A question on Stoke's theorem and the divergence theorem

Question

Consider the following parameterization $φ$ of the parametric surface $S = φ(R)$ with$$ R = [0,2π] × [0,1] ⊂ \mathbb{R}^2, $$\begin{align*} φ:&& R &\longrightarrow \mathbb{R}^3\\ && (u,v) &\longmapsto (x(u,v),y(u,v),z(u,v)) = (v \cos u, v \sin u, v^2), \end{align*} and the vector field\begin{align*} \boldsymbol{F}: && \mathbb{R}^3 &\longrightarrow \mathbb{R}^3\\ && (x,y,z) &\longmapsto (x,y,x+y+z). \end{align*}

1. Find a unit normal $\boldsymbol{n}$ on $S$ from the given parameterization.
2. With the orientation given by $\boldsymbol{n}$, compute both sides of Stoke's theorem for the vector field $\boldsymbol{F}$. (We have to remember here that the orientation of $∂S$ is given by the condition that $\boldsymbol{n} × \boldsymbol{T}$ points towards the surface, where $\boldsymbol{T}$ is the unit tanget vector of $∂S$ defining its orientation.
3. Let $W = \{ (x,y,z) \mid x^2 + y^2 \leqslant z,\ z ∈ [0,1]\}$ be the solid enclosed by $S$ and the disk $D = \{ (x,y,1) ∈ \mathbb{R}^3 \mid x^2 + y^2 \leqslant 1 \}$. For the vector field $\boldsymbol{F}$ and the solid $W$ verify the divergence theorem.