# Exams

## First Midterm Exam Questions

1. A regular parametrized curve $\alpha$ has the property that all its tangent lines pass through a fixed point. Prove that the trace of $\alpha$ is a (segment of a) straight line.
2. Let $\alpha: I \rightarrow R^3$ be a regular curve paramterized by its arclength $s \in I$ . Prove that the oscillating plane at $s$ is the limiting position of the plane determined by the tangent line at $s$ and the point $\alpha(s+h)$ when $h \rightarrow 0$.
3. Given the parametrized curve $\alpha(y)=(3t,3t^2,2t^3)$, where $t \in R$. Prove that this curve is a helix (that is $k/\tau=constant$).
4. Let $\alpha: I \rightarrow R^2$ be a simple closed curve with length l. Let $A$ be the area of the region bounded by this curve. Prove that $A \le l^2/4\pi$.
5. Find a necessary and sufficient condition that a curve lie upon a sphere.

## Second Midterm Exam Questions

1. Prove that the torus a is regular surface.
2. Show that the paraboloid ,$S=\{ (x,y,z) \in R^3; z=x^2+y^2 \}$ , is diffeomorphis to a plane.
3. Show that if all normals to a connected surface pass through a fixed point , the surface is contained in a sphere.
4. Prove that if $L: R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under L, i.e., $L(S) \subset S$ , then the restriction $L/S$ is differentiable and $dL_{p}(w)=L(w)$ where $p \in S$ and $w \in T_{p}(S)$.
5. Let $S$ be a regular surface covered by coordinate neighborhoods $V_{1}$ and $V_{2}$. Assume that the set $W=V_{1} \cap V_{2}$ is connected. Prove that $S$ is an orientable surface.

## Third Midterm Exam Questions

1. (a). Let S be a regular and orientable surface and let $p \in S$. Let $w_{1}, w_{2} \in T_{p}(S)$. Prove that $$dN_{p}(w_{1}) \wedge dN_{p}(w_{2}) =K(p) w_{1} \wedge w_{2}$$. (b). If S is a minimal surface ($H=0$) then  =-K(p) .
2. Show that if w is differentiable vector field on a surface and $w(p) \ne 0$ for some $p \in S$ , then it is possible to parametrize a neighborhood of p by x(u,v) in such a way that $x_{,u}=w$
3. Shoe that at the origin $(0,0,0)$ of the parabolic hyperboloid $z=a xy$ we have the Gauss curvatuture , $K=-a^2$ and the mean curvature , $H=0$. here a is a nonzero constant.
4. Let $x(u,v)$ be a parametrization at $p \in S$ , with $p =x(0,0)$ , and let $e(u,v)=e, f(u,v)=f, g(u,v)=g$ be the coefficients of second fundamental form in this parametrization. Prove that a necessary and sufficient conditions for a parametrization in a neighborhood of a hyperbolic point to be such that the coordinate curves of the parametrization are asymptotic curves is that $e==g=0$.
5. Let $\alpha: I \rightarrow S \subset R^3$ be a curve on a regular surface S and consider the ruled surface generated by the family {\alpha(t), N(t)}, where $N(t)$ is the normal to the surface at $\alpha(t)$. Prove that $\alpha(I) \subset S$ is a line of curvature in S if and only if this ruled surface is developable.

## Final Exam Questions

1. Let S be a regular and connected surface. Let E, F, G and e,f,g be the coefficients of the first and second fundamental forms of S respectively. Let $e=\lambda E, f=\lambda F$ and $g=\lambda G$ , where $\lambda$ is an arbitrary nonzero constant (at all points on S). Prove that S is a sphere.
2. Assume that the oscullating plane of a line of curvature $C \subset S$, which is nowhere tangent to an asymptotic direction , makes a constant angle with the tangent plane of S along C. Prove that C is a plane curve.
3. Let $\al : Ipha \rightarrow R^3$ be a regular parametrized curve with nonzero curvature everywhere and arclength s as parameter. Let
$x(s,v)=\alpha(s)+r[n(s) cos v+ b(s) sin v],$
where r = constant different from zero , $s \in I$ be a parametrized surface (a tube of radius r around $\alpha$) where n is the normal and b is the binormal vector of $\alpha$. (a). Find the singular points of this surface. (b). Find the Gaussian curvature for the case where no singular points exist on the surface and classify its points. (c). Find necessary and sufficient conditions for this surface to be a torus.
4. Prove that the parametrized surface defined by
$x(u,v)= (u-u^3/3+uv^2, v-v^3/3+vu^2,u^2-v^2), (u,v) \in R^2$
is minimal.
5. Prove that if a regular surface S meets a plane in a single point $p \in S$, then this plane coincides with the tangent plane $S$ at p.

## First Midterm Exam

1. let $\alpha: I \rightarrow R^3$ be a regular curve with priciple normal vector $n(s)=(cos as, sin as, 0). Here a is a nonzero constant and$s \in I$is the arclength parameter. Determine the curve completely (find$\alpha$,$k(s)$the curvature and$\tau(s)$the torsion). 2. If a closed plane curve$C$is contained inside a disc of radius$r$prove that there exist a point$p \in C$such that the curve$k$of$C$at$p$satisfies$|k|> 1/r$. 3. Show that the curvature$k(t) \ne 0$of a regular paramterized curve$\alpha: I \rightarrow R^3$is the curvature at$t$of the curve$\pi o \alpha$, where$\pi is the normal projection of $\alpha$ over the oscullating palne at $t$.

## Second Midterm Exam

A Tubular Surface: Let $\alpha: I \rightarrow R^3$ be a regular parametrized curve with nonzero curvature everywhere and arclength as parameter. let
X(s,v)=\alpha(s)+r(n(s) \cos v+ b(s) \sin v)
where $r=constant$, $s \in I$ and $v\in (0,2\pi)$, be a parametrized surface (the tube of radius $r$ around $\alpha(s))$, where $n$ is the normal and $b$ is the binormal vector of $\alpha$.
1. Discuss the regularity of this surface. 2. Show that, when $X$ is regular, its unit normal vector filed is
N(s,v)=-(n(s) cos v+b(s) sin v)
3. Discuss the diffeomorphism between this surface and a right cylinder. 4. Find the coefficients of the first fundamental form. 5. Find the area of this surface when $I=[0,L]$. 6. Discuss the orientibility of this surface.

## Third Midterm Exam

1.a. Let $S$ be aregular oriented surface. Prove that an umbilical point on $S$ satisfies $<{d(fN) \over dt} \times {d \alpha \over dt}>=0$ for all nonzero function $f$ and for every curve on $\alpha$ on $S$.
1.b. Find the umbilical points of the ellipsoid ${x^2 \over a^2}+ {y^2 \over y^2}+{z^2 \over c^2}=1$. 1.c. Discuss the limiting cases $b=c$ and $a=b=c$.
2. Find the eigenvectors and eigenvalues of differential of the Gauss map of an ellipsoid at the points $p_{1}=(\pm a,0,0), p_{2}=(0,\pm b,0), p_{3}=(0,0,\pm c)$.