Let $U$ be some contractible neighbourhood of $0\in\mathbb{R}^n$ and let $X=\sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ be a (smooth) vector field on $U$. This vector field can be thought as a differential 1-form $X=\sum_{i=1}^nX_i\,dx_i$ on $U$.
By Poincaré's lemma, "every closed form is exact".
Does this mean that "every vector field on $U$ is conservative (due to exactness) and satisfies $\frac{\partial X_i}{\partial x_j}=\frac{\partial X_j}{\partial x_i}$ (due to closeness)"?
Then, should I interpret this as "every v.f. on $U$ is the gradient of a potential function (0-form) $V:U\to\mathbb{R}$"
It seems to me that the word every is too strong. So, is my interpretation incomplete?
What about non-conservative vector fields? Are they "always" defined in non-contractible domains?
Not all one form on $U$ are closed (e.g. $y dx$ in $\mathbb R^2$). So not all vector fields are conservative (e.g. $y \frac{\partial}{\partial x}$.