Past exam of the mathematical methods II course of the University of Oxford / 2016 / 8 / b / Solution

Apply Stokes theorem to $\mathbf{F}=\psi \mathbf{B}$, with $\mathbf{B}$ an arbitrary constant vector. The curl product rule for a scalar field reduces to $\nabla \times (\psi \mathbf{B})=\nabla \psi \times \mathbf{B}$. Using the scalar triple product,Since these scalar products agree for every $\mathbf{B}$, the vectors agree. This proves the vector-valued Stokes identityIn particular, the ordering of the cross product is normal crossed with gradient; reversing it would change the sign.