Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / f / Solution

If $\nabla \times \mathbf{F}=0$ on a simply connected region containing the bounded trajectory, then the Poincare lemma gives a scalar potential $U$ with $\mathbf{F}=-\nabla U$. The work iswhich remains bounded when $\mathbf{x}$ and $\mathbf{F}$ remain bounded. Since $H$ is also bounded, $\Delta Q=W-\Delta H$ cannot grow linearly with the observation time.

Thus sustained linear heat dissipation in this setting requires a nonconservative force, and, under the stated simply connectedness assumption,Such a force can perform nonzero work on repeated bounded cycles. On a multiply connected domain, a curl-free force can have nonzero circulation, so the topological assumption is essential.