I need some help completing the last part of this problem.
Problem:
For the position vector $r$ such that $r = x^2 + y^2 + z^2 = \sum_{i=1}x_i^2$, show that the trace of the following expression $$\frac{\partial{}}{\partial{x}_i}\frac{\partial{}}{\partial{x}_j}\left(\frac{1}{r}\right)$$ vanishes.
Attempt at a Solution:
The trace can be found by summing up the diagonal elements $(i = j)$ $$ \sum_i \frac{\partial{}^2}{\partial x_i^2}\left(\frac{1}{r}\right) = \sum_i\frac{\partial{}}{\partial x_i}\left(-\frac{x_i}{r^3}\right) = \sum_i\frac{3x_i^2}{r^5}-\frac{1}{r^3}$$
I am not sure how this sum cancels out to zero. Thank you.
