Let take a square constant matrix $A\in \mathbb{R}^{d\times d}$, and a vector field $v:\mathbb{R}^d \to \mathbb{R}^d$. Is there a way to write the divergence of their product in terms of the divergence of $v$?
That is can we write
$$ \text{div}(Av) $$ as something involving $A$ and $\text{div}(v)$?
Using index notation, divergence is interpreted as the trace of the Jacobian matrix of a vector i.e.
$$ \operatorname{div} \mathbf{v} = \partial_i v^i$$
with Einstein summation convention. Therefore we can correctly interpret the divergence of the product as follows
$$\partial_i(A_j^i v^j) = (\partial_i A_j^i)v^j + A_j^i (\partial_i v^j)$$
The object on the left is the dot product of $\mathbf{v}$ with a vector whose elements are the divergences of the column vectors of $\mathbf{A}$. The object on the right is the "matrix dot product" (element wise multiplication then adding) of $\mathbf{A}$ with the Jacobian matrix of $\mathbf{v}$.