I learnt that taking the divergence of a vector $F(x,y)=(F_x,F_y)$ gives a sum of first order partial derivatives: $$ \nabla\cdot F = (\frac{\partial}{\partial x},\frac{\partial}{\partial y} )\cdot(F_x,F_y)=\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}. $$
Is there a concise notation like the above to write the following sum of second order partial derivatives of a vector? $$ \frac{\partial^2 F_x ^2}{\partial x^2 } + \frac{\partial^2 F_x F_y}{\partial x \partial y} + \frac{\partial^2 F_y F_x}{\partial y \partial x} + \frac{\partial^2 F_y ^2}{\partial y^2 } $$
I may have missed something trivial, but I could not find anything of the sort besides using summation indices: $$ \Sigma_{ij} \frac{\partial^2 F_{ij}}{\partial i \partial j}, $$ where $i,j$ run over all possible choices of $x,y$.