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Math Help - Green's theorem

  1. #1
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    Green's theorem

     \int_{C}[f(x,y)dy - g(x,y)dx] = \int_{A}\int[\frac{\partial f}{\partial x} , \frac{\partial g}{\partial {y}}] dA

    Im trying to verify this theorem for the following line integral

     \int_{C}(x + y^{2})dy + (xy^{2}-y)dx where  C is the triangle with vertices  (1,1), (3,1), (3,3)

    I had no problem calculating the right hand side of green's theorem. However im struggling on the left hand side of the theorem.

    Im thinking i need to get the line integral in terms of either x or y only ...?
    Last edited by Tekken; April 14th 2010 at 07:00 AM. Reason: typo
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  2. #2
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    Quote Originally Posted by Tekken View Post
     \int_{C}[f(x,y)dy - g(x,y)dx] = \int_{A}\int[\frac{\partial f}{\partial x} , \frac{\partial g}{\partial {y}}] dA

    Im trying to verify this theorem for the following line integral

     \int_{C}(x + y^{2})dy + (xy^{2}-y)dx where  C is the triangle with vertices  (1,1), (3,1), (3,3)

    I had no problem calculating the right hand side of green's theorem. However im struggling on the left hand side of the theorem.

    Im thinking i need to get the line integral in terms of either x or y only ...?
    Let  A=(1,1), B=(3,1), D=(3,3)
    along the line segment AB, y=1 and x=t with 1 \leq t \leq 3, hence dy = 0 and dx = dt, calculate the integral.

    do the same thing for the line segment BD and DA

    regard

    DD
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  3. #3
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    Quote Originally Posted by dedust View Post
    Let  A=(1,1), B=(3,1), D=(3,3)
    along the line segment AB, y=1 and x=t with 1 \leq t \leq 3, hence dy = 0 and dx = dt, calculate the integral.

    do the same thing for the line segment BD and DA

    regard

    DD
    Thanks alot,

    I think i may have calculated the right hand side of green's theorem incorrectly.

    i let  f(x,y) = x + y^{2} and  g(x,y) = xy^{2} - y

    this gave me  \frac{\partial f}{\partial x} = 1 and  \frac{\partial g}{\partial y} = 2xy - 1

    so filling these into the equation i got

     \int^{3}_{1}\int^{3}_{1}[(x+y^{2})-(xy^{2}-y)]dx.dy

    However im not sure if these limits are correct, aren't the inside limits of a double integral normally  x's or  y's ?

    If you could answer this, it would be great
    Last edited by Tekken; April 14th 2010 at 09:32 AM. Reason: latex trouble..
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