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Math Help - verify proof of Green's theorem

  1. #1
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    verify proof of Green's theorem

    I have a question on Page 4 of the link you provided, the formula on the top of page 4:http://www.math.mcgill.ca/jakobson/c...a265/green.pdf
    \int_{c}\vec {F}\cdot(\hat {T}\times\hat {k})ds=\int_{c }(\hat {k} \times \vec {F})\cdot \hat {T} ds

    I want to verify how to get

    \int_{c} (\hat {k} \times \vec {F})\cdot \hat {T} ds=\int-Qdy+Pdx

    This is my work:

    \vec{F}=\hat{x}P+\hat{y}Q and \;\hat{k}=\hat{z}

    \vec {s}=\hat {x} x +\hat y {y}\Rightarrow\;d\vec {s}=\hat {x} dx +\hat {y} dy=\hat {T} ds

    \hat {k}\times \vec {F}=-\hat{x} Q+\hat {y} P\;\Rightarrow\; (\hat {k} \times \vec {F)}\cdot \hat {T} ds=-Qdx+Pdy

    \Rightarrow \;\int_{c} (\hat {k} \times \vec {F})\cdot \hat {T} ds=\int_{c} -Qdx+Pdy

    Thanks
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  2. #2
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    Re: verify proof of Green's theorem

    Also, the Tangential form of Green's Theorem is

    \int_{c}\vec{F}\cdot \hat{T}ds=\int_{a}\nabla \times\vec {F}\;da

    \nabla\times \vec{F}=\hat{z} \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\;\Rightarrow \int_{c}\nabla\times \vec{F}\;da=\hat{z} \int_{a}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)da

    \int_{c}\vec{F}\cdot \hat{T}ds is a scalar, BUT \int_{c}\nabla\times \vec{F}\;da=\hat{z} \int_{a}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)da is a vector and they cannot be equal!!!

    What's the story?
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