State Stokes' theorem.

Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=x^{2}+y^{2}+1-\lambda$, where $0 \leqslant z \leqslant 1$ and $\lambda$ is a positive constant. Sketch the surface $S$ for representative values of $\lambda$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

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

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

and verify Stokes' theorem for $\mathbf{F}$ on the surface $S$ for every value of $\lambda$.

Now compute $\nabla \times \mathbf{G}$ for the vector field

$$\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)$$

and find the line integral $\int_{\partial S} \mathbf{G} \cdot \mathbf{d x}$ for the boundary $\partial S$ of the surface $S$. Is it possible to obtain this result using Stokes' theorem? Justify your answer.