# The orientation-preserving diffeomorphism of $\mathbb R^n$

If $f$ is an orientation-preserving diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that:

(1)$f=\tilde f$ on a neighborhood of $K$. (2)There is a bounded set $V$ and $\tilde f=id$ outside $V$?

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For reference: math.stackexchange.com/q/137892/264 – Zev Chonoles Apr 28 '12 at 9:25

To Anton. You are right, this is my first answer on MO and I still have to get familiar with its style. I should have added at least 2 things: 1) The fact that every orientation preserving diffeomorphism of $\mathbb{R}^n$ is isotopic to the identity is proved in Milnor, "Topology from the differentiable viewpoint", Chapter 6, Lemma 2. 2) The desired diffeomorphism is obtained by integrating up to time 1 the modified vector field. – Alberto Abbondandolo Apr 29 '12 at 8:15