Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Question

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector.

Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?

Answer 1

If $M=(m_{ij})$ and $v=(v_j)$ then \begin{align} \nabla\cdot (Mv)=\sum_{k=1}^n\partial_k(Mv)_k= \sum_{k=1}^n\partial_k\Big(\sum_{j=1}^nm_{kj}v_j\Big)=\sum_{k=1}^n\sum_{j=1}^n \big((\partial_km_{kj})v_j+m_{kj}(\partial_kv_j)\big). \end{align}