There is a lemma in John Lee's book called Gluing Lemma for Smooth Maps, that says that if we have a collections of smooth maps each defined on the open subsets $\{U_{\alpha}\}_{\alpha \in A}$ that cover $M$, $F_{\alpha} : U_{\alpha} \rightarrow N$ , for some smooth manifold N, such that they agree on their common domain, then there exist a unique smooth map $F : M \rightarrow N$ such that $F|_{U_{\alpha}} = F_{\alpha}$ for each $\alpha \in A$.
Suppose that i have a smooth manifold $M$ and a smooth vector field $V : U \rightarrow TM$ defined on a open subset $U\subset M$ such that $supp \, V \subset U$. Can i use the Lemma above to construct a smooth vector field that agree with $V$ on $U$ and vanish outside it ? Can i use the same construction to any tensor field ? And how's this different (or related) to this Proving The Extension Lemma For Vector Fields On Submanifolds
Thank you
(Remark : I already do that but i don't really sure its correct. First i find an open cover for M, that is $\{U\} \cup \{M \smallsetminus \text{supp} (V) \}$ and then defined a zero vector field $W : M \smallsetminus \text{supp} (V) \rightarrow TM$.)
