In Evans and Gariepy (p174) there is a proof showing the senses in which smooth functions approximate functions of bounded variation. There is the following computation -- the last step of which I'm confused by. They also say that $\sum D\zeta_k = 0$, how are they obtaining this fact?
Some notation: $\zeta_k$ is a smooth partition of unity on the set $U$. Imagine dividing $U$ into rings -- $\zeta_k$ is supported on 3 of these rings (precise statement given in reference). $\eta_k$ is a standard mollifier:
The computation:
