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Thread: Question on normal derivative of Green's function.

  1. #1
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    Question on normal derivative of Green's function.

    For circular region, why is $\displaystyle \frac{\partial}{\partial n}G(r,\theta,r_0,\phi)= \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) $ ?
    Where $\displaystyle \; \hat{n} \:$ is the outward unit normal of $\displaystyle C_R$.
    Let circular region $\displaystyle D_R$ with radius $\displaystyle R \hbox { and possitive oriented boundary }\; C_R$. Let $\displaystyle u(r_0,\theta)$ be harmonic function in $\displaystyle D_R$.

    The Green's function for Polar coordinate is found to be:

    $\displaystyle G(r,\theta,r_0,\phi) = \frac{1}{2} ln[R^2 \frac{r^2+r_0^2 -2rr_0 cos(\theta-\phi)}{r^2r_0^2 + R^4 - 2rr_0R^2 cos(\theta-\phi)}] $

    Where $\displaystyle \; \theta \;$ is the angle of $\displaystyle \; u(r_0,\theta_0) \;$ and $\displaystyle \; \phi \;$ is the angle of the two points used in Steiner Invertion.
    Next I want to solve the Dirichlet problem using Green's function. For any value of a hamonic function $\displaystyle u(r_0,\theta_0) in D_R$. The standard formular for Dirichlet problem is:

    $\displaystyle u(r_0,\theta_0) = \frac{1}{2}\int_{C_R} u(r,\theta) \frac{\partial}{\partial n}G(r,\theta,r_0,\phi) ds$

    Where $\displaystyle \frac{\partial}{\partial n}G(r,\theta,r_0,\phi)= \nabla G(r,\theta,r_0,\phi) \;\cdot \widehat{n} $

    But the book just simply use $\displaystyle \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) $ Which is only a simple derivative of G respect to $\displaystyle \; r_0 \;$ where in this case $\displaystyle \; r_0 = R \;$ !!!

    $\displaystyle u(r_0,\theta_0) = \frac{1}{2}\int_{C_R} u(r,\theta) \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) ds$

    I don't understant how:

    $\displaystyle \frac{\partial}{\partial n}G(r,\theta,r_0,\phi)= \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) $

    How can a normal derivative become and simple derivative respect to $\displaystyle \; r_0 \;$ only? I know $\displaystyle \widehat{r}_0 \;\hbox { is parallel to outward normal of }\;\; C_R \;$ but the magnitude is not unity like the unit normal. Can anyone explain to me?

    Thanks

    Alan
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  2. #2
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    Did I put this question in the wrong section? I don't know where to put this as this is beyond ODE or maybe PDE. Please move this to the correct sub forum.
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  3. #3
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    Anyone please? Even if you don't have the answer, point me where to look. I am really out of ideas. I have five PDE book and I can't find any help!!!
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  4. #4
    Super Member Rebesques's Avatar
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    I believe $\displaystyle (r,\theta)$ are polar coordinates. So $\displaystyle r_0=r/|r|$ is a unit vector. Now, since the region is a circle, the normal $\displaystyle n$ is exactly $\displaystyle r_0$, and we get
    $\displaystyle \frac{\partial}{\partial n}G=\langle \nabla_{(r,\theta)} G,n\rangle=\langle \nabla_{(r,\theta)} G,r_0\rangle= \frac{\partial}{\partial r_0}G$.
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