Angular Jacobian of a rational map ($J_R={\left[\frac{1+|z{|}^2}{1+|R{|}^2}|R^{\prime }|\right]}^2$)

For a nonconstant rational map between round unit Riemann spheres, $R^{\ast }d\Omega =J_{R}d\Omega$. Counting inverse images with multiplicity gives $\int J_{R}d\Omega =4\pi \mathrm{deg}R$. The Jacobian determinant vanishes at ramification points of a holomorphic map, but the integral still records the global degree of a holomorphic map.