Let $\varphi:\mathbb{R}^2\to\mathbb{R}^3$ be the inmersed surface (Enneper's surface) given by, $$\varphi(u,v)=\left(u-\frac{u^3}{3}+uv^2,v-\frac{v^3}{3}+vu^2,u^2-v^2\right).$$
Prove that a connected component $S$ of $\varphi(\mathbb{R}^2)\smallsetminus(\{x=0\}\cup\{y=0\})$ is a regular surface.
My effort:
Let $p\in S$ and define the set $U=\{(u,v)\in \mathbb{R}^2|u^2+v^2<1\}$, then consider the function $\varphi:U \to S$ . Hence, $$\dfrac{\partial (x,y)}{\partial (u,v)}=1-(u^2+v^2)^2\neq 0,$$ so $d\varphi_p$ is injective.