Vtyjkyuk

Let $S subseteq mathbbR^3$ be a surface and $p in S$. Given any basis $v_1,v_2$ of the tangent plane, there exists some parameterization $varphi: U to S$ centered at $p$ such that $fracpartial varphipartial u=v_1$ and $fracpartial varphipartial v=v_2$?

In general, if I take $v_1$, there exists a curve $alpha: (-epsilon,epsilon) to S$ such that $alpha(0)=p$ and $alpha'(0)=v_1$. Similarly for $v_2$. If $psi$ is a given parameterization whose image contains the trace of the curve $alpha$, I consider $dpsi^-1(v_1)=w_1$ that clearly belongs to $T_p(U)=mathbbR^2$. Then $w_1$ may be written as linear combination of the basis of $T_q(U)$, that we denote by $fracpartial partial x,fracpartialpartial y$. Repeating the argument for $v_2$, we obtain a matrix $A$ whose rows represent the coordinates of $w_1$ and $w_2$ with respect to the basis of $T_q(U)$. Is the map $varphi:=psi circ A$ the desired parameterization? May you express the direct computations for $fracpartial (psi circ A)partial u$, please?