= Solution
A regular surface is a <minimal surface> when its <mean curvature> vanishes identically, equivalently when its two <principal curvature>[principal curvatures] sum to zero everywhere.
The <catenoid>
$$
\phi(u,v)=(\cosh v\cos u,\cosh v\sin u,v)
$$
is an example. In part (a), take $f(v)=\cosh v$ and $g(v)=v$, so $q=\cosh v$. The formula there gives
$$
H=\frac12\left(\operatorname{sech}^2v-\operatorname{sech}^2v\right)=0,
\qquad
K=-\operatorname{sech}^4v<0.
$$
A plane and a <circular cylinder> have zero <Gaussian curvature>, as does a <flat cone> away from its vertex. Because a <local isometry> preserves Gaussian curvature, \b[the catenoid is minimal and is not locally isometric to any of those surfaces]; this is the <catenoid is a minimal surface>.
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