Let
$$dx^2+dy^2+dz^2 = E(u,v)du^2+2F(u,v)dudv+G(u,v)dv^2$$
and
$$dx^2+dy^2+dz^2 = \widetilde{E}d\widetilde{u}^2+2\widetilde{F}d\widetilde{u}d\widetilde{v}+\widetilde{G}d\widetilde{v}^2$$
Then apparently
$$\widetilde{E}=E\frac{\partial u}{\partial\widetilde{u}}\frac{\partial u}{\partial\widetilde{u}}+2F\frac{\partial u}{\partial\widetilde{u}}\frac{\partial v}{\partial\widetilde{u}}+G\frac{\partial v}{\partial\widetilde{u}}\frac{\partial v}{\partial\widetilde{u}}$$
How has this been derived?