I am reading Differential Geometry of Curves and Surfaces from M.P. Do Carmo. In the chapter about minimal surfaces he says that,
(1) A regular parametrized surface is called minimal if its mean curvature vanishes everywhere.
(2)A regular surface $S \subset \mathbb{R}^3$ is minimal if each of its parametrizations is minimal
So I wanted to show that the catenoid given by $$x(u,v)=(\cosh(u)\cosh(v),\cosh(u)\sin(v),u)$$ is minimal.
Do I just need to show that the mean curvature $H(p)$ is zero everywhere? Which I would assume according to (1). But why does then (2) state that this needs to be done for every parametrization?