Let us show that the Poincaré $1$ -form is globally defined. That is, $\alpha$ has the same expression in all local coordinates. Suppose $x^i, \tilde{x}^i$ are overlapping coordinates for $M$ . Then we have overlapping local coordinates $(x^i, y_i)$ , $(\tilde{x}^i, \tilde{y}_i)$ for $T^\ast M$ with the transformation rule $$ \tilde{y}_i = \frac{\partial \tilde{x}^j}{\partial x^i} y_j. $$ Hence \begin{eqnarray*} \sum_{i=1}^n \tilde{y}_i d\tilde{x}^i &=& \sum_{i=1}^n \tilde{y}_i \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{i=1}^n \frac{\partial \tilde{x}^j}{\partial x^i} y_j \frac{\partial \tilde{x}^i}{\partial x^k} dx^k \\ &=& \sum_{k=1}^n y_k dx^k. \end{eqnarray*}

Properties

1. The Poincaré $1$ -form play a crucial role in symplectic geometry. The form $d\alpha$ is the canonical symplectic form for $T^\ast M$ .
2. Suppose $\pi\colon T^\ast M\to M$ is the canonical projection. Then $$ \alpha(w) = \xi( (D\pi)(w) ),\quad w\in T_\xi(T^\ast M), $$ which is an alternative definition of $\alpha$ without local coordinates.
3. The restriction of this form to the unit cotangent bundle, is a contact form.