A question on tangent vector fields I'm currently taking an introductory course in Differential geometry of curves and surfaces. 
I have a question on vector field, $\textbf{w}$:
Given $p \in S, \textbf{w}(p)=f(u,v)\textbf{x}_u+g(u,v)\textbf{x}_v=(X(u,v),Y(u,v),Z(u,v)),$ where $\textbf{x}$ is a coordinate function of $S$ defined on some open set $U \subseteq \mathbb{R}^2$ and $X,Y,Z$ are some functions defined on $U.$ So, if $X,Y,Z$ are smooth, how should I prove that $f,g$ are also smooth?
Hints will suffice, thank you. 
 A: I might be missing something with the quantifiers here, but I read this as you want your condition on your vector field $\mathbf{w}$ to hold at a single point $p$, not a neighborhood of $p$.  (The answer\strategy changes if this is not the case.)  Under this assumption . . . 
The vector $\left(X(u, v), Y(u, v), Z(u, v)\right)$ is to be tangent to a surface $S$ at point $p$ when $S$ is embedded/immersed in $\mathbb{R}^{3}$.  It seems that it would suffice to take any other (tangent) vector in $\mathbb{R}^{3}$ and use that vector in combination with  $\left(X(u, v), Y(u, v), Z(u, v)\right)$ to define the tangent plane to $T_p S$.  
Now, to find the surface $S$, when can simply take the plane in $\mathbb{R}^{3}$ defined  by your point and the two vectors.  Now you merely need to parametrize said plane appropriately.
A: We are given a surface
$$S:\quad(u,v)\mapsto{\bf x}(u,v)=\bigl(x_1(u,v),x_2(u,v),x_3(u,v)\bigr)\in{\mathbb R^3}$$
and are assuming that ${\bf p}={\bf x}(0,0)$ is a regular point of $S$, i.e., that
$${\bf n}(u,v):={\bf x}_u\times {\bf x}_v\ne{\bf 0}$$
in a neighborhood $U$ of $(0,0)$. So let's assume that, e.g., $$n_3(u,v)=x_{1.u}x_{2.v}-x_{1.v}x_{2.u}\ne0\qquad\bigl((u,v)\in U\bigr)\ .\tag{1}$$
In addition there is talk of a tangent vector field ${\bf w}$ on $S$ which is given in the form
$${\bf w}(u,v)=f(u,v){\bf x}_u+ g(u,v){\bf x}_v\ .$$
Its ${\mathbb R}^3$ components $W_i$ can be written as
$$W_i(u,v)=f(u,v) x_{i.u}(u,v)+g(u,v)x_{i.v}(u,v)\qquad(1\leq i\leq 3)\ .$$
It is obvious that the smoothness of ${\bf x}(\cdot,\cdot)$, $f$, and $g$ implies the smoothness of the $W_i\>$. Conversely, assume that the $W_i$ are smooth. Then $(1)$ implies that the linear system
$$\eqalign{f(u,v) x_{1.u}+g(u,v)x_{1.v}(u,v)&=W_1(u,v) \cr
f(u,v) x_{2.u}+g(u,v)x_{2.v}(u,v)&=W_2(u,v) \cr}$$
can be solved for $f$ and $g$ and produces smooth functions $f$, $g:\>U\to{\mathbb R}$.
But note that the $W_i$ cannot be described independently. Together they have to fulfill the condition $\sum_{i=1}^3 n_i(u,v) W_i(u,v)\equiv0$, in order to make sure that the vector field ${\bf w}$ is indeed tangent to $S$ at all points.
