Solution
= Solution
The overlap $U_0\cap U_1$ is $\mathbb C^*$ 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.