First, calculate the partial derivatives appearing in Green's theorem, via the product rule : ∂ ∂ P u ∂ ∂ v = ∂ ∂ ( F ∘ ∘ ψ ψ ) ∂ ∂ v ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ u + ( F ∘ ∘ ψ ψ ) ⋅ ⋅ ∂ ∂ 2 ψ ψ ∂ ∂ v ∂ ∂ u ∂ ∂ P v ∂ ∂ u = ∂ ∂ ( F ∘ ∘ ψ ψ ) ∂ ∂ u ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ v + ( F ∘ ∘ ψ ψ ) ⋅ ⋅ ∂ ∂ 2 ψ ψ ∂ ∂ u ∂ ∂ v