State a necessary and sufficient condition for a vector field $\mathbf{F}$ on $\mathbb{R}^{3}$ to be conservative.

Check that the field

$$\mathbf{F}=\left(2 x \cos y-2 z^{3}, 3+2 y e^{z}-x^{2} \sin y, y^{2} e^{z}-6 x z^{2}\right)$$

is conservative and find a scalar potential for $\mathbf{F}$.