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\title{Math 23a Problem Set 10}
\begin{document}

We say that a set $C\in \bR ^2$ is a smooth curve if for any $(a,b)\in
C$ our set $C$ is a smooth curve at  $(a,b)$. Analogously we say that 
a set $ \Sigma \in \bR ^3$ is a smooth surface if for any
$(a,b,c)\in \Sigma$ the set  $\Sigma$ is a  smooth surface at $(a,b,c)$.

1.Let $F(x,y)=x^3+y^3-3xy$. For any $r\in \bR$ we define  $C_r=\{ (x,y)\in
\bR ^2 |F(x,y)=r\}$ 

a)Find for which $r$ the level curve $C_r$ is not smooth.

b)For any $r$ such that the level curve $C_r$ is not smooth find all
points $(a,b)\in C_r$ such that the curve  $C_r$
is singular [not smooth] at the point  $(a,b)$.

c)Find all points $(a,b)\in \mathbb C ^2$ such that the curve
$F(x,y)=F(a,b)$ is singular [not smooth] at the point  $(a,b)$.

d)Draw the curve $C_r \subset  \bR ^2$ for   $r=0$ and the case when $r$
is a small positive and negative number.

e)Show that for any $t\in \bR ,t\neq 0,-1$ there exist unique point
$x(t),y(t) \in C_0\setminus\{(0,0)\}$ such that $y(t)=tx(t)$.

f)Show that the map $t\rightarrow \phi (t)=(x(-1+1/t),y(-1+1/t)),t\in \bR
-\{ 0,1\} $ extends to a
parameterization $\phi :\bR \rightarrow C_0$ of the curve $ C_0$

g) Show that for small $\epsilon >0$ the intersection $C_0\cap
U_\epsilon$is a union of $C' \cup C''$ where $C', C''$ are
 smooth curves which intersect the point  point $(0,0)$ where 
 $U_\epsilon =(-\epsilon , \epsilon )\times (-\epsilon , \epsilon )$

h)Find the tangent lines to  $C'$ and $C''$ at $(0,0)$.

2.Let $\Sigma =\{ (x,y,z)\in \bR ^3 |x^2+y^2-z^2=1\}$

a)Prove that  $\Sigma$ is a  smooth surface. Are you familiar with this
surface ?

b) For any $r\in \bR$ define $C_r =\{ (x,y,z)\in \Sigma |x+y+z=r\}$. In
other words  $C_r$ is the intersection of  $\Sigma$ with the plane
$x+y+z=r$. 

Find for which  $r\in \bR$ the curve  $C_r$ is not smooth. Describe the
curve $C_r$ for such  $r\in \bR$ and find singular points on  $C_r$.

c)Do the same problem for the surface  $\Sigma '=\{ (x,y,z)\in \bR ^3
|x^2+y^2+z^2=1\}$ 

3.Let  $\Sigma \subset  \bR ^3 $ be a smooth surface such that
,$(0,0,0)\in \Sigma,f: \bR ^3 \rightarrow \bR$ be a linear function,
$C_f =\{ (x,y,z)\in \Sigma |f(x,y,z)=0\}$ .

a)Prove that $C_f$ is a smooth curve at $(0,0,0)$ if and only if the
restriction $f_\Sigma$ of $f$ to the tangent plane $T_\Sigma (0,0,0)$ is
not equal to $0$.

b)Show that in this case the tangent line to  $C_f$ at $(0,0,0)$ is
equal to $Ker f_\Sigma$.

c)EXTRA CREDIT Formulate a statement which generalizes this result.
\end{document}
