Could anyone help me with the following problem? We have:
Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in [0, \varepsilon]$, $h(t) = 1 − \frac{1}{\ln t}$ for all $t$ large enough.
Consider the map $f : \mathbb{R}^n → \mathbb{R}^n$, $f(a) = \frac{h(\lVert a \rVert)}{\lVert a \rVert} a$, where $\lVert a \rVert = \sqrt{ (a^1)^2 + \cdots + (a^n)^2}$ is the usual Euclidean norm. We have that $f$ is a diffeomorphism onto the open unit ball $B$.
Define smooth vector fields $X_i$ on $B$ by $X_i = f_*(\frac{\partial}{\partial x^i})$, that is, $$X_i(b) = (d_{f^{-1}(b)}f)(\frac{\partial}{\partial x^i} \lvert _{f^{-1}(b)}).$$ Put $X_i(p) = 0$ for $p \notin B$. Then the vector fields $X_i$ on $\mathbb{R}^n$ are smooth and $X_i(p) = \frac{\partial}{\partial x^i} \lvert_{p}$ for $\lVert p \rVert \leq \varepsilon.$
The problem
Explain why the vector fields $X_i$ and $X_j$ pairwise commute.
What I have so far
I'm using Spivak's A Comprehensive Introduction to Differential Geometry I, and the definition of $f_∗$ (for a differentiable map $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$) is defined as the union of the maps $f_{∗p}:\mathbb{R}^n_{\ \ p} \rightarrow \mathbb{R}^m_{\ \ f(p)}$ defined as $f_{∗p}(v_p) = [Df(p)(v)]_{f(p)}$.
So, my thoughts are that $X_i = f_*(\frac{\partial}{\partial x^i}) = Df(\frac{\partial}{\partial x^i})$, so I fix a coordinate system $(y,U)$ and then my $Df$ is equal to the Jacobi matrix of $f$: $[\frac{\partial f_s}{\partial y^t}]_{s,t = 1,\dots, n}$.
Then I want to find out what $Df(\frac{\partial}{\partial x^i})$ is. Do we just get that $$Df(\frac{\partial}{\partial x^i}) = \begin{bmatrix} \frac{\partial f_1}{\partial y^i}\frac{\partial}{\partial x^i} \\ \vdots \\ \frac{\partial f_n}{\partial y^i}\frac{\partial}{\partial x^i} \\ \end{bmatrix}? $$ I know that I want to find out what $X_j(X_i(g))$ is, and show that that is equal to $X_i(X_j(g))$.
Any help is much appreciated! Anyone?