Solution
= Solution
{c}
Apply the <product rule> to each <vector component> and sum:
$$
\begin{aligned}
\nabla\cdot(\phi\mathbf F)
&=\sum_{j=1}^3\partial_{x_j}(\phi F_j)\\
&=\phi\sum_{j=1}^3\partial_{x_j}F_j+\sum_{j=1}^3(\partial_{x_j}\phi)F_j\\
&=\boxed{\phi\nabla\cdot\mathbf F+\nabla\phi\cdot\mathbf F.}
\end{aligned}
$$
Thus the <divergence> of a <scalar field> times a <vector field> contains a <divergence> term and a <dot product> with the <gradient> of the <scalar field>.