Let \(X=\text{Spec}\;A\) be a smooth affine variety over the reals with \(\dim X\geq 2\). Let \(M\) be the underlying real manifold of real points of \(X\) and assume that \(M\) is non-empty. Further assume that the module of \(n\)-forms over \(X\) is a free module of rank one (which in this paper is referred to as \(X\) being oriented). As usual, let \(\mathbb{R}(X)=S^{-1}A\) where \(S\) is the set of elements of \(A\) not vanishing at any point of \(M\). The authors define a natural isomorphism \(\zeta: E(\mathbb{R}(X),\mathbb{R}(X))\to \mathrm{H}^n(M,\mathbb{Z})\) where \(E\) denotes the Euler class group.