(i) State what it means for surfaces $f: U \rightarrow \mathbb{R}^{3}$ and $g: V \rightarrow \mathbb{R}^{3}$ to be isometric.

Let $f: U \rightarrow \mathbb{R}^{3}$ be a surface, $\phi: V \rightarrow U$ a diffeomorphism, and let $g=f \circ \phi: V \rightarrow$ $\mathbb{R}^{3} .$

State a formula comparing the first fundamental forms of $f$ and $g$.

(ii) Give a proof of the formula referred to at the end of part (i). Deduce that "isometry" is an equivalence relation.

The catenoid and the helicoid are the surfaces defined by

$$(u, v) \rightarrow(u \cos v, u \sin v, v)$$

and

$$(\vartheta, z) \rightarrow(\cosh z \cos \vartheta, \cosh z \sin \vartheta, z)$$

Show that the catenoid and the helicoid are isometric.