Let $\sigma: U→R^3$ be a regular surface patch, let $(u_0,v_0)∈U$ and let $\sigma(u_0,v_0)=(x_0,y_0,z_0)$.
Suppose that the unit normal $N(u_0,v_0)$ is not parallel to the $xy$-plane.
Show that there is an open set $V∈R^2$ containing $(x_0,y_0)$, an open subset $W∈U$ containing $(u_0,v_0)$ and a smooth function $\phi:V→R$ such that $\Sigma(x_0,y_0)=(x,y,\phi(x,y))$ is a reparametrisation of $\sigma: W→R^3$. Thus, 'near' $p=\sigma(u_0,v_0)$, the surface is part of the graph $z=\phi(x,y)$.
What happens if $N(u_0,v_0)$ is parallel to the $xy$-plane?
How can I interpret the condition "the unit normal is not parallel to the $xy$-plane"? The gradient of $\sigma$ is non-zero? How can I apply the implicit function theorem here? Thanks. :)