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##(adsbygoogle = window.adsbygoogle || []).push({});

I want to get to

[tex]\oint _cMdy-Ndx=\oint_{s}\left(\frac{\partial M}{\partial x}-\frac{\partial N}{\partial y}\right)dxdy[/tex]

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)##

[tex]\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[/tex]

[tex]=\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[/tex]

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

Please help

Thanks

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# Derivation of Green's Function

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