Proof of the product rule for the divergence How can I prove that
$\nabla \cdot (fv) = \nabla f \cdot v + f\nabla \cdot v,$ 
where $v$ is a vector field and $f$ a scalar valued function?
Many thanks for the help!

$$
\begin{align}
\nabla \cdot (fv) &= \sum \partial_i (f v_i) \\
&= \sum (f \partial_i v_i + v_i \partial_i f) \\
&= f \sum \partial_i v_i + \sum v_i \partial_i f \\
&= f \nabla \cdot v + v \cdot \nabla f
\end{align}
$$
