Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / a / ii / Solution

Periodic integration by parts givesAdding one half of the divergence term to both transport forms yieldsDensity extends the identity from smooth fields to all $u,v,w\in \tilde{V}$. In particular,without requiring $\nabla \cdot u=0$.