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Vector circulation. Stokes, Gauss and maybe more?

  1. Apr 9, 2014 #1
    Hello. My first time posting here. So... My question is kinda hard to explain but I will try to. So we all know about the Kelvin-Stokes theorem (not talking about manifolds here) :
    5f65e93751487f9350c194aa5f2bb8de.png
    And we also know about Ostrogradsky/Gauss Theorem :
    fbd758cc6576dd8296bf36e6b7403b02.png [PLAIN]http://upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/23px-OiintLaTeX.svg.png[PLAIN]http://upload.wikimedia.org/math/a/b/9/ab9fd5a4aaa36e402c98cbd36af3a70d.png [Broken]

    I don't have a problem with these 2. I perfectly get them. What bothers me is: can there be found a relationship between the circulation of a vector 5c8f6ef66e3c5a145a37a4e0b7219ce3.png
    and the volume integral ? Just like Stokes relates circulation and curl and Gauss relates flux and div . Now how can a contour enclose a volume ? If you can modify is shape you can(see my attachment ) THat's just an idea, I dont know if it's been done or it can be done. And while you can modifiy the shape of the contour to get different surfaces of different areas you can only get one sphere of only one volume, speciffic to the length of the contour. I'm sorry if I was inchorent, just ask me if you want me to try and formulate my question in a different way although I tried to explain it as good as I could.
     

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    Last edited by a moderator: May 6, 2017
  2. jcsd
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