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?

I don't know that anyone calls this the nabla operator.  The symbol is nabla, but the operator is called the gradient.  There is no need for a special notation for an individual partial derivative, you can just write $\partial f/\partial y$ or $\partial/\partial y$ for the operator.
You do sometimes see $\nabla_x f$ for a function of many variables $f(x_1, x_2,..., x_n, y_1, y_2, ..., y_n)$ to mean the gradient w.r.t. the $x$ variables.  (But this isn't what the question was asking about)
