How to show there exists a smooth local frame $(\tilde{X_1},\cdots,\tilde{X_n})$ on some neighborhood of $A$ such that $\tilde{X_i}|_A=X_i$? Let $M$ be a smooth $n$-manifold with or without boundary. If $(X_1,\cdots,X_n)$ is a linearly independent $n$-tuple of smooth vector fields along a closed subset $A\subset M$, how to show there exists a smooth local frame $(\tilde{X_1},\cdots,\tilde{X_n})$ on some neighborhood of $A$ such that $\tilde{X_i}|_A=X_i$ for $i=1,\cdots,n$?
 A: This is an exercise in Lee's book on smooth manifolds. Here are some hints to get you started: 
$1).\ $ the result follows from this more general result (also an exercise in the same book!): let $\pi : E \to M$ be a smooth vector bundle over a
smooth manifold $M$ and $A$ a closed subset of $M$. If $\sigma: A \to E$ is a smooth section of $E|_A$ then for each open subset $A\subseteq U \subseteq M$ there exists a global smooth section $\tilde \sigma$ such that $\tilde \sigma|_A = \sigma$ and
supp $\tilde \sigma\subseteq U.$ To show this, follow this sketch:
$2).\ $ Each $p\in A$ has a neighborhood $W_p\overset{wlog}\subseteq U$ such that there is a $\tilde \sigma:W_p\to E$ satisfying $\tilde \sigma _{W_p\cap A}=\sigma$.
$3).\ $ Using $2).$, find a cover of $M$ (the sets will be indexed by $p\in A$ except for one of them, $W_0$ which will be a different but obvious choice) and a partition of unity $\{\psi_0,\psi_{p}\}_p $ subordinate to it.
$4).\ $ Apply the extension theorem for functions to $\psi_p\tilde \sigma_p$.
$5).\ $ Define $\tilde \sigma$ to be the obvious sum, and check that it is the desired section.
