TITLE: What is the divergence of a matrix valued function?
QUESTION [5 upvotes]: According to Wikipedia:

The divergence of a continuously differentiable tensor field $\underline{\underline{\epsilon}}$ is:
$$\overrightarrow{\operatorname{div}}\,(\mathbf{\underline{\underline{\epsilon}}}) = 
\begin{bmatrix}
\frac{\partial \epsilon_{xx}}{\partial x} +\frac{\partial \epsilon_{xy}}{\partial y} +\frac{\partial \epsilon_{xz}}{\partial z} \\
\frac{\partial \epsilon_{yx}}{\partial x} +\frac{\partial \epsilon_{yy}}{\partial y} +\frac{\partial \epsilon_{yz}}{\partial z} \\
\frac{\partial \epsilon_{zx}}{\partial x} +\frac{\partial \epsilon_{zy}}{\partial y} +\frac{\partial \epsilon_{zz}}{\partial z}
\end{bmatrix}
$$

How do you get this formula from the definition of divergence? Either formally, or with some abuse of notation?

REPLY [4 votes]: If S a matrix, with columns $S^{j}$, $j=1$, $n$ then $\mathrm{div}(S)_{j} = \mathrm{div}(S^{j})$.