(Solved): Let be the surface parameterized by (u,v)=(2usin(2v),2ucos(2v),3v) for 0u1 and 0 ...

Let $\square$ be the surface parameterized by $\Phi (u,v)=\left(2u\sin \left(\frac{v}{2}\right),2u\cos \left(\frac{v}{2}\right),3v\right)$ for $0\le u\le 1$ and $0\le v\le 2\pi$. Calculate the tangent vectors ${\mathbf{T}}_u$ and ${\mathbf{T}}_v$ and the normal vector $\mathbf{N}(u,v)$ at $P=\Phi \left(1,\frac{3\pi }{2}\right)$. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) $\mathbf{T}$ Incorrect ${\mathbf{T}}_v($ Incorrect $\mathbf{N}(1$ Incorrect Find an equation of the tangent plane at $P$. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the equation in terms of $x,y$, and $z$.) eq Incorrect Compute the surface area of $\square$. (Express numbers in exact form. Use symbolic notation and fractions where needed.)