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Need help with line integral

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the parameterized curve gamma and vector field F so that
    the [itex]\int[/itex][itex]\gamma[/itex] F ds = [itex]\int[/itex][itex]\int2xy dx dy[/itex] by Green's Theorem.
    where -2<y<2
    1-sqrt(4-y2) < x < 1+sqrt(4-y2)

    3. The attempt at a solution


    x = 1 + sqrt(4-y2)
    (x-1)2=4-y2
    (x-1)2+y2=4

    so the path is a circle centered at (1,0) with radius of 2.
    parametrize this by setting x = 1+2Cos(t) and y = 2Sin(t)

    for the vector field I got F = (xy2,2x2y)
    but is there more than one possible vector field?
     
  2. jcsd
  3. Feb 28, 2012 #2

    LCKurtz

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    Yes, there are infinitely many such vector fields. For example, you can modify yours like this:$$
    \vec F = \langle xy^2+g(x), 2x^2y+h(y)\rangle$$where h and g can be any differentiable functions.
     
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