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Calculus post Vector Calculus?

  1. May 6, 2015 #1
    Are there fields of "pure" Calculus that follow Vector Calculus? I mean fields that are not primarily ODE, PDE etc, Real Analysis or Complex Analysis, topology etc. Does Vector Calculus and its prerequisites fully encompass the basics of Calculus as a field?
     
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  3. May 6, 2015 #2

    jedishrfu

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    Theres tensor analysis and differential forms.
     
  4. May 6, 2015 #3
    Concise, thank you jedishrfu. May the force be with you.
     
  5. May 6, 2015 #4

    jedishrfu

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    It was particularly strong two days ago :-)
     
  6. May 6, 2015 #5

    micromass

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    Calculus of variations might qualify.

    And of course, there are many calculus subjects that can be explored more indepth than what is usually seen in calculus courses. For example, there are other techniques of solving integrals than the ones usually seen (ie substitution and by parts). One that comes to mind immediately is differentiation under the integral sign.
     
  7. May 6, 2015 #6
    I see micromass. Would you consider the Residue Theory from Complex Analysis as applied to real integrals to fit this description as well?
     
  8. May 6, 2015 #7

    micromass

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    Perhaps. But I think it has a very different flavour than usual calculus. So I wouldn't consider it part of calculus.
     
  9. May 6, 2015 #8

    jedishrfu

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    And Calculus of Variations...
     
  10. May 6, 2015 #9

    SteamKing

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    Why not? The Residue Theorem has got it all: integrals, infinite series, limits, Stokes' Theorem, etc. Seems like those topics are "usual" calculus. :wink:
     
  11. May 6, 2015 #10

    micromass

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    I guess it's a matter of taste. I understand why you would include them in usual calculus, but I don't feel it that way.
     
  12. May 9, 2015 #11
    Would vector calculus also include differential geometry?
     
  13. May 10, 2015 #12

    FactChecker

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    You might get better answers by specifying areas of interest and applications than by trying to classify subjects this way.
     
  14. May 14, 2015 #13
    Just trying to get an idea of what micromass was talking about here:

    From my perspective it seems as though the edges of "pure" calculus get a bit fringy as they head off into other fields; some applications are not clear cut as one discipline or another.
     
  15. May 17, 2015 #14

    mathwonk

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    just restricting to differential calculus, this is the theory of approximating non linear functions locally by linear ones. the term vector calculus' usually means to me this theory carried out in a finite number of dimensions. if this is your experience then the enxt stage woulod be infinite dimensional (banach space) differential calculus,. which also encompasses calculus of variations.

    then there is integral calculus, which also can be done in finite dimensionalm vector spaces or on m anifolds, or in general measure spaces.
     
  16. May 18, 2015 #15
    Interesting. This is how you would generally define differential calculus?
     
  17. May 18, 2015 #16

    hunt_mat

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    There are Sobolev spaces, measure theory and something which I have been involved in called geometric algebra and geometric calculus.
     
  18. May 18, 2015 #17

    phion

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    Complex analysis is, in my humble opinion, one of the coolest, and beautiful branches of mathematics that subsequently requires the entire calculus sequence to fully comprehend. It's a tremendously powerful and useful mathematics with many applications relevant to calculus inside pure math as well as physics and engineering.
     
  19. May 19, 2015 #18

    hunt_mat

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    It is however constrained to two dimensions only.
     
  20. May 19, 2015 #19

    micromass

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    There is complex analysis in multiple variables too.
     
  21. May 19, 2015 #20

    hunt_mat

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    Quaternion analysis? Or several complex variables?
     
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