Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume that $\{U_\alpha\}$ is locally finite and countable whose members all have compact closure. My question is:
Q: Does there exist a uniform constant $C>0$ (which only depends on the cover $\{U_\alpha\} $ and the chosen coordinate system) and two sets of smooth partition of unity $\{\rho_\alpha\}$ and $\{\phi_\alpha\}$ such that
(1) $\mathrm{supp}\,\rho_\alpha \subset \mathrm{int}(\mathrm{supp}\phi _\alpha)\subset \mathrm{supp}\,\phi_\alpha \subset U_\alpha$ for every $\alpha$ and
(2) $|\nabla \rho_\alpha|\leqslant C\phi_\alpha$ on $U_\alpha$ for every $\alpha$, where the gradient is computed with respect to THE COORDINATE SYSTEM $(x_\alpha^1,\dots,x_\alpha^n)$?
I have found a seemingly related post Existence of a partition of unity with uniformly bounded derivatives. but there the gradient is taken with respect to some riemannian metric on $M$. I wonder whether my question, which seems more elementary and local, could have an affirmative answer.
Many thanks! Anar
