Final Exam

1. Let $\vec{x}(u,v)$ be a parametrization of a surface and suppose that $E=1$, $F=0$, and $G$ is some function of $u$ and $v$. Find the surface's Gaussian curvature, $K$, in terms of $G$.
2. A parametrization $\vec{x}(u,v)$ of a surface such that $E=G=1$ is called a Tchebycheff parametrization. It follows that $\vec{x}_u$ and $\vec{x}_v$ are unit vectors, and therefore $F = \vec{x}_u \cdot \vec{x}_v = \cos\theta$, where $\theta$ is the angle between those vectors. Find the surface's Gaussian curvature $K$ in terms of $\theta$.
3. Consider the spacecurve $\vec{\alpha}(s)$ parameterized by arclength, and let $\vec{b}(s)$ be its binormal vector. Define the ruled surface $S$ through $\vec{x}(s,v) = \vec{\alpha}(s) + v\,\vec{b}(s)$. Show that $\vec{\alpha}$ is a geodesic on $S$.
4. Let $R$ be the part of the graph of the paraboloid $z=x^2+y^2$ in Cartesian coordinates $x, y, z$ that lies between the planes $z=0$ and $z=a^2$ for some $a>0$. For convenience, let's express $R$ in the familiar parametric form $\vec{x}(u,v) = \langle u, v, u^2 + v^2 \rangle$
   1. Calculate the Gaussian curvature $K$ of $R$ as a function $u$ and $v$.
   2. Evaluate $\ds\int_R K \,dA$.
   3. Evaluate $\ds\int_C \kappa_g \,ds$, where $\kappa_g$ is the geodesic curvature of $R$'s boundry curve $C$.
   4. Conclude that the Gauss–Bonnet theorem holds on $R$.
5. In the subfigure (a) below, the angle $\theta$ is $15^\circ = \frac{\pi}{12}$ radians. The slanted line is rotated about the $z$ axis to generate the right circular cone of base radius $1$ and height $h$, shown in subfigure (b).
   1. Find the parametrization $\vec{x}(u,v)$ of the cone, where $u \in [-\pi,\pi]$ and $v \in [0,h]$.
   2. Calculate and plot the geodesic $\vec{\alpha}(t) = \vec{x}\bigl( u(t), v(t) \bigr)$ that takes off from $\vec{x}(0,1)$ with $u'(0)=v'(0)=1$ as seen in subfigure (b),

   (a)	(b)