Confused about the meaning of a differantial map in baby do Carmo.
For both $S_1, S_2$ there are differentiable maps $\phi_1:\Omega_1\to S_1$ and $\phi_2:\Omega_2\to S_2$ with $\Omega_1,\Omega_2$ open sets in $\mathbb R^2$, so a differentiable map $f:\Omega_1\to\Omega_2$ can be used to define $\varphi$, this is $$\phi_2\cdot f\cdot\phi_1^{-1}.$$