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Stokes theorem

  1. Feb 13, 2007 #1
    For the surface S (helicoid or spiral ramp) swept out by the line segment joining the point (2t, cost, sint) to (2t,0,0) where 0 is less than or equal to t less than or equal to pi.

    (a) Find a parametrisation for this surface S and of the boundary A of this surface.

    I can only guess that
    x=rcost
    y=rsint
    z=t
    0 less than or equal to t less than or equal to pi
    0 less than or equal to r less than or equal to 1
    Is this right or totally bogus?

    (b) For the vector field F=(x,y,z) compute the flux of F through the surface S. Assume the normal to the surface has a non-negative k component at t=0.
    No idea because what is the surface S? It has no equation.

    (c) Compute integral (F*dr) where A is the boundary curve of the surface S and F is the force field (x,y,z).
    I think I just use Stokes Theorem for (c), but I'm having trouble setting it up since again I have no equation for the surface S. I also don't know how the whole line segment joining works into it.

    I can evaluate the integrals, I just have trouble setting them up. If anyone could help me with these three I would appreciate it. Just explain what's going on.
     
  2. jcsd
  3. Feb 13, 2007 #2

    Dick

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    You have just given a parametric form for the surface, x(r,t)=r*cos(t), y(r,t)=r*sin(t),
    z(r,t)=t. How can you say S has no equation?
     
  4. Feb 13, 2007 #3
    Well is that right for a helicoid?
    And how does the line segment joining those two points play into the helicoid?
     
  5. Feb 13, 2007 #4

    Dick

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    The parameter r sweeps out the line segment. r=0 and you're on the z-axis, r=1 and you are on the helix. r=in between and you are in between.
     
  6. Feb 13, 2007 #5
    So does it factor into the bounds of integration?

    I guess I don't see what the 2t means for the x part also?

    It just seems when I do the integrations out, it's going to be simply
    t between 0 and pi
    r between 0 and 1
    for both parts (b) and (c) if I use parametrisation.

    Does the line segment joining the point (2t, cost, sint) and (2t,0,0) somehow give me the unit normal n which I have to find?
     
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