What is $\nabla \cdot \mathbf{A}$ when $\mathbf{A} \in \mathbb{R}^{m \times n}$ is a matrix, and where is there a consise definition of this notation?
The Euler equations on Wikipedia contain terms on the form $\nabla \cdot (\mathbf{u} \otimes \mathbf{u} - w\mathbf{I})$ where $\mathbf{I}$ is the identity matrix. Other material on the Euler equations simply writes $\nabla \cdot (\mathbf{u} \mathbf{u} - w\mathbf{I})$. I assume $\mathbf{u} \otimes \mathbf{v} = \mathbf{u}\mathbf{v}^T \in \mathbb{R}^{m \times n}$ for $\mathbf{u} \in \mathbb{R}^m$ and $\mathbf{v} \in \mathbb{R}^n$.
Is $\nabla \cdot \mathbf{A}$ defined when $\mathbf{A}$ is not square / not symmetrical?
Clicking "show" next to "Demonstration of the conservation form" reveals more context: the matrix $$\begin{pmatrix}{\mathbf {u}}\otimes {\mathbf {u}}+w{\mathbf {I}}\\{\mathbf {u}}\end{pmatrix}$$ is written out in components as $$ {\begin{pmatrix}u_{1}^{2}+w&u_{1}u_{2}&u_{1}u_{3}\\u_{2}u_{1}&u_{2}^{2}+w&u_{2}u_{3}\\u_{3}u_{1}&u_{3}u_{2}&u_{3}^{2}+w\\u_{1}&u_{2}&u_{3}\end{pmatrix}}$$ Other formulas clarify that $\nabla$ is applied to this matrix by differentiating the first column with respect to $x_1$, the second with respect to $x_2$, the third with respect to $x_3$, and then adding the results. The result being $$ \nabla\cdot {\begin{pmatrix}u_{1}^{2}+w&u_{1}u_{2}&u_{1}u_{3}\\u_{2}u_{1}&u_{2}^{2}+w&u_{2}u_{3}\\u_{3}u_{1}&u_{3}u_{2}&u_{3}^{2}+w\\u_{1}&u_{2}&u_{3}\end{pmatrix}} =\begin{pmatrix} (\operatorname{div} \mathbf u)\mathbf u + \operatorname{grad} \mathbf w \\ \operatorname{div} \mathbf u \end{pmatrix} $$ where I used explicit names to avoid any further $\nabla$-confusion.
If we think of $\nabla$ as a symbolic vector of partial derivatives, $\nabla = \begin{pmatrix} \partial /\partial x_1 \\ \partial /\partial x_2 \\ \partial /\partial x_3 \end{pmatrix}$, then the above is more properly $$ \begin{pmatrix}{\mathbf {u}}\otimes {\mathbf {u}}+w{\mathbf {I}}\\{\mathbf {u}}\end{pmatrix} \nabla = {\begin{pmatrix}u_{1}^{2}+w&u_{1}u_{2}&u_{1}u_{3}\\u_{2}u_{1}&u_{2}^{2}+w&u_{2}u_{3}\\u_{3}u_{1}&u_{3}u_{2}&u_{3}^{2}+w\\u_{1}&u_{2}&u_{3}\end{pmatrix}} \begin{pmatrix} \partial /\partial x_1 \\ \partial /\partial x_2 \\ \partial /\partial x_3 \end{pmatrix} $$ This makes sense for any matrix where the number of columns matches the dimension.