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Math Help - Surface Integration Problem

  1. #1
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    Surface Integration Problem

    I'm stuck on this problem,

    Consider the domain in the u - v plane bounded by the circle u^2+v^2=1 and the surface S in R^3 defined parametrically by
    r(u,v)=(u^2+v^2)i + (uv)j + (u+v)k
    where the positive sense around the boundary is determined by the positive sense around the boundary of the region in the u - v plane.

    Set F= (yz)i + (xz)j + (xy)k

    Compute SS(curlF).n dS. It is a double integral where . is the dot product between the curlF and the normal n.

    I've done problems similar to this, but never with different variables as the boundary. How do I set this up as an integral?
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  2. #2
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    Quote Originally Posted by Five Star View Post
    I'm stuck on this problem,

    Consider the domain in the u - v plane bounded by the circle u^2+v^2=1 and the surface S in R^3 defined parametrically by
    r(u,v)=(u^2+v^2)i + (uv)j + (u+v)k
    Let us examine this surface.
    We have, parametrically,
    x=u^2+v^2
    y=uv
    z=u+v

    Note that,
    z^2=(u+v)^2=u^2+2uv+v^2=(u^2+v^2)+2(uv)=x+2y.
    Thus, the surface is, a plane,
    z=x+2y
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  3. #3
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    Is it z^2 = x+2y, not z=x+2y ?
    This would not make it a plane right?
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  4. #4
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    Quote Originally Posted by Five Star View Post
    Is it z^2 = x+2y, not z=x+2y ?
    This would not make it a plane right?
    Sorry.
    No, that would not be a plane.

    Remember when you use a triple intergral and need the upper and lower surface you get the plus/minus when you take the square root.
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  5. #5
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    I was thinking. Is there a name for this figure?
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  6. #6
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    Quote Originally Posted by Five Star View Post
    I was thinking. Is there a name for this figure?
    It is not on a standard list of solids provided in thy Method of Fluxions (Calculus) scroll.

    It is however a general quadric surface, thus maybe it has a name.
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