I am studying differential geometry with do carmo's book. In his book, on page 136, the definition of the Gauss map is given,
He then proceeds to say it's straightforward to verify the Gauss map is differentiable. Actually, I have no idea how to prove this, can someone help me understand why? Any explanation will be very welcome.
Well, this depends on the smoothness of $S$. We need $S$ to be at least $C^2$. Locally, say $S$ is parametrized by $x(u,v), y(u,v), z(u,v)$. Then we have two tangent vectors $T_u = (x_u, y_u, z_u)$ and $T_v = (x_v, y_v, z_v)$ everywhere. The normal vector is $T_u\times T_v$ (or $-T_u\times T_v$ which depends on the orientation). This needs to be normalized to be $$N(u,v)=\frac{T_u\times T_v}{\|T_u\times T_v\|}$$
Now to show $N$ is differentiable, note that $N_u, N_v$ only depends on $T_u, T_v, T_{uu},T_{uv}, T_{vv}$, and since $T$ is $C^2$, we know $N_u, N_v$ are continuous and hence $N$ is at least $C^1$.
However, I would suggest not to worry too much about smoothness as a beginner. You may assume everything is $C^{\infty}$ and gain the key geometric intution first.