Exact local charges and independent values

Choose $t_{\ast }>0$ small enough that the compact swept sets

lie in disjoint open subsets of $U$. Here $e_1$ is the coordinate vector in the $x^1$ direction. Choose smooth compact cutoffs ${\chi }_i,{\eta }_i$ with disjoint supports for different $i$, such that both equal one near these swept sets and $\mathrm{supp}{\eta }_i\subset \{{\chi }_i=1\}$. Set