I. Consider surfaces presented by the following sets of equations, with parameters $u$ and $v$ in a three-dimensional orthogonal coordinate system $x y z$. Show the equations for the surfaces without the parameters and sketch them. Here, $a , b$, and $c$ are non-zero real constants.

\begin{enumerate}
  \item $x = a u \cosh v , y = b u \sinh v , z = u ^ { 2 }$.
  \item $x = a \frac { u - v } { u + v } , y = b \frac { u v + 1 } { u + v } , z = c \frac { u v - 1 } { u + v }$.
\end{enumerate}

II. In a three-dimensional orthogonal coordinate system $x y z$, consider the surface $S$ represented by the following equation, where $a$ and $b$ are real constants.

$$z = x ^ { 2 } - 2 y ^ { 2 } + a x + b y$$

\begin{enumerate}
  \item Determine the normal vector at a point $( x , y , z )$ on the surface $S$.
  \item Determine the equation for the surface $T$ which is obtained by rotating the surface $S$ around the $z$-axis by $\pi / 4$. Here, the positive direction of rotation is counter-clockwise when looking at the origin from the positive side of the $z$-axis.
  \item Consider the surface $S ^ { \prime }$, which is the portion of the surface $S$ in $- 1 \leq x \leq 1$ and $- 1 \leq y \leq 1$. Determine the area of the projection of the surface $S ^ { \prime }$ onto the $y z$ plane.
  \item Calculate the length of the perimeter for the surface $S ^ { \prime }$ when $a = b = 0$.
  \item Calculate the Gaussian curvature of the surface $S$ at the point $\left( 0 , \frac { 1 } { 4 } , - \frac { 1 } { 8 } \right)$ when $a = b = 0$.
\end{enumerate}