We all know that the Euler characteristic is a topological invariant. But let's suppose that we don't know this or anything else about algebraic topology for that matter. We are given only the Gauss-Bonnet theorem, which expresses the Euler characteristic in geometrical terms. In his string theory text, Polchinski claims that this expression is a "total derivative" (actually, the integral of one), which I believe is physics-speak for "exact form". He claims that this observation demonstrates that the Euler characteristic does indeed depend only on topology. How does this argument go? It seems that, if what we are given is true, Stoke's theorem should rewrite the Euler characteristic in terms of the value of some form on the boundary. But this seems wrong... no?