Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and $\mathbb{R}^n[dx_i]$ the tangent and cotangent space at each point. By definition $dx_i(\partial/\partial x_j)=\delta^i_j$.
I can push vectors and covectors to the manifold $M$.
We define $v=\varphi_*(\frac{\partial}{\partial x_i}|_p)$ so that $\varphi_*(\frac{\partial}{\partial x_i})|_p(f)=\frac{\partial}{\partial x_i}|_{\varphi(p)}(f\circ\varphi^{-1})$
We define $f=\varphi^*(dx_j):=\frac{\partial\varphi}{\partial x_j}dx_j$
Do you agree with my definitions? Are this vectors still dual? Is there a general expression for $f(v)$? Is my last question sensible?