Angular Jacobian of a rational map
= Angular Jacobian of a rational map
{title2=$J_R=\left[\frac{1+|z|^2}{1+|R|^2}|R'|\right]^2$}
For a nonconstant <rational map> between round unit <Riemann spheres>, $R^*d\Omega=J_Rd\Omega$. Counting inverse images with multiplicity gives $\int J_Rd\Omega=4\pi\deg R$. The <Jacobian determinant> vanishes at <ramification points of a holomorphic map>, but the integral still records the global <degree of a holomorphic map>.