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Taking a course in calculus on manifolds.

  1. Sep 5, 2007 #1
    im thinking of taking in 2008 the second semester a course in analysis of manifolds.
    now some of the preliminaries although not obligatory, are differnetial geometry and topology, i will not have them at that time, so i think to learn it by my own, will baby rudin and adult rudin books will suffice, perhaps also for the course itself?

    here's the syllabus of the course:

    don't worry it's in english.
  2. jcsd
  3. Sep 6, 2007 #2


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    there is no differential geometry nor topology in rudin. nor manifolds.

    dieudonne's book, fundations of modern analysis will have all the needed metric space theory and calculus, and shifrin of uga has free diff geom notes online.


    easier books to read than dieudonne also exist on topology for anaysis, like simmons, or kelley.

    advanced calc by loomis sternberg is also free on sternbergs website i believe, and covers also manifolds.
    Last edited: Sep 6, 2007
  4. Sep 6, 2007 #3
    I gave a look at baby rudin, there's a chapter on basic topology, which from the syllabus i think that should be enough, although i guess i need to email the lectrurer on this matter, on how much one needs to know on topology before learning manifolds.

    well, i guess i only need a brief overview on differnetial geometry which the notes might come handy.

    btw, i guess you already took some courses on manifolds, what were your univ preliminaries when you learned it?
  5. Sep 6, 2007 #4


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    when i was a student, calculus on manifodls was part of the sophomore honors advanced calc course, for which dieudonne was one textbook, foundations of modern analysis, and then also loomis and sternberg, and calculus of several variables by fleming.

    i had only one variable calc and linear algebra.

    dieudonne is much more recommended than baby rudin in my opinion. indeed as a book to learn from, or even teach from, i rather dislike baby rudin, although the big rudin has a good solid collection of topics.

    rudin is a pure analyst with no taste for or like for geometry or topology, and what you learn there will not be that useful for understanding topology of manifolds in my view. compare his treatment of differential forms to that of spivak for instance.

    a well written but pricey book covering much of your course topics is the one by guillemin and pollack.
  6. Sep 7, 2007 #5

    Chris Hillman

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    I take it that the course will be taught from the instructor's notes, with no assigned textbook? So you want both some background reading and recommendations for a supplementary textbook?

    The two books by Rudin are "real analysis" textbooks, dealing with analytic topics such as convergence of series, measure theory, and so forth. These books certainly will not come close to serving as suitable supplementary textbooks for a course on calculus on manifolds, nor will they be particularly useful as background reading.

    From your "handle" I assume you are interested in gravitation physics (why not statistics? that's much more interesting and important for the 21st century!), so try these supplementary textbooks to see what's involved:

    Flanders, Differential Forms With Applications to the Physical Sciences, Dover reprint of 1963 classic.

    Isham, Modern Differential Geometry for Physicists, World Scientific, 2006.

    Frankel, Geometry of Physics, Cambridge University Press.

    You probably already have sufficient background to start reading these books. The last named book covers pretty much all the topics mentioned in the syllabus you cited.

    If you have some general topology books handy, one topic to look for might be partitions of unity, but most of that stuff won't be directly needed. It's probably more important to review differential equations and linear algebra to prepare for the course, and getting a leg up on grappling with the notion of an atlas of coordinate charts would be a good idea (see any of the three books above for that).
  7. Sep 7, 2007 #6
    I guess that we will learn from the lectrurer's notes, and he will supply us with recommended textbooks at the first day of the course, as was in every course iv'e learnt so far.
    I'll give a look at the books, although I'm not that akin for applications in physics cause for that i have another book in gravitation waiting for just that.

    I will be taking a course this coming semester in probability, the only course from the statistics department which might be intersting is (although i don't have much knowledge to appreciate it or not) intro to stochastic process, but i have other priorities in courses in maths, and statistics and probability is in the bottom of this list of priorities.
  8. Sep 10, 2007 #7

    Chris Hillman

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    Priorities for Gravitation Theory

    Am I wrong in assuming from your user name that you hope to make a career in gravitation theory, then?
  9. Sep 11, 2007 #8
    well, I want if it would be possible to have a career in pure maths, and also to be knowledgeable in theoretical physics, which includes gravitation as well, if everything goes as planned then i would learn in my third year a course intro to GR which i highy want to take, but because i also take courses in maths im not sure that in my third year it will be possible.
    I learn maths and physics a combined b.sc degree.

    I'm now entering my second year at univ, so because Iv'e postponed the labs a year I would finish this degree after some three years or so from now.
    Last edited: Sep 11, 2007
  10. Sep 11, 2007 #9

    Chris Hillman

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    OK, your interests are broad, that's good. If a chance to study serious statistics comes your way, I still say you should go for it. If you are very ambitious and very good at spotting dubious assertions I can't imagine any career more important to society as a whole than fixing the problems with statistical practice. (Maybe I should have said: I can't imagine any career goal more quioxtic!)
  11. Sep 11, 2007 #10
    the question is what should be in the syllabus of such a course?
    as i said this semester ther's a course in probability which i must take anyway.
    and in the list for elected courses there's a course called statistical theory which its preliminary is the probability course.

    As I said, statistics and probability are in my low priorities, i.e i wont be dissapointed if i will not take an advanced course in statistic and probability besides the one im obliged to take.
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