Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 8

For differentiable vector fields, expand the divergence using the Levi-Civita symbol and the product rule:The first contraction uses the cyclic identity ${ϵ}_{ijk}={ϵ}_{kij}$; the second uses ${ϵ}_{ijk}=-{ϵ}_{jik}$. This fixes the relative minus sign without guessing the orientation of the cross products.

To obtain the volume integral of a curl, test its components with an arbitrary constant vector $c$. By part (a), $\nabla \cdot (G\times c)=c\cdot (\nabla \times G)$. Applying the divergence theorem to the closed boundary, with outward unit normal $n$, givesThe last step is cyclic symmetry of the scalar triple product. Since this holds for every $c$, all vector components agree:The same regularity conditions as the divergence theorem apply: for example a continuously differentiable field on a neighbourhood of a bounded volume with piecewise smooth boundary.

For constant $b$, the curl product rule for a scalar field givesUsing part (b) and the order of the cross product,Here $dS$ inside a vector product denotes the outward vector area element $ndS$. Closedness is the condition that permits replacing the entire surface by the boundary integral of a volume.

The closed boundary includes the curved unit hemisphere and the disk at $y=0$. On the disk, $F=(0,y,0)=0$, so that contribution vanishes. On the curved part the outward normal is $n=(x,y,z)$, giving $F\times n=(yz,0,-xy)$.

Use spherical coordinates with the polar axis along positive $y$:The area element is $\sin \theta d\theta d\varphi$. ConsequentlyThis directly performs the surface integral and includes the planar cap, rather than applying the closed-surface theorem to only an open hemisphere.