State the formula of Stokes's theorem, specifying any orientation where needed.

Let $\mathbf{F}=\left(y^{2} z, x z+2 x y z, 0\right)$. Calculate $\boldsymbol{\nabla} \times \mathbf{F}$ and verify that $\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \times \mathbf{F}=0$.

Sketch the surface $S$ defined as the union of the surface $z=-1,1 \leqslant x^{2}+y^{2} \leqslant 4$ and the surface $x^{2}+y^{2}+z=3,1 \leqslant x^{2}+y^{2} \leqslant 4$.

Verify Stokes's theorem for $\mathbf{F}$ on $S$.