臺灣大學數學系

八十九學年度第二學期碩博士班資格考試試題

幾何

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1.

　

　

Let $P$ be the pole of polar coordinate. $r = 1 - \sin \theta$ is a cardioid. $Q = (x,y) = (0,-2)$. Arclength ${\stackrel{\frown}{PQ}} = L =$? A string of length $L$ has its one end fixed at $P$ and winds around the cardioid so that its other end generates the involute of the cardioid. Is this involute a cardioid, too? (25/100)

2.

Can you find a surface in $\mathbb{R}^3$ passing through the origin $(x,y,z)=(0,0,0)$ so that both its mean curvature $H$ and Gauss curvature $K$ vanish at the origin yet the surface is not a plane? If not, explain why not. (25/100)

3.

Can you find a closed differential 2-form ω in $\mathbb{R}^3 - (0,0,0)$ which is not exact? If yes, $\omega=? dx \wedge dy+ ? dy \wedge dz + ? dz \wedge dx$ (25/100)

4.

　

$\mbox{Cone}=\{x^2+y^2=z^2\}, \ P=(1,0,1),Q=(-1,0,1)$
$\gamma_1=\mbox{cone} \cap \{z=1\} \cap \{y \ge 0\}$
$\gamma_2=\mbox{cone} \cap \{z=1\} \cap \{y \le 0\}$
$\vec{V}=(-1,0,-1)$ is a tangent vector to the cone at $P$.
Parallel translate $\vec{V}$ from $P$ to $Q$ along $\gamma_1=(?,?,?)$. If we translate along $\gamma_2$ instead of $\gamma_1$, do we get the same vector? (25/100)

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