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Solve using Green's Theorem

  1. Jun 26, 2011 #1
    1. The problem statement, all variables and given/known data
    Use Green’s theorem to find the integral
    [itex]\oint_{\gamma} \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy[/itex]
    along two different curves γ: first where γ is the simple closed curve which goes along x = −y2 + 4 and x = 2, and second where γ is the square with vertices (−1, 0), (1, 0), (0, 1), (0, −1).


    2. Relevant equations



    3. The attempt at a solution
    I'm bit confused b/c
    [itex]d(\frac{-y}{x^2+y^2})/dy = \frac{y^2-x^2}{(x^2+y^2)^2}[/itex]
    [itex]d(\frac{x}{x^2+y^2})/dx = \frac{y^2-x^2}{(x^2+y^2)^2}[/itex]

    Then by Green's theorem one gets

    [itex]\int_{A}\int (d(\frac{x}{x^2+y^2})/dx-d(\frac{-y}{x^2+y^2})/dy) dx dy = \int_{A}\int 0 dx dy = 0[/itex]

    What am I missing?
     
  2. jcsd
  3. Jun 26, 2011 #2
    You are not missing anything. The answer IS 0.

    Here you have a line integral of a conservative field (you can tell it's conservative from the equality of the partial derivatives).
    A line integral of a conservative field will always be zero for any path that begins and ends at the same point, i.e any closed curve.
     
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