ケルビン・ストークスの定理
( ( J F ) ψ ( u , v ) − t ( J F ) ψ ( u , v ) ) x = ( ∇ × F ) × x , for all x ∈ R 3 {\displaystyle ({(J\mathbf {F} )}_{\psi (u,v)}-{}^{t}{(J\mathbf {F} )}_{\psi (u,v)})\mathbf {x} =(\nabla \times \mathbf {F} )\times \mathbf {x} ,\quad {\text{for all}}\,\mathbf {x} \in \mathbb {R} ^{3}}
さらに、スカラー三重積を考慮すると、以下の等式を得る。
∂ P 1 ∂ v − ∂ P 2 ∂ u = ⟨ ∂ ψ ∂ u | ( ∇ × F ) × ∂ ψ ∂ v ⟩ = det [ ( ∇ × F ) ( ψ ( u , v ) ) ∂ ψ ∂ u ( u , v ) ∂ ψ ∂ v ( u , v ) ] {\displaystyle {\begin{aligned}{\frac {\partial {P}_{1}}{\partial v}}-{\frac {\partial {P}_{2}}{\partial u}}&=\left\langle {\frac {\partial \psi }{\partial u}}\ |(\nabla \times \mathbf {F} )\times {\frac {\partial \psi }{\partial v}}\right\rangle \\&=\det {\begin{bmatrix}(\nabla \times \mathbf {F} )(\psi (u,v))&{\frac {\partial \psi }{\partial u}}(u,v)&{\frac {\partial \psi }{\partial v}}(u,v)\\\end{bmatrix}}\end{aligned}}}