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Teaching Myself Vector Calculus, need some advice.

  1. May 13, 2010 #1
    I am a physics major currently finished with my second year and I am trying to teach myself vector calculus since the Calculus III class at my university did not include it and I am taking upper level electromagnetic physics courses at another univeristy this coming up fall semester. So far I have completed Calculus I, II, and III, Ordinary Differential Equations, and Linear Algebra all with A's. I am pretty avid at mathematics and have been able to teach myself most of what the teachers could not with fair success.
    The book I am using for vector calculus is Thomas' Calculus 11th edition, which includes a chapter dedicated to Vector Analysis. In order to make sure I am understanding everything right and can solve some problems I would like to have a supplementary book on vector analysis but I don't know what to pick, what is your advice? The Thomas book leaves off a lot of proofs by stating "in more advanced texts..." which is a pain most of the time. I would not like to buy a separate calculus textbook but rather a specialized book on vector analysis that will help me as a physicist and a mathematician. The topics covered in the Thomas book are:
    1. Line Integrals
    2. Vector Fields, Work, Circulation, and Flux
    3. Path Independence, Potential Functions, and Conservative Fields
    4. Green's Theorem in the Plane
    5. Surface Area and Surface Integrals
    6. Parametrized Surfaces
    7. Stoke's Theorem
    8. The Divergence Theorem and a Unified Theorem

    Advice please! Thank you.
  2. jcsd
  3. May 13, 2010 #2

    you could try Stewart Calculus. I learned vector calc out of that, and it was a fairly straightforward read. plenty of examples, which really helped for me.
  4. May 13, 2010 #3


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  5. May 16, 2010 #4
    this is an free online lecture website of vector calculus
    its from MIT - Massachusetts Institute of Technology
    they post their lectures online, so people can learn by themselves for free.
    there are many other lectures posted by them if you are interested in it.

    for vector calculus click on the link:
  6. May 16, 2010 #5
    There are two separate things here....

    1) if you want more practice at the practical parts of doing problems (i.e. an engineering approach) get the "Vector Analysis Problem Solver"

    2) if you want a stronger foundation on the theory behind vector calculus. I don't know of any great books, but one good book is "Advanced Calculus: A Differential Forms Approach" by Edwards. Also "A Course in Mathematical Physics" by Szekeres is good as an encyclopedia of advanced mathematics, but I don't know if it makes a good textbook.

    Edwards is great because you'll learn how most of what you've learned about vector calculus thus far was taught incorrectly. :-) :-) :-) Most vector calculus courses are designed for engineers and applied physicists to solve problems and they end up using different set techniques and language than what theoretical physicists use. One problem with the "engineering" language of vector calculus is that it doesn't generalize well to higher dimensions.

    There may be better books out there, so I'll be happy is someone points me to them.

    Also I'd like to start a wiki page at wikiversity on this sort of thing. One thing that I've noticed is that there are some great "gold nuggets" out there on the web, but no one has gathered them up.
  7. May 16, 2010 #6
    Something else that you might or might not want to do is to take a look at text books at different decades and see how vector calculus has been taught at different times. It's interesting to see how mathematicians develop new "technology" and this technology makes it into the textbooks.

    Right now what it looks like people are trying to do is to start teaching in the language of differential forms.

    (I just caught the reference to Hubbard and it looks like a good textbook.)
  8. May 16, 2010 #7
    Thanks to everyone thus far with your advice. I have taken a look at MIT's opencourseware before and find it very helpful. And that Advanced Calculus textbook looks intriguing. Has anybody used the book: Introduction to Vector Analysis by Harry F. Davis and Arthur Snider? I hear its more on the applied side but a friend recommended it to me, any thoughts?
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