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Math Help - Integration; work

  1. #1
    s7b
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    Integration; work

    I started this question;

    Find the work done by F=e^(yz)i + (xze^(yz) + zcosy)j + (xye^(yz) + siny)k over the following paths from (1,0,1) to (1,pi/2, 0)

    a) The line segment x=1 , y=(pi/2)t , z=1-t t between 0 and 1
    b) The line segment from (1,0,1) to the origin followed byt he line segment from the origin to (1,pi/2,0)
    c) The line segment from (1,0,1) to (1,0,0) followed by the x-axis from (1,0,0) to the origin, followed by the parabola y=pix^2/2 , z=0 from there to (1,pi/2,0)


    ***I started by solving the potential function which I got to be
    f(x,y,z)=xye^(yz) + siny + C ----> F=gradient(xye^(yz) + siny)


    I think dr/dt for part A is : pi/2j - k

    This is where I'm stuck...
    Help!!!
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  2. #2
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    Fortunately F is a conservative field because the vector product of
    F and the operator is zero . Therefore, W = F(a,b,c) - F(p,q,r)
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  3. #3
    s7b
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    How do you do it when it's split into two segments like part b)
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  4. #4
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    since curl F is zero
    in part b , the total work done is
    work done over the first segment from (1,0,1) to (0,0,0) +
    work done over the second segment from (0,0,0) to (1,pi/2,0)
    = [F(0,0,0) - F(1,0,1)] + [F(1,pi/2,0) - F(0,0,0)]
    = F(1,pi/2,0) - F(1,0,1)
    we can see that the shape of the path is not important unless curl F is
    not zero
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  5. #5
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    We may ensure that the path is differentiable .
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