In coordinate space, consider two vectors $\vec{u}$ and $\vec{v}$ satisfying the dot product $\vec{u} \cdot \vec{v} = \sqrt{15}$ and the cross product $\vec{u} \times \vec{v} = (-1, 0, 3)$. Select the correct options.
(1) The angle $\theta$ between $\vec{u}$ and $\vec{v}$ (where $0 \leq \theta \leq \pi$, $\pi$ is the circumference ratio) is greater than $\frac{\pi}{4}$
(2) $\vec{u}$ could be $(1, 0, -1)$
(3) $|\vec{u}| + |\vec{v}| \geq 2\sqrt{5}$
(4) If $\vec{v}$ is known, then $\vec{u}$ can be uniquely determined
(5) If $|\vec{u}| + |\vec{v}|$ is known, then $|\vec{v}|$ can be uniquely determined

In coordinate space, there are three mutually perpendicular vectors $\vec { u } , \vec { v } , \vec { w }$. Given that $\vec { u } - \vec { v } = ( 2 , - 1,0 )$ and $\vec { v } - \vec { w } = ( - 1,2,3 )$. What is the volume of the parallelepiped spanned by $\vec { u } , \vec { v } , \vec { w }$?
(1) $2 \sqrt { 5 }$
(2) $5 \sqrt { 2 }$
(3) $2 \sqrt { 10 }$
(4) $4 \sqrt { 5 }$
(5) $4 \sqrt { 10 }$

Problem 4

For the real numbers $\theta$ and $\alpha$ within the regions $0 \leq \theta < 2 \pi$ and $0 \leq \alpha \leq \pi$, consider the line L that passes through two points: point $\mathrm { P } ( \cos \theta$, $\sin \theta , 1 )$ and point $\mathrm { Q } ( \cos ( \theta + \alpha ) , \sin ( \theta + \alpha ) , - 1 )$ in a three-dimensional Cartesian coordinate system $x y z$. I. Represent the line L as a linear function of a parameter $t$. Here, the point on the line L at $t = 0$ should represent the point Q and the point at $t = 1$ should represent the point P. II. Find the surface S swept by the line L as an equation of $x , y$ and $z$ when $\theta$ varies in the region $0 \leq \theta < 2 \pi$. Let C be the intersection lines of the surface S with the plane $y = 0$. Find the equation of C in terms of $x$ and $z$, and sketch the shape of C.
Next, examine the Gaussian curvature of the surface S. Generally, when the position vector $r$ of a point R on a curved surface is represented using parameters $u$ and $v$ by
$$\boldsymbol { r } ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) ,$$
the Gaussian curvature $K$ is represented as the following equation:
$$K = \frac { \left( \boldsymbol { r } _ { u u } \cdot \boldsymbol{e} \right) \left( \boldsymbol { r } _ { v v } \cdot \boldsymbol { e } \right) - \left( \boldsymbol { r } _ { u v } \cdot \boldsymbol { e } \right) ^ { 2 } } { \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { u } \right) \left( \boldsymbol { r } _ { v } \cdot \boldsymbol { r } _ { v } \right) - \left( \boldsymbol { r } _ { u } \cdot \boldsymbol { r } _ { v } \right) ^ { 2 } } ,$$
where $\boldsymbol { r } _ { u }$ and $\boldsymbol { r } _ { v }$ are first-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$, and $\boldsymbol { r } _ { u u } , \boldsymbol { r } _ { u v }$ and $\boldsymbol { r } _ { v v }$ are second-order partial differentials of $\boldsymbol { r } ( u , v )$ with respect to the parameters $u$ and $v$. $( \boldsymbol { a } \cdot \boldsymbol { b } )$ represents the inner product of two three-dimensional vectors $a$ and $b$, and $e$ is the unit vector of the normal direction at the point R. III. Let the point W be the intersection of the surface S and the $x$ axis in the region $x > 0$. Calculate the Gaussian curvature of S at the point W for $\alpha$ within the region $0 \leq \alpha < \pi$. IV. For $\alpha$ within the region $0 \leq \alpha < \pi$, prove that the Gaussian curvature is less than or equal to 0 at arbitrary points on the surface $S$.

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.

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

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

1. Determine the normal vector at a point $( x , y , z )$ on the surface $S$.
2. 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.
3. 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.
4. Calculate the length of the perimeter for the surface $S ^ { \prime }$ when $a = b = 0$.
5. Calculate the Gaussian curvature of the surface $S$ at the point $\left( 0 , \frac { 1 } { 4 } , - \frac { 1 } { 8 } \right)$ when $a = b = 0$.

Problem 4
Answer the following questions on shapes in the three-dimensional orthogonal coordinate system $xyz$.
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 $\mathrm { A } ( 2, 0, 2 )$.
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.

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

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.

1. Calculate the area of the cross section obtained by cutting the solid $V$ with the $xz$-plane.
2. Calculate the area of the cross section obtained by cutting the solid $V$ with the plane $T$ obtained in Question I.

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