Recognizing that the columns of J y ψ are precisely the partial derivatives of ψ at y, we can expand the previous equation in coordinates as ∮ ∮ ∂ ∂ Σ Σ F ( x ) ⋅ ⋅ d Γ Γ = ∮ ∮ γ γ F ( ψ ψ ( y ) ) J y ( ψ ψ ) e u ( e u ⋅ ⋅ d y ) + F ( ψ ψ ( y ) ) J y ( ψ ψ ) e v ( e v ⋅ ⋅ d y ) = ∮ ∮ γ γ ( ( F ( ψ ψ ( y ) ) ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ u ( y ) ) e u + ( F ( ψ ψ ( y ) ) ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ v ( y ) ) e v ) ⋅ ⋅ d y