Let $M$ be a smooth manifold and $T_p^*M$ be its cotangent space at $p$. I wonder if any element $w \in T_p^*M$ (a functional in $T_pM \to \mathbb R$) can be realized as the differential of a smooth function on $M$. i.e. $w=df_p$ for some $f:M\to \mathbb R$, smooth function.
For $X_p\in T_pM$, by definition $$df_p(X_p)(g)=X_p(g\circ f), g\in C^\infty(\mathbb R).$$
I don't know how to find such an $f$ given $w$. Also, will this $f$ be independent of the choice of local coordinate?