= Solution
{c}
For $r>0$, the <chain rule> gives $\partial r/\partial x_i=x_i/r$. Applying the <product rule> to the components of the <vector field> yields
$$
\begin{aligned}
\nabla\cdot\mathbf u
&=\sum_{i=1}^3\frac{\partial}{\partial x_i}(x_i r^{-\alpha})\\
&=\sum_{i=1}^3\left(r^{-\alpha}-\alpha x_i^2r^{-\alpha-2}\right)\\
&=3r^{-\alpha}-\alpha r^{-\alpha-2}\sum_{i=1}^3x_i^2
=\boxed{\frac{3-\alpha}{r^\alpha}}.
\end{aligned}
$$
The calculation is pointwise away from the origin; any extension of the <vector field> to the origin must be considered separately.
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