In a recent post I introduced the following differential operator:
\[\vec \nabla \dot \star \vec v(x,y) \triangleq \begin{bmatrix} \frac{\partial}{\partial x} g(x,y) \\ \frac{\partial}{\partial x}h(x,y) + \frac{\partial}{\partial y}g(x,y) + \frac{\partial}{\partial y}h(x,y) \end{bmatrix}\]It has a divergence of:
\(\nabla \cdot \left[ \vec \nabla \dot \star \vec v(x,y) \right] = \nabla \cdot \begin{bmatrix} \frac{\partial}{\partial x} g(x,y) \\ \frac{\partial}{\partial x}h(x,y) + \frac{\partial}{\partial y}g(x,y) + \frac{\partial}{\partial y}h(x,y) \end{bmatrix}\) \(\nabla \cdot \left[ \vec \nabla \dot \star \vec v(x,y) \right] = \frac{\partial^2}{\partial x^2} g(x,y) + \frac{\partial^2}{\partial x \partial y} h(x,y) + \frac{\partial^2}{\partial y^2} g(x,y) + \frac{\partial^2}{\partial y^2} h(x,y)\)