I recently stumbled upon the linear partial differential operator $$\mathcal{L}A\colon=\nabla \mathrm{tr} A\quad\text{for matrix-valued fields } A\colon\mathbb{R}^d\rightarrow\mathbb{R}^{d\times d}.$$
In coordinates, this is $(\mathcal{L}A)_i=\partial_i(\sum_j A_{jj})$. In fact, the operator that arose was $$\nabla \mathrm{tr}-\mathrm{div},$$where the divergence is taken row wise, i.e., $(\mathrm{div}A)_i=\sum_j \partial_j A_{ij}$; also, my matrix fields are symmetric.
I could swear I've seen a fluid mechanics reference where the second operator appeared, but I cannot remember what it was. I could, of course, be wrong.
Could you please let me know if you've encountered any of these operators in applications or in any place in the literature?
Thank you!
Best Answer
I suspect this is the same thing as the other answer but I lack the fluency in differential geometry to verify. Note that if you apply trace to a jacobian, you end up with div. So in the context of fluid mechanics, I bet it arose from the double curl formula $$ \nabla\times(\nabla\times u) = [\nabla,\text{div}]u = \nabla \operatorname{div} u - \Delta u= \nabla\operatorname{tr} (\nabla u) - \operatorname{div} (\nabla u)$$ For divergence free fields one can invert the Laplacian to recover the "velocity" $u$ from the "vorticity" $\omega=\nabla \times u$ (this is called the "Biot-Savart law")