What is the local trivialization $(\pi^{-1}(U),\Phi)$ associated with a coordinate chart $(U,\varphi)$?

Question

In this set of notes on Vector Bundles: http://www.math.toronto.edu/mgualt/MAT1300/week10.pdf (example 3.13), they say that given a coordinate chart $(U,\varphi)$, there is an associated local trivialization $(\pi^{-1}(U),\Phi)$.

What is the function $\Phi$ explicitly? Is it simply, $\Phi(p,\varphi(p)) = (p,\phi(v))$, for $p \in U$?

Edit: it should be $\Phi(p,\varphi(p)) = (p,\varphi(v))$.

Answer 1

The local trivialization is the map $\Phi : \pi ^{-1}(U) \to U \times \mathbb{R}^n$ defined locally by

$$\Phi(v^i \frac{\partial}{\partial x^i}|_p) = (p,(v^1, \dots, v^n))$$

Clearly this map is linear on fibers and satisfies $\pi _U \circ \Phi = \pi$.