Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 8

Apply the ordinary product rule to the components of the curl. For example, the first component isThe last two terms are the first component of $\nabla \varphi \times \mathbf{F}$. The other components follow cyclically, soThe sign in the last form follows from antisymmetry of the cross product, not from a different curl convention. Continuous first derivatives suffice for this identity.

For a twice continuously differentiable scalar function $\psi$, equality of mixed partial derivatives gives $\nabla \times \nabla \psi =0$. Substitute $\mathbf{F}=\nabla \psi$ into the preceding product rule:Stokes theorem on a spanning surface with consistent boundary orientation then givesThis is a weighted gradient circulation identity. All fields must be defined and sufficiently smooth on a neighborhood of the surface; a scalar prefactor can create nonzero circulation even when the unweighted field is a gradient.

Choose counterclockwise traversal viewed from $+z$, and span the curve by the disk $z=1$, $x^2+y^2\le 2$ with upward normal. HereThe surface side is therefore $-{\iint }_D{x}^{2}dxdy$. In polar coordinates it isFor the direct boundary integral, $dz=0$ gives $\nabla \psi \cdot d\mathbf{l}=dx$ on $z=1$. Set $x=\sqrt{2}\cos \theta$, $y=\sqrt{2}\sin \theta$, soThe two independent evaluations agree. Clockwise traversal and downward normal give $+\pi$ on both sides.