Poincaré-Hopf theorem using Stokes The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one in Milnor's Topology from the Differentiable Viewpoint there is no integration.
Does the proof become easier by using Stokes' theorem? Is there a good reference?

In step 3 of the scketch you read "the degree of the Gauss map". Now one has to look for the definition of the degree of a map $ f:M\to N$. But that map gives a linear map in de Rham cohomology $f^*:H^m(N)\to H^m(M), n=\dim(M)=\dim(N)$. Stokes theorem says that the integral of $m$-forms induces an isomorphism $\int:H^m\to R$ an this transforms  $f^*$ (by change of variables in the integral) into a linear map $R\to R$. Such a linear map is the product times an scalar $d$: this is the degree of $f$. 
