Consider $\mathbb{R}^n$ with the standard volume form $\omega_{0}:=dx_1\wedge ...\wedge dx_n$ . Let $r$ and $R$ be two positive real numbers, with $R > r$ . Show that for every $a, b \in B (0, r )$, there exists a diffeomorphism $f$ of $\mathbb{R}^n$ such that $f(a) = b , f(x)= x$ for $|| x || > R$ and $f ^{*}\omega_{0} = \omega_{0}$.
One-dimensional case is fine. I have tried to extend that idea to the higher dimensional case. I have an exam. I don't understand how to construct $f$. Hints are welcomed. Can anyone please help me?
$\newcommand{\Reals}{\mathbf{R}}$Hint: Let $(r, \theta)$ denote polar coordinates on $\Reals^{2}$. If $\phi:(0, \infty) \to \Reals$ is an arbitrary smooth function, the formula $$ f(r, \theta) = \bigl(r, \theta + \phi(r)\bigr) $$ defines a "rotational shear" mapping that is smooth and area-preserving.