I am an undergraduate taking course on geometry of curves and surfaces. While going through the book "Elementary differential geometry" by "Andrew Pressley" I came across the Proof of prop.8.4.3 page 203 which says
Let $\tilde σ : \tilde U → R^3$ be a surface patch, and suppose that for all $(\tilde u, \tilde v) ∈ \tilde U$ we are given tangent vectors $e_1(\tilde u, \tilde v) = a(\tilde u, \tilde v)\tilde σ_{\tilde u} + b(\tilde u, \tilde v)\tilde σ_{\tilde v}, e_2(\tilde u, \tilde v) = c(\tilde u, \tilde v)\tilde σ_{\tilde u} + d(\tilde u, \tilde v)\tilde σ_{\tilde v},$ whose components $a, b, c, d$ are smooth functions of $(\tilde u, \tilde v)$. Assume that, at some point $(\tilde u_0, \tilde v_0) ∈ \tilde U$ , the vectors $e_1(\tilde u_0, \tilde v_0)$ and $e_2(\tilde u_0, \tilde v_0)$ are linearly independent. Then, there is an open subset $\tilde V$ of $\tilde U$ containing $(\tilde u_0, \tilde v_0)$ and a reparametrization $σ(u, v)$ of $\tilde σ(\tilde u, \tilde v)$, for $(\tilde u, \tilde v) ∈ \tilde V$, such that $σ_u$ and $σ_v$ are parallel to $e_1$ and $e_2$, respectively.
In the proof of the theorem a function lambda is defined $\lambda(s_1,s_2)$ as shown in the image and the dependence of $\lambda(s_1,s_2)$ on $s_2$ everywhere is understood but the dependence on $s_1$ is not clear to me, except on the curve $\gamma_1(t)$. Hence, I can't convince myself that the function $\lambda(s_1,s_2)$ is smooth on the open set on which it is defined. I think I am missing something. Theory of ordinary differential equation is also being used. I just want to know how is the function $\lambda(s_1,,s_2)$ smooth.
The principal idea is that the solution curves of the vector fields $e_1$ and $e_2$ span a grid (locally), and that one can use this grid to define a new coordinate system (locally). Name the flows $\phi_k(s;p)$ for the vector field $e_k$ for the solution starting in a point $p$ of the surface.
To define the new grid coordinates one has to fix the parametrization along two reference lines, take them to be the unaltered solutions through the given center point $ p_0=σ( u_0, v_0)$. Then every other point can be obtained by following these grid lines back to the reference lines. Thus one has $$ σ(u,v)=\phi_2(t_2;\phi_1(s_1;p_0))=\phi_1(t_1;\phi_2(s_2;p_0)) $$ The grid coordinates for $σ(u,v)$ are then the implicitly defined $(s_1,s_2)$. The speed along a grid line as defined by its vector field will in general be different from the speed induced by the parametrization via the transversal grid lines, that is, $s_k$ will in general be different from $t_k$, the $t_k$ are locally smooth functions of $(s_1,s_2)$, again implicitly defined.
This complexity of the situation appears to be missing in the cited text.