I know that the operator $\nabla$ denotes $$ \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{pmatrix} $$
Is there some kind of similar notation to denote
$$ A = \begin{pmatrix} \frac{\partial}{\partial u_x} \\ \frac{\partial}{\partial u_y} \end{pmatrix} $$
where $u_x = \frac{\partial}{\partial x} u$, likewise $u_y$. Here $u$ denotes some differentiable function in $\Omega \subset \mathbb{R}^2$.
It is fairly common to use the notation $\nabla_{\mathbf{u}}$ for gradient with respect to the velocity field. An alternative notation is $\partial/\partial{\mathbf{u}}$ so that the usual gradient is written as $\partial/\partial{\mathbf{x}}$.