I want to compute $$\vec{n}\cdot \nabla^\prime G(\,\vec{r},\,\vec{r}^\prime)$$ with $$G(\,\vec{r},\,\vec{r}^\prime) = \frac{1}{|\,\vec{r}-\vec{r}^\prime|}-\frac{\frac{R}{r^\prime}}{|\,\vec{r}-\left(\frac{R}{r^\prime}\right)^2\,\vec{r}^\prime|},\quad 0<R\in\mathbb{R},\quad r^\prime = |\,\vec{r}^\prime|.$$
Now, since $\vec{\,n}$ is the outer normal of a sphere, I want to compute this in spherical coordinates. However, writing down nabla in spherical coordinates and applying it to the above expression results in a quite tedious task. Is there any other, quicker way of doing this?
This is how far I get: We have $$ \frac{\partial}{\partial x_i'} \left[ x_i - \left(\frac{R}{|r'|}\right)^2 x_i' \right] = -R^2 \frac{\partial}{\partial x_i'} \left[ \left(\sum_i x_i'^2\right)^{-1}x_i' \right] = -R^2\left( -2\frac{x_i'^2}{|r'|^4} +\frac{1}{|r'|^2} \right) = \left(\frac{R}{|r'|}\right)^2\left( 2\frac{x_i'^2}{|r'|^2} - 1 \right) \quad (1) $$
and so $$ \frac{\partial}{\partial x_i'} \left|r -\left(\frac{R}{|r'|}\right)^2 r'\right| = \frac{\partial}{\partial x_i'} \sqrt{\sum_i \left(x_i -\left(\frac{R}{|r'|}\right)^2 x_i'\right)^2} = \\ \frac{\left(x_i -\left(\frac{R}{|r'|}\right)^2 x_i'\right) \left(\frac{R}{|r'|}\right)^2\left( 2\frac{x_i'^2}{|r'|^2} - 1 \right) } {\left|r -\left(\frac{R}{|r'|}\right)^2 r'\right|} \quad (2) $$
and thus $$ \left[\nabla' G(r, r')\right]_i = \frac{\partial}{\partial x_i'} G(r, r') = \\ \frac{\partial}{\partial x_i'} \frac{1}{|r -r'|} - \frac{\partial}{\partial x_i'} \frac{\frac{R}{|r'|}}{\left|r -\left(\frac{R}{|r'|}\right)^2 r'\right|} = \\ \frac{x_i - x_i'}{|r -r'|^3} - \frac{-R\frac{x_i'}{|r'|^3} \left|r -(\frac{R}{|r'|})^2 r'\right| -\frac{R}{|r'|} \left(\frac{R}{|r'|}\right)^2\left( 2\frac{x_i'^2}{|r'|^2} - 1 \right) }{\left|r -\left(\frac{R}{|r'|}\right)^2 r'\right|^2} = \\ \frac{x_i - x_i'}{|r -r'|^3} + \frac{\frac{R}{|r'|}\frac{x_i'}{|r'|^2}}{\left|r -\left(\frac{R}{|r'|}\right)^2 r'\right|} - \frac{ \left(\frac{R}{|r'|}\right)^3\left( 2\frac{x_i'^2}{|r'|^2} - 1 \right) }{\left|r -\left(\frac{R}{|r'|}\right)^2 r'\right|^2} \quad (3) $$ where I probably made errors. :-)
The rest depends on how your $n$ is defined, is it $n = r/|r|$ or $n = r'/|r'|$?