= Degree by integration of a pullback volume form
{title2=$\deg f=\int_Mf^*\omega,\quad\int_N\omega=1$}
For a smooth map $f:M\to N$ between oriented connected <closed manifolds> of the same dimension, a normalized <volume form> on the target gives $\deg f=\int_Mf^*\omega$. The formula agrees with the <degree as a sum of local degrees>. A change of normalized top-form adds an exact form by top-dimensional <de Rham cohomology>, whose pullback has zero integral by <Stokes theorem>.
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