Show that the map $\varphi:\mathbb{R}^2\to \mathbb{R}^3$ defined by, $$\varphi(u,v)=\left(u-\frac{u^3}{3}+uv^2,v-\frac{v^3}{3}+vu^2,u^2-v^2\right),$$ is an injective immersed surface.
The problem is from the book Curves and Surfaces by Abate and Tovena.