\textbf{Problem 4}

Answer the following questions on shapes in the three-dimensional orthogonal coordinate system $xyz$.

\textbf{I.} Consider the surface $S _ { 1 }$ represented by the equation $x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } = 0$. Find the equations expressed in $x , y$, and $z$ of the normal line and the tangent plane $T$ to the surface $S _ { 1 }$ at the point $\mathrm { A } ( 2, 0, 2 )$.

\textbf{II.} Consider the surface $S _ { 2 }$ represented by the following set of equations with the parameters $u$ and $v$:
$$\left\{ \begin{array} { l } 
x = \frac { 1 } { \sqrt { 2 } } \cosh u \cos v \\
y = \frac { 1 } { 2 } \cosh u \sin v - \frac { 1 } { \sqrt { 2 } } \sinh u \\
z = \frac { 1 } { 2 } \cosh u \sin v + \frac { 1 } { \sqrt { 2 } } \sinh u
\end{array} \right.$$
where $u$ and $v$ are real numbers, and $0 \leq v < 2 \pi$.

Let $S _ { 3 }$ be the surface obtained by rotating the surface $S _ { 2 }$ around the $x$-axis by $- \pi / 4$. Here, the positive direction of rotation is the direction of the semi-circular arrow on the $yz$-plane shown in Figure 4.1.

Answer the following questions.
\begin{enumerate}
  \item Find the matrix $\boldsymbol { R }$ that represents the linear transformation rotating a shape around the $x$-axis by $- \pi / 4$.
  \item Find an equation expressed in $x , y$, and $z$ for the surface $S _ { 3 }$.
  \item Find an equation expressed in $x , y$, and $z$ for the surface $S _ { 2 }$.
\end{enumerate}

\textbf{III.} Consider the solid $V$ that is enclosed by the surface $S _ { 3 }$ obtained in Question II.2 and by the two planes $z = 1$ and $z = - 1$. Answer the following questions.
\begin{enumerate}
  \item Calculate the area of the cross section obtained by cutting the solid $V$ with the $xz$-plane.
  \item Calculate the area of the cross section obtained by cutting the solid $V$ with the plane $T$ obtained in Question I.
\end{enumerate}