Let $M$ be an $n$-dimensional manifold and $p \in U \subset M$. Let $k: O \rightarrow U$ be a coordinate chart where $O \subset \mathbb{R}^n$ (here I follow Weintraub's Differential Forms: Theory and Practice in defining a coordinate chart as a map from a Euclidean space to the manifold). Let $r_1$ and $r_2$ be the parametrizations of two curves on $M$ with $r_1(0) = r_2(0) = p$. We say that these two curves are equivalent if $(k{}^{-1} \circ r_1)'(0) = (k{}^{-1} \circ r_2)'(0)$. Then the tangent vectors to $M$ at $p$ are defined as the equivalence classes of curves on $M$ passing through $p$.
Now according to Weintraub, there is a one-one correspondence between the tangent vectors on $M$ and those on $O$. I wonder if there is also a one-one correspondence between the covectors on $M$ and those on $O$.