As in § Theorem, we reduce the dimension by using the natural parametrization of the surface. Let ψ and γ be as in that section, and note that by change of variables ∮ ∮ ∂ ∂ Σ Σ F ( x ) ⋅ ⋅ d Γ Γ = ∮ ∮ γ γ F ( ψ ψ ( γ γ ) ) ⋅ ⋅ d ψ ψ ( γ γ ) = ∮ ∮ γ γ F ( ψ ψ ( y ) ) ⋅ ⋅ J y ( ψ ψ ) d γ γ where J y ψ stands for the Jacobian matrix of ψ at y = γ (t).