# Derivation of Green's Function

1. Aug 17, 2013

### 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

2. Aug 17, 2013

### LCKurtz

3. Aug 17, 2013

### SteamKing

Staff Emeritus
4. Aug 17, 2013

### yungman

5. Aug 18, 2013

### yungman

6. Aug 18, 2013

### yungman

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

Last edited: Aug 18, 2013
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