State the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

The surface $S_{1}$ is defined by the equation $z=3-2 x^{2}-2 y^{2}$ for $1 \leqslant z \leqslant 3$; the surface $S_{2}$ is defined by the equation $x^{2}+y^{2}=1$ for $0 \leqslant z \leqslant 1$; the surface $S_{3}$ is defined by the equation $z=0$ for $x, y$ satisfying $x^{2}+y^{2} \leqslant 1$. The surface $S$ is defined to be the union of the surfaces $S_{1}, S_{2}$ and $S_{3}$. Sketch the surfaces $S_{1}, S_{2}, S_{3}$ and (hence) $S$.

The vector field $\mathbf{F}$ is defined by

$$\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)$$

Evaluate the integral

$$\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}$$

where the surface element $\mathrm{d} \mathbf{S}$ points in the direction of the outward normal to $S$.