Let (mathcal{S}) be the surface parametrized by [ Phi(u, v)=left(2 u sin frac{v}{2}, 2 u cos frac{v}{2},

Question:

Let $\mathcal{S}$ be the surface parametrized by
\[
\Phi(u, v)=\left(2 u \sin \frac{v}{2}, 2 u \cos \frac{v}{2}, 3 vight)
\]
for $0 \leq u \leq 1$ and $0 \leq v \leq 2 \pi$
(a) Calculate the tangent vectors $\mathbf{T}_{u}$ and $\mathbf{T}_{v}$ and the normal vector $\mathbf{N}(u, v)$ at $P=\Phi\left(1, \frac{\pi}{3}ight)$.
(b) Find the equation of the tangent plane at $P$.
(c) Compute the surface area of $\mathcal{S}$.

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