What Is the Divergence of a Matrix Valued Function?
$\Begingroup$ According to Wikipedia: the Divergence of a Continuously Differentiable Tensor Field $\Underline{\Underline{\Epsilon}}$ Is...
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?
1 Answer
If S a matrix, with columns $S^{j}$, $j=1$, $n$ then $\mathrm{div}(S)_{j} = \mathrm{div}(S^{j})$.
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