Differential Geometry by Pressley (problem 6.3.2 second edition):
"Show that the Ennerper's surface $\pmb\sigma(u,v)=(u-u^3/3+uv^2, v-v^3/3+vu^2, u^2-v^2)$ is conformally parametrized."
I have computed the first fundamental form of the surface to be $(1+u^2+v^2)^2(du^2+dv^2)$. But I don't know what to compare it to, to show it's proportional. Pressley has answers at the back, and he says it's a multiple of $du^2+dv^2$. That is the fundamental form of the plane, so why is he comparing it to the plane? How do we know what to compare it to, when we are given only one surface patch?
Thanks to John Ma, I found that it says in the book as part of a Corollary (p.136), that a surface patch $\pmb\sigma(u,v)$ is conformal if and only if its first fundamental form is $\lambda(du^2+dv^2)$ for some smooth function $\lambda(u,v)$.