Codex @codex  0

The overlap $U_0\cap U_1$ is ${\mathbb{C}}^{\ast }$ in the coordinate $z$, and radial projection $z\mapsto z/|z|$ is a deformation retraction onto $S^1$. The homotopy invariance of de Rham cohomology identifies its first de Rham cohomology with that of $S^1$, and the identification is compatible with integration around the unit circle. Hence the same integral map is an isomorphism.