On the image of an embedding of $\mathbb{R}$ in $\mathbb{R}^3$

Question

The function $f:\mathbb{R}\to\mathbb{R}^3$ given by $t\mapsto(t,t^2,t^3)$ is clearly an injection and also an immersion. Also, using Heine Borel theorem one can show that it is a proper map, and thus is an embedding. Thus, the image $Z$ of this map is a submanifold of $\mathbb{R}^3$, of codimension $2$. Thus there exist two real functions $g_1$ and $g_2$ on $\mathbb{R}^3$ which cut out $Z$ and are locally independent, i.e., $Z$ is the set of all common zeros of $g_1$ and $g_2$ and the differentials of $g_1$ and $g_2$ are linearly independent on an open neighbourhood of $Z$. I want to find out two such functions explicitly. Any help will be appreciated.

Answer 1

$g_1(x)=x_2-x_1^2$ and $g_2(x)=x_3-x_1^3$ do that.