A basic theorem in differential geometry is
Suppose that $X$ is a vector field on a smooth manifold $M^n$ and $X(p)\neq 0$. Then there exists a neighborhood $U$ of $p$ and a coordinate system $x$ on $U$ such that $X = \frac{\partial}{\partial x^1}$.
The proof that I am reading uses uniqueness of solutions to differential equations, and I'm wondering about another method.
Take any coordinate system $y$ on a neighborhood $V$ of $p$. This provides a bundle equivalence $TM|_V \to V \times \mathbb{R}^n$. We should be able to choose smoothly for each $p$ a complement $X_2(p), \dotsc, X_n(p)$ linearly independent from $X_1(p)$. This would get us a bundle map $V \times \mathbb{R}^n \overset{\Delta}{\to} V \times \mathbb{R}^n$ such that $\Delta\circ y_*(X(p))=(e_1)_{y(p)}$ for all $p$. But I suppose the problem is that this bundle map does not necessarily arise as the differential of some diffeomorphism. Am I correctly identifying the problem with this line of thinking?
What is an example of a local trivialization of $TM$ (or a bundle automorphism of $U \times \mathbb{R}^n$ for some trivialization) that is demonstrably not $x_*$ for some diffeomorphism $x$?
Thanks for clearing up my confusion, just learning this stuff.
Given any smooth map $f: \Bbb R^n \to GL_n(\Bbb R)$, you can construct a bundle map $T_f: T\Bbb R^n \to T\Bbb R^n$ by $T_f(x,v) = (x,f(x)v)$. This is not induced by a diffeomorphism, because it lies over the identity! This construction (roughly) proves that bundle maps induced by diffeomorphisms are of infinite codimension in the space of all bundle automorphisms.
Two points are worth bringing up. First, this is related to the fact that, given any two Riemannian structures $g_1$ and $g_2$ on a vector bundle $E$, there is a bundle isomorphism $f: E \to E$ that takes $g_1$ to $g_2$. (That is, 'all Riemannian structures on a vector bundle are isomorphic.') But it is far from true that any two diffeomorphic Riemannian manifolds are isometric!
Second, the desire to take a random bundle map and promote it to one induced by a smooth map is clearly quite desirable when doing differential topology or geometry - you can take algebraic-topological information (the space of bundle maps) to geometric structure (the space of embeddings or what have you). This idea is usually known as the h-principle. One famous example is the Smale-Hirsch immersion theory, which states that the space of bundle maps $TM \to TN$ deformation retracts onto the space of immersions $M \looparrowright N$. (Corollary: sphere eversion.)