Manipulating Vector Identity I would like to expand then simplify (if possible) the following quantity. 
$\nabla (a \cdot (C\, a))$
Where $a = a(x)$ is a vector valued function of $x$ and $C$ is a constant matrix. 

I personally find that using indices makes things like this way easier.
Write  the $i$th component as $\sum_{j,k}\partial_i (a_j C_{jk} a_k)$. You can now just use the product rule etc. as normal and reconvert into matrix notation if you wish.
