Vector bundle trivialization

A local trivialization of a rank-$r$ vector bundle $\pi :E\to M$ is a bundle isomorphism ${\pi }^{-1}(U)\cong U\times {\mathbb{R}}^r$ over an open set $U$, linear on each fiber. Overlapping trivializations differ by $(p,v)\mapsto (p,g(p)v)$ for a smooth $G{L}_r(\mathbb{R})$-valued function $g$. A tangent bundle chart uses the derivative of its base manifold chart for this fiber identification.