Show that $\Phi(u,v)= (3 u+6 ,u-v,5 u+v)$ parametrizes the plane $2x-y-z=12$. Then:

(a) Calculate ${\mathbf{T}}_u$ , ${\mathbf{T}}_v$, and ${\mathbf{n}}(u,v)$.

(b) Find the area of $\mathcal{S}=\Phi(\mathcal{D})$, where $\mathcal{D}= (u,v) : 0\le u\le 3, 0\le v \le 8$.

(c) Express $f(x,y,z)=yz$ in terms of $u$ and $v$ and evaluate $\iint_{\mathcal{S}} f(x,y,z)\,d \mathcal{S}$.

(a) ${\mathbf{T}}_u=$ , ${\mathbf{T}}_v=$ , ${\mathbf{n}}(u,v)=$

(b) $Area(S)=$

(c) $\iint_{\mathcal{S}} f(x,y,z)\,d \mathcal{S}=$