Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / h / Solution

Let $({\rho }_0,{\rho }_1)$ be a smooth partition of unity subordinate to $(U_0,U_1)$. On $U_0$, extend ${\rho }_1\phi$ by zero away from the overlap, and on $U_1$ extend ${\rho }_0\phi$ similarly. DefineOn the overlap,and therefore