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Derivation of Green's Function

  1. Aug 17, 2013 #1
    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
    [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
     
  2. jcsd
  3. Aug 17, 2013 #2

    LCKurtz

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  4. Aug 17, 2013 #3

    SteamKing

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  5. Aug 17, 2013 #4
  6. Aug 18, 2013 #5
  7. Aug 18, 2013 #6
    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:

    [tex]\int_c\vec F\cdot(\hat T\times\hat k)ds=\int_c (\hat k \times \vec F)\cdot \hat T ds[/tex]

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

    Thanks
     
    Last edited: Aug 18, 2013
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