Eigenvalues of the differential of Gauss application.

Question

I am trying to understand how to obtain the eigenvalues of the differential of Gauss application (also called Weingarten application) in the case in which we consider the hyperbolic paraboloid.

This is an excerpt of Do Carmo's textbook on curves and surfaces:

I do not understand the step in which they say "Restricting $N(u,v)$ to this curve" and they conclude that $N'(0)=(2u'(0),-2v'(0),0)$. Would anyone be so kind to explain what is going on in that part?

In case it helps, this is how the differential of Gauss application is defined in my book:

Thank you.

Answer 1

Restricting $N$ to the curve means considering $N\circ\alpha$. They are computing $(N\circ\alpha)'(0)$. By the chain rule, this is $dN_p(\alpha'(0))$.