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Homework Help: Use Stokes Theorem to evaluate the integral

  1. Jul 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Use Stokes Theorem to evaluate the integral[itex]\oint_{C} F.dr[/itex] where F(x,y,z) = [itex]e^{-x} i + e^x j + e^z k [/itex] and C is the boundary of that part of the plane 2x+y+2z=2 in the first octant

    2. Relevant equations

    [itex]\oint_{C} F.dr = \int\int curlF . dS[/itex]

    3. The attempt at a solution

    So first out i calculated the curl and i got [itex]e^x [/itex] K

    Also z=1-x-[itex]\frac{1}{2}[/itex]y
    and[itex]\frac{\partial z}{\partial x} = -1[/itex]
    and[itex]\frac{\partial z}{\partial y} = -\frac{1}{2}[/itex]
    and [itex]\sqrt{(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2 + 1}[/itex] = [itex]\sqrt{\frac{9}{4}}[/itex] = [itex]\frac{3}{2}[/itex]

    To get my limits. when Z=0 the image of the plane on the xy plane is a triangle and so my limits will be x=0 to 1 and y=0 to 2-2x

    so putting all this together i get

    [itex]\int^{1}_{0}\int^{2-2x}_{0} (e^x k). (\frac{2i+j+2k}{3})(\frac{3}{2}) dydx[/itex]

    [itex]\int^{1}_{0}\int^{2-2x}_{0} (e^x)dydx[/itex]
    i have worked out these integrals and i get 2([itex]e^1 +2[\itex])
    this doesnt look right but i dont know where i went wrong. i've gone over it twice
    anyone throw some light on where im going wrong here?
    Thanks for reading!
     
  2. jcsd
  3. Jul 28, 2012 #2
    Everything looks right. I don't think you evaluated the integral correctly though. I get 2e - 4.
     
  4. Jul 28, 2012 #3

    chiro

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    Hey gtfitzpatrick.

    I haven't done these kinds of problems in a while, but I'm wondering if you are trying to normalize the curl, do you have to divide by 3/2?

    I can see you have normalized the plane with the division by 3 (SQRT(2^2 + 2^2 + 1)) and I see how you derived the limits for the triangle in the first octant, but the only thing I'm wondering about is this 3/2 factor.
     
  5. Jul 28, 2012 #4
    The 3/2 factor comes from the dA factor.
     
  6. Jul 28, 2012 #5

    chiro

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    Science Advisor

    Thanks for that.
     
  7. Jul 28, 2012 #6
    Thanks a million,yes your right i got a sign wrong, it should be 2e-4
     
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