Let Σ Σ be a smooth oriented surface in R 3 with boundary ∂ ∂ Σ Σ ≡ ≡ Γ Γ . If a vector field F ( x , y , z ) = ( F x ( x , y , z ) , F y ( x , y , z ) , F z ( x , y , z ) ) is defined and has continuous first order partial derivatives in a region containing Σ Σ , then ∬ ∬ Σ Σ ( ∇ ∇ × × F ) ⋅ ⋅ d Σ Σ = ∮ ∮ ∂ ∂ Σ Σ F ⋅ ⋅ d Γ Γ . More explicitly, the equality says that ∬ ∬ Σ Σ ( ( ∂ ∂ F z ∂ ∂ y − − ∂ ∂ F y ∂ ∂ z ) d y d z + ( ∂ ∂ F x ∂ ∂ z − − ∂ ∂ F z ∂ ∂ x ) d z d x + ( ∂ ∂ F y ∂ ∂ x − − ∂ ∂ F x ∂ ∂ y ) d x d y ) = ∮ ∮ ∂ ∂ Σ Σ ( F x d x + F y d y + F z d z ) .