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Math Help - Circulation

  1. #1
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    Circulation

    Since the field vectors F(x,y,z) = (x-y)i + (y-z)j + (z-x)k and the curve \alpha that is the intersection of surfaces:
    S1: x+y+z =1 and S2: x^2+y^2=1
    Find the circulation of F arrond the \alpha

    My solution:

    curl F = (1,1,1)

    Surface S1:

    Normal: N = (0,0,1)

    \int_0^1 \int_0^{1-v} dudv = \frac{1}{2}

    Surface S2:

    Normal: N = (cosu, sinu, 0)

    \int_0^{2 \pi} \int_0^1 (rcos \theta + rsin \theta)rdrd \theta = 0

    Circulation:

    \int_C F.dr = \int \int_S F.N.da = S1 + S2 = \frac{1}{2}

    It is correct ?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    No-- you kind of missed the point

    alpha is the curve of intersection of the plane and the cylinder

    See the attachment for the details where the line integral is computed directly

    If you use Stokes theorem then your surface is z= 1-x-y over a circle of radius 1

    N = i + j + k

    curl F*N = 3 int(curlF*NdA) = 3*area = 3 pi which is what we obtain computing the line integral directly
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    Last edited by Calculus26; June 9th 2009 at 08:36 AM.
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  3. #3
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    Quote Originally Posted by Calculus26 View Post
    No-- you kind of missed the point

    alpha is the curve of intersection of the plane and the cylinder

    See the attachment for the details where the line integral is computed directly

    If you use Stokes theorem then your surface is z= 1-x-y over a circle of radius 1

    N = i + j + k

    curl F*N = 3 int(curlF*NdA) = 3*area = 3 pi which is what we obtain computing the line integral directly

    Ok. Thank you
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