and nodded at in Loomis's book, (but not his course).

    But if you are curious, look at all these books, including ones I did not like. We are all different, and you may appreciate something I did not. But I suggest you not pay a large sum of money for an out of print and rare copy sight unseen.
    Last edited: Jan 2, 2012
  14. Jan 2, 2012 #13


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    Spivak's Calc on manifolds is based on the idea tht the most important theorems are:
    i) interchange of order of limits, ii) inverse function theorem, iii) fubini's theorem, and iv) stokes' theorem. so he does a careful job on all of those.

    In fact maybe fubini is equivalent to change of order of limits, so there may be only three main theorems. I myself learned several variables calc from spivak's book.
  15. Jan 6, 2012 #14


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    here is a fair review of the nickerson et al. book by professional differential geometer:


    and here is a rave review of another book i do not know, by barbara and john hubbard:


    here is a copy of NCC for under $15:


    the comment I have always remembered by a famous mathematician about NCC was that it was concerned "more with the ride than the destination".
    Last edited: Jan 6, 2012
  16. Jan 11, 2012 #15


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  17. Jan 11, 2012 #16


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    a professor i respected very much used buck as his text for advanced calculus, and it is certainly one of the well regarded classical texts.
  18. Jan 13, 2012 #17
    Found another one:
    Allendoerfer - Calculus of Several Variables & Differentiable Manifolds
    Probably a bit too tough though but worth looking in
  19. Jan 14, 2012 #18
    I've looked through this one before. Nothing of particular interest: it's a fairly brief book with a horrible typesetting ( before LaTeX ). I would stick to Spivak's Calculus on Manifolds if you were going to resort to this book
  20. Jan 14, 2012 #19
    When you reach the final chapters of Spivak's Calculus on Manifolds, a good book to read in conjunction is:
    "Differential Forms" - Henri Cartan
  21. Jan 14, 2012 #20
    I would recommend Analysis on Manifolds by Munkres. I have that book and Spivak's Calculus on Manifolds, but I find that Spivak's book is too terse.
  22. Jan 14, 2012 #21
    The second chapter is an exposition on existence & uniqueness theorems in ordinary
    differential equations, proven both by Picard iteration & contraction maps, as well as
    a discussion on global integrability criterion for vector fields & a generalization by Frobenius.
    Third chapter contains two proofs of the inverse function theorem, one using Frobenius'
    theorem. After that there's a hell of a lot of a discussion of manifolds I can't recall.
    To say there's nothing of particular interest is a bit much, the typesetting is off-putting
    but it's definitely worth a look if accessible.
  23. Jan 14, 2012 #22
    Hmm..I don't remember that stuff being in the copy I saw. Perhaps there was a difference of editions or my memory has just failed me, sorry. ( I don't mind bad type setting either, another good "advanced calculus" book is the one by Nickerson, except the typesetting is pre-LaTeX as well )
  24. Jan 14, 2012 #23
    No problem, dying to get the Nickerson book you mentioned.
  25. Jan 14, 2012 #24
    Last edited: Jan 14, 2012
  26. Jan 14, 2012 #25
    all things considered, would you guys say that munkres's book is better than spivak's?
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