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Math Help - Line Integral

  1. #1
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    Line Integral

    Hi, could you help me out with this line integral?

    Calculate the line integral of the field f=(y,z,x) around the surfaces x+y=2 and x^2+y^2+z^2=2(x+y)

    Thanks!!!
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  2. #2
    Senior Member
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    Mar 2010
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    Sphere may rewriten
    (x-1)^2+(y-1)^2+z^2=2
    with centre (1,1,0).
    Intersection with plane y=2-x gives circle
    centre (1,1,0) radius \sqrt{2}.
    In parametic form it is
    x=1+cost
    y=1-cost
    z=\sqrt{2}sint
    t[0,2pi].
    \int{f(x,y,z)ds}=\int{f(x(t),y(t),z(t))\sqrt{(\fra  c{dx}{dt})^2+({\frac{dy}{dt}})^2+({\frac{dz}{dt}})  ^2}dt}
    This \sqrt{}= \sqrt{2} so we get
    \sqrt{2}\int{f(1+cost,1-cost,\sqrt{2}sint)dt} from 0 to 2\pi.
    Please check.
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  3. #3
    Senior Member
    Joined
    Mar 2010
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    Beijing, China
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    Thanks
    23
    I think f is a vector field (y,z,x) and the line integral should be
    \int f.dl

    Quote Originally Posted by zzzoak View Post
    Sphere may rewriten
    (x-1)^2+(y-1)^2+z^2=2
    with centre (1,1,0).
    Intersection with plane y=2-x gives circle
    centre (1,1,0) radius \sqrt{2}.
    In parametic form it is
    x=1+cost
    y=1-cost
    z=\sqrt{2}sint
    t[0,2pi].
    \int{f(x,y,z)ds}=\int{f(x(t),y(t),z(t))\sqrt{(\fra  c{dx}{dt})^2+({\frac{dy}{dt}})^2+({\frac{dz}{dt}})  ^2}dt}
    This \sqrt{}= \sqrt{2} so we get
    \sqrt{2}\int{f(1+cost,1-cost,\sqrt{2}sint)dt} from 0 to 2\pi.
    Please check.
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