Let $\displaystyle U$ be an open bounded subset of $\displaystyle \mathbb{R}^{n}$, assume we have a finite open covering of $\displaystyle U$.

Consider the partition of unity subordinated: $\displaystyle \theta_{0}, \theta_{1}...\theta_{k} \in C^{\infty}(\mathbb{R}^{n})$ and $\displaystyle \sum_{i=0}^{k}\theta_{i} = 1$ where $\displaystyle 0 \leq \theta_{i} \leq 1$.

How does it follow that $\displaystyle \nabla(\theta_{0}) = - \sum_{i=1}^{k}\nabla(\theta_{i})$ ?

Thanks!