Let $\phi : U \subseteq \mathbb{R}^2 \to M \subset \mathbb{R}^k$ be a parametrization of a surface M. Suppose $p \in M$ and (u,v) and $(\bar{u},\bar{v})$ are 2 sets of local coordinates around p. We know that any vector $\vec{w} \in T_p(M)$ can be represented by
$$\vec{w} = A^1 \frac{\partial\vec{\phi}}{\partial u} + A^2 \frac{\partial\vec{\phi}}{\partial v} = \bar{A^1} \frac{\partial\vec{\phi}}{\partial u} + \bar{A^2} \frac{\partial\vec{\phi}}{\partial v} $$
Using the chain rule, show that $\vec{w}$ is a contravariant vector.
What I have so far is this:
Let $A^1 = \frac{dx^1}{dt}$ and $A^2 = \frac{dx^1}{dt}$
$\bar{A^1} = \frac{d\bar{x^1}}{dt} = \frac{d\bar{x^1}}{dx^1}\frac{dx^1}{dt}+\frac{d\bar{x^1}}{dx^2}\frac{dx^2}{dt}+...+\frac{d\bar{x^1}}{dx^n}\frac{dx^n}{dt} $= $A^1 \frac{d\bar{x^1}}{dx^1} + A^2 \frac{d\bar{x^1}}{dx^2}+...+A^n\frac{d\bar{x^1}}{dx^n}$
Is this correct and if so how can I change it to correct notation with $\phi$ and u,v