Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 2 / 11 / a / ii / Solution

The curl of a gradient vanishes because the mixed second partial derivatives of the smooth function $\varphi$ agree. Therefore $\nabla \times \mathbf{v}=\nabla \times \nabla \varphi =0$ everywhere, andUnlike part (i), this integrand does not involve the problematic auxiliary field.