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Best books for learning differential forms?

  1. Apr 1, 2012 #1
    Can someone recommend a good textbook for learning differential forms for someone with an understanding of calculus at the level of Spivak?

  2. jcsd
  3. Apr 1, 2012 #2
    The obvious choice would be Spivak's Calculus on Manifolds. Since it's rather brief, a "decompressed" alternative that I also recommend is Edwards' Advanced Calculus of Several Variables (a Dover, so it's cheap).
  4. Apr 1, 2012 #3
    How about Cartan's differential forms?
    Other than calling manifold "variety", I heard this is good book.

    Actually, it depends on how much you want to learn differential forms.
    If you just want to learn differential forms living in R^n, spivak is perfect.
    If you want to learn differential forms in general manifolds,
    read Lee, doCarmo, Lang, Abraham/Marsden's manifolds book
    (not Foundations of Mechanics).

    (Lols to translator for not noticing that English-speaking people use the term manifolds instead of variety)
    (variety is okay to stick in algebraic geometry, becuase that's the universal term
    that mathematician uses)
  5. Apr 2, 2012 #4
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  6. Apr 2, 2012 #5
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  7. Apr 2, 2012 #6
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  8. Apr 2, 2012 #7
    Thanks for the suggestions. I'm trying to learn differential forms in preparation for a differential topology class that uses Pollack. I think I'll go with Spivak's Calc on Manifolds.
  9. Apr 3, 2012 #8
    Spivak is the way to go (maybe not for first learning), but you won't regret that you bought it.

    One thing to mention is that Gullemin/Pollack is more concerned with topological aspects of smooth manifolds rather analytical aspects.

    After Spivak (for smooth manifolds books focusing on analytical aspects):
    -For nice but slow spoonfeeding intro, read John Lee (still good to get it because you won't have too much headache and nevertheless this will be your standard textbook).
    -More physical but still rigorous intro would be Abraham/Marsden's Manifolds book.
    -Hardcore treatment would be Lang
  10. Apr 3, 2012 #9
    Would Spivak's Caclulus book prepare one for the exercises in his Calculus on Manifolds?

    I think the other books mentioned assume less knowledge of topology and analysis.

    Also, just in case it's not clear, there are two advanced calculus books by different Edwards, Advanced Calculus: A Differential Forms Approach by Harold M. Edwards and Advanced Calculus of Several Variables by C. Henry Edwards. Henry doesn't cover differential forms until about chapter 5, while Harold starts right off with them.
  11. Apr 3, 2012 #10
    For Calculus on Manifolds, I would say the necessary and sufficient prerequisite knowledge is his Calculus as well as linear algebra and some familiarity with metric spaces. To add to what chhan92 said, after Calculus on Manifolds you may want to look at Spivak's A Comprehensive Introduction to Differential Geometry Vol. 1.
  12. Apr 3, 2012 #11
    Oh I completely forgot about it.

    It is rival of John Lee, but has distinctive pros & cons relatively
    (meaning that both books are somewhat masterpiece)

    John Lee:
    -pros: Relatively easy to learn despite being in GTM (reads like UTM), nice set of examples to work through, topological aspects are not too shallow (he uses smooth covering maps), one of the best book for Lie groups (except books specializing in Lie groups of course)
    -cons: too slow in some sense, not a good reference, relatively not too good to give good insights

    Spivak Vol 1:
    -pros: Nice insights (just like all of his books! (except little spivak, but oh well you need to deduce it!)), Master of exercises!
    -cons (not my opinion, but amazon people's): relatively outdated in terms of teaching

    I have John Lee, but maybe some day I should get Spivak Volume 1
  13. Apr 10, 2012 #12
    You will still want to read spivak afterwards, but a good first book is:
    A geometric approach to differential forms
    It explains in much greater detail (but less rigorously) the concepts of differential forms than Spivak/Munkres. I love Spivak's writing, but calculus on manifolds as a first exposure to the material made absolutely no sense to me (I'm referring to chapters 4 and 5. I had no qualms with Chapters 1-3).
  14. Apr 15, 2012 #13


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    If you are referring to the book on differential topology by guillemin and pollack, there is no prerequisite of differential forms for reading that book. In fact chapter 4 of that book contains an elementary introduction to forms similar to that in spivak's calculus on manifolds.

    still, all these recommendations of other sources seem excellent. I myself would probably read cartan, and i agree bachmann's book succeeds at giving a geometric feel for forms. As one amazon reviewer points out, chapter 7 of arnol'd's mathematical methods of classical mechanics seems to be a precursor of bachmann's book, so arnol'd might merit a look as well.
    Last edited: Apr 15, 2012
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