Let $\vec{F}=\langle f(x,y,z), g(x,y,z), h(x,y,z)\rangle $ be a vector field, and all of its components are continuously differentiable. Determine which of the follow statements are correct:

$$\begin{align*} I: &\text{curl }(\nabla f)=\vec{0} \\\ II: &\langle \frac{\partial f}{\partial x}, \frac{\partial g}{\partial x}, \frac{\partial h}{\partial x}\rangle \text{ is also a vector field} \\\ III: &\frac{\partial f}{\partial y} = \frac{\partial g}{\partial x} \implies \vec{F} \text{ is conservative} \end{align*}$$