# verify proof of Green's theorem

• Aug 18th 2013, 12:09 PM
Alan0354
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
$\displaystyle \int_{c}\vec {F}\cdot(\hat {T}\times\hat {k})ds=\int_{c }(\hat {k} \times \vec {F})\cdot \hat {T} ds$

I want to verify how to get

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

This is my work:

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

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

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

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

Thanks
• Aug 18th 2013, 08:16 PM
Alan0354
Re: verify proof of Green's theorem
Also, the Tangential form of Green's Theorem is

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

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

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

What's the story?