Gluing vector bundles with isomorphic restrictions I am dealing with the following situation.
We have two vector bundles $E \xrightarrow{\pi_1} U$ and $F \xrightarrow{\pi_2} V$ over some open subsets of a smooth manifold whose restrictions on the intersection $U \cap V$  are isomoprhic, i.e. we have a vector bundle isomorphism $\phi : E|_{U\cap V} \rightarrow F|_{U\cap V}$.
I am trying to glue $E$ and $F$ to a vector bundle over $U\cup V$ and so, here is my idea.
First, I consider the disjoint union $G=E_p \displaystyle\bigsqcup_{p \in U\cup V} F_p$ equipped with the disjoint union topology and then define an equivalence relation: $u \sim v \Longleftrightarrow$ u and v belong to to $E_p$ or $F_p$ for $p\in U \cap V$.  Then, we pass to the quotient space
\begin{equation*}
\tilde{G}=\big (E_p \displaystyle\bigsqcup_{p \in U\cup V} F_p\big)mod\sim
\end{equation*}
and consider the projection map
\begin{equation*}
\pi_(v)=
\begin{cases}
\pi_1(v), \text{if}~ v\in E_p~ \text{for}~ p\in U\setminus U\cap V\\
\pi_2(v), \text{if}~ v\in F_p~ \text{for}~ p\in V\setminus U\cap V\\
\pi_1(v)=\pi_2(v), \text{if}~ v\in E_p\simeq F_p~ \text{for}~ p\in U\cap V
\end{cases}
\end{equation*}
As I see it, there is a well-defined vector space structure on the fibres of $\pi$ so my problems begin with the local trivialization of $\tilde{G}$. There are some straightforward local trivializations charts over a point p, coming from $E$ and $F$ , when $p \in U \setminus \overline{U \cap V}$ or $p \in V \setminus \overline{U \cap V}$ respectively. I could even define such charts over points in $U \cap V$.
However, I just cannot see how things work out in the boundary $\partial(U \cap V)$ and more generally, I do not feel comfortable with the simplicity of this approach.
I would be thankful, If someone could shed some light to this question!
 A: It may help to consider a simple analogy: Suppose we have two open sets $U$ and $V$ of real numbers, and two continuous functions, $f$ on $U$ and $g$ on $V$, such that $f = g$ in $U \cap V$. The "patched" formula
$$
(f \cup g)(x) = \begin{cases}
  f(x) & x \in U, \\
  g(x) & x \in V,
\end{cases}
$$
is well-defined in the intersection $U \cap V$, and therefore represents a function on the union $U \cup V$. Moreover (in relation to your question), $f \cup g$ is continuous: If $p$ is an arbitrary point of $U \cup V$, there exists a neighborhood $W$ of $p$ such that either

*

*$W \subset U$, so $f \cup g = f$ throughout $W$, or

*$W \subset V$, so $f \cup g = g$ throughout $W$.

In either case, $f \cup g$ is continuous. The same argument works for any local property a function might satisfy: smoothness, real-analyticity, ....

In the situation at hand we have two vector bundles, and we know there exists a vector bundle isomorphism $\phi:E|_{U \cap V} \to F|_{U \cap V}$. The total space of the vector bundle over $U \cup V$ may be constructed as the disjoint union of fibres just as you say, but it may be desirable to write the fibre equivalence over $U \cap V$ explicitly:
$$
\bigl[(p, v) \in E|_{U \cap V} \bigr] \sim \bigl[(p, \phi(v)) \in F|_{U \cap V} \bigr].
$$
The point is now to see that the quotient space $\widetilde{G} = E \cup F$ is locally trivial over $U \cup V$. Conceptually, this is precisely the simple analogy above. At risk of taking a joke past its punch line, for each point $p$ of $U \cup V$, there exists a neighborhood $W$ of $p$ such that either $W \subset U$ trivializes $E$, or $W \subset V$ trivializes $F$, or both (and the notions of trivialization agree).
Whether or not $p$ is a boundary point of the intersection $U \cap V$ turns out to be immaterial, though it's also perfectly understandable why that contingency might seem to require attention.
