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

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

     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 \; \theta \; is the angle of \; u(r_0,\theta_0) \; and \; \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 u(r_0,\theta_0) in D_R. The standard formular for Dirichlet problem is:

    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 \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 \frac{\partial}{\partial r_0}G(r,\theta,r_0,\phi) Which is only a simple derivative of G respect to \; r_0 \; where in this case \; r_0 = R \; !!!

    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:

    \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 \; r_0 \; only? I know  \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 (r,\theta) are polar coordinates. So r_0=r/|r| is a unit vector. Now, since the region is a circle, the normal n is exactly r_0, and we get
    \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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