Write the differential 1-form associated to a function F as ω F. Then one can calculate that ⋆ ⋆ ω ω ∇ ∇ × × F = d ω ω F , where ★ is the Hodge star and d is the exterior derivative. Thus, by generalized Stokes' theorem, ∮ ∮ ∂ ∂ Σ Σ F ⋅ ⋅ d γ γ = ∮ ∮ ∂ ∂ Σ Σ ω ω F = ∫ ∫ Σ Σ d ω ω F = ∫ ∫ Σ Σ ⋆ ⋆ ω ω ∇ ∇ × × F = ∬ ∬ Σ Σ ∇ ∇ × × F ⋅ ⋅ d Σ Σ