Let $R > r > 0$ and $a,b\in B(0,r)\subset R^n$ where $n \geq 2$. Construct a diffeomorphism $f$ of $R^n$ satisfying $f(a)=b, f(x)=x $ for all $||x|| > R$ and $det(Df_x) =1 $ for all $x\in R^n$.
I was trying with $f(x)=\rho(x) g(x) + x$ where $\rho$ is a bump function and $g$ is smooth function with $g(a)=b-a$. But problem is that this type of function may not be diffeomorhism and also jaccobian of $f$ is not necessarily 1.