If $\{v_1,v_2\}$ is a basis for $T_P(S)$, give a parameterization whose partial derivatives agree with $v_1$ and $v_2$

Question

Given a regular surface $S \subseteq \mathbb{R}^3$, a point $p \in S$ and a basis $\{v_1,v_2\}$ of the tangent plane $T_P(S)$, I want to prove the existence of a parameterization $\varphi: U \to S$ centered at $p$ for which $\frac{\partial \varphi}{\partial u}=v_1$ and $\frac{\partial \varphi}{\partial v}=v_2$.

I tried to give an argument in Existence of a parameterization whose differential has columns $v_1,v_2 \in T_p(S)$

However, I'm not sure it is correct. May you help me, please?