Here is my question:
Does the following limit exist? $ \lim_{y\to\xi}\frac{(\xi_i-y_i)(\xi_j-y_j)({{\xi}-y})\cdot n(y)}{|\xi-y|^5},\quad 1\leq i,j\leq 3,\tag{*} $ where $S\subset{\mathbb R}^3$ is a surface which has a continuously varying normal vector, $\xi=(\xi_1,\xi_2,\xi_3)\in S$, $y=(y_1,y_2,y_3)\in S$, $n(y)$ is the [EDITED: unit] normal vector at point $y$. Here $(\xi-y)\cdot n(y)$ is the dot product.
In the spirit of Polya, I find a simpler case where $S$ is a unit sphere. Then we have $n(y)=y$. But I don't have a strategy to go on.