Suppose there are two $E^3$ surfaces, $$S:\mathbf{r}(u,v)=(au,bv,\frac{au^2+bv^2}{2})$$ $$\widetilde{S}:\widetilde{\mathbf{r}}(\tilde{u},\tilde{v})=(\tilde{a}\tilde{u},\tilde{b}\tilde{v},\frac{\tilde{a}\tilde{u}^2+\tilde{b}\tilde{v}^2}{2})$$The question is, what's the relation between $(a,b)$ and $(\tilde{a},\tilde{b})$ when there exists a isometric transform between $S$ and $\widetilde{S}$?
The answer provided by the book is $(a,b)=(\tilde{a},\tilde{b})$ or $(a,b)=(\tilde{b},\tilde{a})$. But there is no proof in detail.
The matrix of change of basis is an isometry for all tangent plane at the surface. Infacr, you note that an isometry is a map $f:S\rightarrow \tilde {S}$ such that for all $x\in S$ exist a neithborhod $U (x)\subseteq S$ for wich $ df:T_x S:\rightarrow T_{f (x)}\tilde {S} $ is an isometry for all $p\in U (x)$.