(a) State Stokes' Theorem for a surface $S$ with boundary $\partial S$.

(b) Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=1+x^{2}+y^{2}$ where $\sqrt{2} \leqslant z \leqslant \sqrt{5}$. Sketch the surface $S$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

(c) Compute $\nabla \times \mathbf{F}$ for the vector field

$$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ x y(x+y) \end{array}\right)$$

and verify Stokes' Theorem for $\mathbf{F}$ on the surface $S$.