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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.

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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.

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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.

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  • $\begingroup$ Thank you for the detailed response. Here is the an excerpt of the original problem quoted from my lecture notes: A vector field $\textbf{w}$ on $S$ is smooth if it is smooth at every point on $S.$ For a point $\textbf{w}(p)$ on $S,$ let $\textbf{w}(p)=f(u,v)\textbf{x}_u+g(u,v)\textbf{x}_v =(X(u,v),Y(u,v), Z(u,v)).$ Then, $f,g$ are smooth functions of $(u,v) \in U$ iff $X,Y,Z$ are smooth functions of $(u,v) \in U.$ $\endgroup$ – Alexy Vincenzo Apr 4 '14 at 17:38
  • $\begingroup$ I apologize for my ambiguous phrasing. $\endgroup$ – Alexy Vincenzo Apr 4 '14 at 17:39
  • $\begingroup$ The hint provided is to assume $ \left|\begin{matrix} f_u & f_v \\ g_u & g_v \end{matrix}\right| \neq 0 $ and work backwards to express $X,Y$ in terms of $f,g $ $\endgroup$ – Alexy Vincenzo Apr 4 '14 at 17:59
  • $\begingroup$ I have made the necessary amendments to the question. $\endgroup$ – Alexy Vincenzo Apr 4 '14 at 18:19

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