Let $u\in C^\infty(\Omega,\mathbf R^3)$ with $\Omega$ open set in $\mathbf R^2$. Can we find $u_k\in C^\infty(\Omega,\mathbf R^3)$ with $\mathrm{rank}(Du_k(x))=2$ for all $x\in\Omega$ such that $u_k \to u$ uniformly?
added. I actually have $\Omega$ bounded and $u\in C^\infty(\Omega)\cap C^0(\overline \Omega)$ with $u=0$ on $\partial \Omega$ and need to approximate uniformaly with functions $u_k$ with the same properties and full jacobian rank. Then I would like to apply Nash-Kuiper theorem to $u_k$ to obtain an isometric map approximating $u$.