Let $\Phi$ be the regular surface at $(u_o,v_o)$ (ie., $\Phi$ is of class $C^1$ and $T_u\times T_v\ne 0$
a)Use the implicit function theorem to show that the image of $\Phi$ near $(u_o, v_o)$ is the graph of a $C^1$ function of 2 variables. If the z components of $T_u\times T_v$ is non zero, we can write it as $z=f(x,y)$
b) Show that the tangent plane at $\Phi(u_o, v_o)$ defined by $T_u\times T_v$ coincides with the tangent plane of the graph of $z=f(x,y)$ at this point.
I do not understand the question very well, what is the image of the surface near the points?