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\newcommand{\bR}{{\mathbb R}}
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\begin{document}
   Let $C$ be a subset of $ \bR ^n, v\in C$. In class we gave three
   criteria for $C$ to be a smooth curve at $\bar v =(v_1,\dots,v_n)$.

First definition:[graph form] there exists $i,1\leq i\leq n$ an open
subset $U$ 
of $\bR,v_i \in U$ and continuously differentiable functions
$f_1,\dots,f_{i-1},f_{i+1},\dots,f_n$ such that
$C=\{f_1(u),\dots,f_{i-1}(u),u,f_{i+1}(u),\dots,f_n(u)\},u\in U$ near $\bar
v$.

 Second definition:[level curve] there exists an open subset  subset $U$
of $\bR ^n,v\in U$ and continuously differentiable
 function $f\from U \to \bR ^{n-1}$  such that $Df(\bar v)$ is onto and
 $C=\{\bar x \in U|f(\bar x)=f(\bar v)\}$ near $\bar v$.

Third  definition:[parametrized curve] there exists a smooth map $r\from
I\to \bR ^n$ where $I$ is an open 
subset of $\bR,t_0 \in I$ such that

a)$ r(t_0 )=\bar v$,

b)$ Dr(t_0 )\neq 0$
and

c)$r\from I\to  \bR ^n$ is an embedding.

1)a)Prove that all three definitions are equivalent.

b)Give three  definitions for the  tangent line $T_C(\bar v )$ to $C$ at
$\bar v$
[that is, give a definition in the case
when  $C$ is given in the graph 
form,  for the case when  $C$ is given as a level curve and  for the case
when  $C$ is given as a parametrized curve] and show  the 
equivalence of these definitions [that is, show that the  tangent line
$T_C(\bar v)$ does not depend on a way you describe the curve $C$].

c) Show that one can omit the condition that $r\from I\to  \bR ^n$
is an embedding in the third definition of smoothness.

2)Let $C\{(x,y) \in \bR ^2|x^2+y^2=1\}$.

a)Prove that for any $(x,y) \in C$, the curve $C$ is smooth at $(x,y)$
using the first definition of smoothness.

b)Find a parametrization $r=(r_1,r_2)\from\bR \to C-(-1,0)$ such that
$r_2(t)=t (r_1(t)+1),t\in \bR$

c)Find all pairs $a,b$ of rational numbers such that $a^2 +b^2 =1$.

d)Use the parametrization $r$ of $C$ to find 
$\int 1/\sqrt {1-x^2}dx$

3)a)Let $f\from\bR \to \bR$  be a twice differentiable function,such that
  $f(0)=0$  and   $f'(0)\neq 0$. As we know that exists an inverse
  function $g\from I\to \bR,g(0)=0$ for a sufficiently small interval
  $I,0\in 
  I$. Prove that $g$ is twice differentiable at $0$ and find the
  expression for  $g''(0)$ in terms of $f'(0)$ and  $f''(0)$.

b)Let $f\from\bR \to \bR$  be a thrice  differentiable function, such that
  $f(0)=0$  and   $f'(0)\neq 0$. As we know that exists an inverse
  function $g\from I\to \bR,g(0)=0$ for a sufficiently small interval $I,0\in
  I$. Prove that $g$ is thrice differentiable at $0$ and find the
  expression for  $g'''(0)$ in terms of $f'(0),f''(0)$ and  $f'''(0)$.
\end{document}
