Is $\nabla^{n}$ a correct terminology for a partial derivative? I'm learning optimization techniques and came accross the gradient (nabla) operator : $\nabla$.
If I'm right, the $\nabla$ operator of a function means the vector of all its partial derivatives. Then, if for example I'm talking about this specific partial derivative $\left(\frac{\partial}{\partial y}\right) F(x,y,z)$ is it a right terminology to write something like : $\nabla^{y}$ (since we can't use the standard i-th-element-in-vector notation like $\nabla^{2}$ as it would mean the Laplacian).
If it is not a correct terminology, is there a correct terminlogy to specify a specific component of the partial derivatives vector?

Nomenclatures are uncountable and is a matter of definition, but I would say that if $y$ is a scalar then it is more common to see $\partial_y F$. Another one I have seen is $F_y$ for simplicity. In this case a double derivative would be $F_{yz}$ for example.
If $y$ was a vector and $F(\textbf{x},\textbf{y},\textbf{z}) = F(x_1,...,x_m,y_1,...,y_l,z_1...,z_k)$ then $\nabla_\textbf{x} F$ would mean $(\dfrac{\partial F}{\partial{x_1}}, ...,\dfrac{\partial F}{\partial{x_m}} )$.
Note that it seems you are thinking about gradients, that is the meaning of nabla in this case, but it can have other meanings.
