\section*{Problem 4}
In the three-dimensional orthogonal coordinate system $x y z$, consider the surface $S$ defined by the following equation:

$$\left( \begin{array} { c } 
x ( \theta , \phi ) \\
y ( \theta , \phi ) \\
z ( \theta , \phi )
\end{array} \right) = \left( \begin{array} { c c c } 
\cos \theta & - \sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array} \right) \left( \begin{array} { c } 
\cos \phi + 2 \\
0 \\
\sin \phi
\end{array} \right)$$

where $\theta$ and $\phi$ are parameters of the surface $S$, and $0 \leq \theta < 2 \pi , 0 \leq \phi < 2 \pi$. Let $V$ be the region surrounded by the surface $S$, and let $W$ be the region satisfying the inequality $x ^ { 2 } + y ^ { 2 } \leq 4$. Answer the following questions for the surface $S$.

I. Find the unit normal vector oriented inward the region $V$ at the point $P \left( \begin{array} { c } \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } \\ 0 \end{array} \right)$ on the surface $S$.

II. Find the area of the surface $S$ included in the region $W$.

III. Find the overlapping volume created by the two regions $V$ and $W$.

IV. Consider the three-dimensional curve $C$ on the surface $S$, which is defined by setting $\theta = \phi$. Find the curvature of the curve $C$ at the point $Q \left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ on the curve $C$. Note that, in general, given a three-dimensional curve defined by $c ( t ) = \left( \begin{array} { c } x ( t ) \\ y ( t ) \\ z ( t ) \end{array} \right)$ represented by a parameter $t$, the curvature $\kappa ( t )$ of the curve at the point $c ( t )$ on the curve is given by the following equation:

$$\kappa ( t ) = \frac { \left| \frac { \mathrm { d } c ( t ) } { \mathrm { d } t } \times \frac { \mathrm { d } ^ { 2 } c ( t ) } { \mathrm { d } t ^ { 2 } } \right| } { \left| \frac { \mathrm { d } c ( t ) } { \mathrm { d } t } \right| ^ { 3 } }$$