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

1. ## 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

2. 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.

3. 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!!!

4. 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
$\displaystyle \frac{\partial}{\partial n}G=\langle \nabla_{(r,\theta)} G,n\rangle=\langle \nabla_{(r,\theta)} G,r_0\rangle= \frac{\partial}{\partial r_0}G$.