Spherical degree by area pullback ($\mathrm{deg}g=\frac{1}{4\pi }{\int }_{S^2}{g}^{\ast }\omega$)

For a smooth map between oriented unit spheres, integrate the pullback of the target area form. Localizing a normalized top-degree form near a regular value, and using Stokes theorem to discard exact-form differences, identifies this integral with the degree as a sum of local degrees.