Vector operator of second derivatives?

Question

Given a vector function $\mathbf{A}=A_{x}\mathbf{i}+A_{y}\mathbf{j}+A_{z}\mathbf{k}$, does the quantity given by $$\mathbf{B}=\frac{\partial^{2}}{\partial x^{2}}A_{x}\mathbf{i}+\frac{\partial^{2}}{\partial y^{2}}A_{y}\mathbf{j}+\frac{\partial^{2}}{\partial z^{2}}A_{z}\mathbf{k}$$ have an special name or significance? can it be written in terms of $\nabla$?

Answer 1

No.

Consider the diagonal vector field $$\mathbf A=(xy)\frac{\mathbf i+\mathbf j}{\sqrt2}$$ for which, clearly, $\mathbf B=0$.

But if we simply rotate the coordinate system by $45^\circ$, $$x=\frac{x'-y'}{\sqrt2},\quad y=\frac{x'+y'}{\sqrt2}$$ $$\mathbf A=\bigg(\frac{x'^2-y'^2}{2}\bigg)\mathbf i'$$ then $\mathbf B=(1)\mathbf i'\neq0$.

So this differential operator depends on the choice of coordinates, unlike $\mathbf\nabla$.