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Geometry Spivak's differential geometry vs calculus on manifolds

  1. Jan 19, 2016 #1

    I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first. Also, how does the exposition between those two books compare? Are they very similar?
  2. jcsd
  3. Jan 20, 2016 #2
    They are extremely similar. In particular, the first volume of Spivak's Differential Geometry saga elaborates on some specific examples from Calculus on Manifolds, which themselves were elaborations of specific examples from his Calculus. So the books follow one another well as a coherent whole. You should probably at the very least get to the section on integration on chains and orientation before starting the first volume, but there is no great necessity. Although they follow well, it is not necessary to finish Calculus on Manifolds first, especially if you have encountered basic aspects of topology and differential geometry before.
  4. Jan 21, 2016 #3
    I am currently using Hubbard/Hubbard as a gentle introduction to the manifolds. Is this book a good replacement for the Spivak's Calculus on Manifolds, or should I still read Spivak first before going into the differential geometry books like Spivak and Lee? Also is the college geometry a necessary prerequisite for any book in the differential geometry?
  5. Jan 21, 2016 #4
    Hubbard/Hubbard is a fine, sort of more verbose, version of Spivak's Calculus on Manifolds, so it's fine to go right to differential geometry introductory texts afterwards. If by college geometry, you mean Euclidean and non-Euclidean geometries, it is nice to know the axiomatic and detailed schematics of different types of geometries, but it's not strictly necessary, as differential geometry takes its own road down that path. As usual, however, studying geometry from more than one point of view will greatly enhance the richness and depth of your understanding.
  6. Jan 21, 2016 #5
    Thank you for the advice. Receiving your advice, I think I will pass the reading of Spivak's calculus on manifolds and read Lee then. I have not yet taken any course in geometry (probably not in my undergraduate due to time), but I am thinking about picking a book that briefly summarizes both the non-Euclidean and Euclidean geometries since they appear on my computational biology.
  7. Jan 21, 2016 #6
    Thanks for the input! This is my first encounter with differential geometry and topology, so I think I will finish calculus on manifolds first. One more question, there were some parts of calculus on manifolds where I thought the explanations were a bit too brief, would it be similar in his other textbooks? I am self teaching so I don't have many other resources.
  8. Jan 22, 2016 #7
    Calculus on Manifolds may seem exceedingly brief because it is essentially just an extension of his Calculus text to multivariable functions, vector-valued functions, and finally functions between smooth manifolds. He won't go over the things he has already gone over in his Calculus text. Also, Spivak tends to find axiomatic explanations, or explanations based in purely mathematical concerns, more satisfying than explanations that allude to physical concerns, as other books might use for connecting the abstract work to physical applications the student may be familiar with. Spivak writes generally for the student who is satisfied with purely mathematical (non-contradictory logical) reasons for mathematical definitions and structures, and leaves applications to other, more specialized textbooks. He also leaves the development of crucial ideas to series of step-by-step problems in his exercises, so skipping an exercise or two can sometimes cause a hole in understanding later on, which may make the text seem too dense.
    While his other texts are not as dense as Calculus on Manifolds, he does continue to include necessary exercises in his problem sets. This is the usual format of mathematics texts, however, as many complicated structures don't make much sense until you get into the convenience of certain ideas yourself when working through a problem.
    If you find something confusing in any mathematics text, however, the internet is a great resource for alternative explanations. This forum itself has a sticky note at the top which links to several excellent free mathematics textbooks on calculus and differential geometry.
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