
\vspace*{.2in}
\noindent{\bf Supplementary problem 2: }
The function $h:\R\to\R^2$ is clearly $C^1$ and parametrizes the set $C$.  
Therefore $C$ is a dimension one (i.e. codimension 1) submanifold of 
$\R^2$ except at points where $h'$ fails to have (maximal) rank one.
Specificly,
$$
h'(t) = (2t,3t^2),
$$
which has rank equal to one unless both entries vanish--i.e. unless $t=0$.
Hence $C$ is a dimension one submanifold at all points other than
$h(0) = (0,0)$.  

The tangent space to $C$ at $h(t)$ is the image of $\R$ under the linear
transformation $h'(t)$.  In particular, since `1' is a basis for $\R$,
the vector $h'(t)\cdot 1 = (2t,3t^2)$ will be a basic for the tangent
space to $C$ at $h(t)$.


