Doubt in differential of Gauss Map

Question

This might be a trivial doubt. It is from the book "do Carmo Differential Geometry of Curves and surfaces"

Here, it is mentioned that "the tangent vector $N'(0) = dN_p(\alpha '(0))$is a vector in $T_p(S)$".
How is that? I would like a geometric as well as mathematical answer.

Later, the author writes the following -

How is $dN_p(x_uu'(0) + x_vv'(0)) = \frac d{dt} N(u(t), v(t))|_{t=0}$?

Answer 1

As you found yourself, the answer to your second question is that the author abuse notation. The same goes for your first question. It is rather $(N\circ\alpha)'(0) = dN_p(\alpha '(0))$, which follows from definition.

As for the answer to your first question, note that $dN_p$ has codomain $T_p(S)$ and so $dN_p(\alpha '(0))\in T_p(S)$.

Answer 2

The analytical explanation of the first fact is simply that, since $$ N(t)\cdot N(t)=1,\quad \forall t, $$ then differentiating this relation you obtain $$ \frac{ dN}{dt}\cdot N=0.$$ Here $N(t)$ is the restriction of the normal field $N$ to the curve $\alpha=\alpha(t)\in S$.