How Do You Derive Green's Function Using Vector Calculus?

[yungman]

The normal form of Green's function is $\oint_c\vec F\cdot \hat n dl'=\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy$

I want to get to
$$\oint _cMdy-Ndx=\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy$$

Let $\vec F=\hat x M(x,y)+\hat y N(x,y)$
Let a rectangle area A with corners: $(x,y),\;(x+\Delta x,y),\;(x+\Delta x,y+\Delta y),\;(x,y+\Delta y)$

$$\oint_c\vec F\cdot \hat n dl'=\int_{right}\vec F\cdot \hat x dy+\int_{left}\vec F\cdot(- \hat x) dy+\int_{top}\vec F\cdot \hat y dx+\int_{bottom}\vec F\cdot(- \hat y) dx$$
$$=\int_c M(x+\Delta x,y) dy-\int_c M(x,y) dy+\int_c N(x,y+\Delta y) dx- \int_c N(x,y)dx\;=\;\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy$$

I can't get $\oint _c Mdy-Ndx$

Please help

Thanks

 

[LCKurtz]

Most calculus books have a proof of Green's theorem for regions as general as oval shapes. Also on the internet there are proofs. One is here:

http://www.math.mcgill.ca/jakobson/courses/ma265/green.pdf

 

[SteamKing]

Be careful with terminology here: Green's Theorem is not the same as Green's function.

http://en.wikipedia.org/wiki/Green's_theorem

http://en.wikipedia.org/wiki/Green's_function

 

[yungman]

Most calculus books have a proof of Green's theorem for regions as general as oval shapes. Also on the internet there are proofs. One is here:

http://www.math.mcgill.ca/jakobson/courses/ma265/green.pdf

Thanks

 

[yungman]

Be careful with terminology here: Green's Theorem is not the same as Green's function.

http://en.wikipedia.org/wiki/Green's_theorem

http://en.wikipedia.org/wiki/Green's_function

Yes, it's a typo. I meant Green's theorem.

 

[yungman]

Most calculus books have a proof of Green's theorem for regions as general as oval shapes. Also on the internet there are proofs. One is here:

http://www.math.mcgill.ca/jakobson/courses/ma265/green.pdf

I know there are tons of proofs using graph of type I and type II regions. But I want to proof without using graph and just by vector calculus. I have a question on Page 4 of the link you provided, the formula on the top of page 4:

$$\int_c\vec F\cdot(\hat T\times\hat k)ds=\int_c (\hat k \times \vec F)\cdot \hat T ds$$

I want to verify how to get
$$\int_c (\hat k \times \vec F)\cdot \hat T ds=\int-Qdy+Pdx$$
$$\vec s=\hat x x +\hat y y\Rightarrow\;d\vec s=\hat x dx +\hat y dy=\hat T ds$$
$$\hat k\times \vec F=-\hat x Q+\hat y P\;\Rightarrow\; (\hat k \times \vec F)\cdot \hat T ds=-Qdx+Pdy$$
$$\Rightarrow \;\int_c (\hat k \times \vec F)\cdot \hat T ds=\int_c -Qdx+Pdy$$

Thanks