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A Stokes' theorem on a torus?

  1. Apr 27, 2017 #1
    I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in the equation, the author says the integration of ##F## over the whole torus is the same thing as the difference between the two line integral along C1 and C2. Is this a case of stokes' theorem? I don't understand how C1 and C2 is the boundary of S. Please help.
    Untitled-1.png
     
  2. jcsd
  3. Apr 27, 2017 #2

    jedishrfu

    Staff: Mentor

    Notice they using radial measure on the horizontal axis and they mention that the difference is ##2\pi##

    That means C1 is a closed loop and C2 is a closed loop.

    Does that make sense?
     
  4. Apr 27, 2017 #3
    Sorry for the confusion, the horizontal axis is parameterized with angle, both C1 and C2 are closed loops because the left and right sides are identified.
     
  5. Apr 27, 2017 #4

    jedishrfu

    Staff: Mentor

    So what don't you understand about the boundary of S? It's like a ring with C2 as the outer boundary and C1 as the inner boundary.

    When they plot it using radial measure S looks like a rectangle.
     
  6. Apr 27, 2017 #5
    So this is not a torus? I though the upper side and bottom side are identified as the same edge.
     
  7. Apr 28, 2017 #6
    Stokes theorem for torus: ##\int_{\mathbb{T}^m} d\omega=0## :) The torus does not have boundary
     
  8. Apr 28, 2017 #7

    lavinia

    User Avatar
    Science Advisor

    I do no think this is a torus. Rather, it seems to be a cylinder. The two vertical edges seem identified but the top and bottom do not. If so, then it is a case of Stokes Theorem because the boundary of this cylinder is ##C_1-C_2## or ##C_2-C_1## depending on the orientation.

    If the top and bottom were also identified then you would have a torus but then the integral of ##dA## would be zero since as zwierz pointed out, the boundary of a torus is empty.

    Another way to think of it is that if ##C_1## and ##C_2## are identified to make a torus then the integrals of ##A## along them would be the same except for a sign and would cancel out to give zero.
     
    Last edited: Apr 28, 2017
  9. Apr 28, 2017 #8
    Hi guys, Thanks for the help, I think it really is a ring not torus.
     
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