Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category):

A $k$-dimensional vector bundle is a triple $(E,B,\pi)$, where $\pi\colon E \to B$, satisfying the following:

a) The map $\pi$ is onto (I don't know if everyone requires this, but I will).

For each $p\in B$:

b) The fiber $E_p=\pi^{-1}(p)$ is a (real) vector space.

c) There's a neighborhood $U\ni p$ and a diffeomorphism $\phi\colon \pi^{-1}(U) \to U \times \mathbb{R}^k$ such that $P_1\circ \phi = \pi$, where $P_1$ is projection onto the first factor.

d) for each $q\in U$, the restriction $\phi\colon E_q \to {q}\times \mathbb{R}^k$ (where $\phi$ is the diffeomorphism from c)) is a linear isomorphism.

## My Question

Can d) be replaced with

d') the restriction $\phi\colon E_p \to {p}\times \mathbb{R}^k$ is a linear isomorphism.

In the original version, the restriction of $\phi$ to *every* fiber over $U$ must be a linear isomorphism. In the alternate version, this is only required for the *single* fiber $E_p$.

(*Note: This was posted at Math.SE a few weeks ago but received no answers.*)