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Thread: Theorem of Green

  1. #1
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    Theorem of Green

    C is the segment of straight the point $\displaystyle (a,b)$ to point $\displaystyle (c,d)$. Calculate

    $\displaystyle \int_C -ydx + xdy$


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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    C is the segment of straight the point $\displaystyle (a,b)$ to point $\displaystyle (c,d)$. Calculate

    $\displaystyle \int_C -ydx + xdy$


    Green's Theorem states that

    $\displaystyle \int_C{(L\,dx + M\,dy)} = \int{\int_D{\left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)}}$

    Where $\displaystyle C$ is a positively oriented, piecewise smooth, simple closed curve in the plane, and $\displaystyle D$ is the region bounded by $\displaystyle C$.


    Here $\displaystyle L = -y, M = x$ so $\displaystyle \frac{\partial L}{\partial y} = -1, \frac{\partial M}{\partial x} = 1$.

    But what is your region? All you've given is a straight line. What bounds the region $\displaystyle D$?
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  3. #3
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    Quote Originally Posted by Prove It View Post
    Green's Theorem states that

    $\displaystyle \int_C{(L\,dx + M\,dy)} = \int{\int_D{\left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)}}$

    Where $\displaystyle C$ is a positively oriented, piecewise smooth, simple closed curve in the plane, and $\displaystyle D$ is the region bounded by $\displaystyle C$.


    Here $\displaystyle L = -y, M = x$ so $\displaystyle \frac{\partial L}{\partial y} = -1, \frac{\partial M}{\partial x} = 1$.

    But what is your region? All you've given is a straight line. What bounds the region $\displaystyle D$?
    I thinks all he wants is the line integral along the straight line from $\displaystyle (a,b)$ to $\displaystyle (c,d)$.
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  4. #4
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    Quote Originally Posted by danny arrigo View Post
    I thinks all he wants is the line integral along the straight line from $\displaystyle (a,b)$ to $\displaystyle (c,d)$.
    The answer is
    $\displaystyle ad - bc$
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