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. 

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.
