A question about Rudin's PMA, definition 10.46 The definition which we need to understand the theorem:
The class $\mathscr C'$ is the class of continuously differentiable functions.


I don't understand how do we get $(135)$
-
$N$ $=$ ($\alpha_2$$\beta_3$ $-$ $\alpha_3$$\beta_2$)$e_1$ $+$ ($\alpha_3$$\beta_1$ $-$ $\alpha_1$$\beta_3$)$e_2$ $+$ ($\alpha_1$$\beta_2$ $-$ $\alpha_2$$\beta_1$)$e_3$
and
$(137)$
$N$ $\cdot$ $(Te_1)$ $=$ $0$ $=$ $N$ $\cdot$ $(Te_2)$
I would be grateful for any kind of help.

$\Phi$ is a surface via both $(u,v)$ and the $N$ representation ($130$).
The equations in ($133$) describe the differentials of the function $\varphi_i$ at the point $p_0$ using the two generic differential operators $D_1$ and $D_2$, which correspond to the differential operators in the numerators of ($129$), i.e. $D_1=y,D_2=z$ etc... for each term., and lead to ($135$) via the Jacobian.
($137$) uses the dot product and is straight-forward as each positive term has a negative partner.
$$N = (\alpha_2\beta_3-\alpha_3\beta_2)e_1+(\alpha_3\beta_1-\alpha_1\beta_3)e_2+(\alpha_1\beta_2-\alpha_2\beta_1)e_3$$
$$T(e_1)= \alpha_1e_1+\alpha_2e_2+\alpha_3e_3$$
$$N\cdot T(e_1)=(\alpha_2\beta_3-\alpha_3\beta_2)\alpha_1+(\alpha_3\beta_1-\alpha_1\beta_3)\alpha_2+(\alpha_1\beta_2-\alpha_2\beta_1)\alpha_3=0$$
Similarly
$$T(e_2)= \beta_1e_1+\beta_2e_2+\beta_3e_3$$
$$N\cdot T(e_2)=(\alpha_2\beta_3-\alpha_3\beta_2)\beta_1+(\alpha_3\beta_1-\alpha_1\beta_3)\beta_2+(\alpha_1\beta_2-\alpha_2\beta_1)\beta_3=0$$
