Prove that $F$ mapping $V(\Omega\subset V)$ to $\partial \Omega$ can not be the identity mapping on $\partial\Omega$

Question

Let $\Omega\subset\mathbb R^n$ be compact and let $\partial\Omega$ be smooth. Let $V\subset\mathbb R^n$ be open and $\Omega\subset V$.Suppose $int\Omega\neq\emptyset$. If $F\in C^1(V,\partial\Omega)$, can $F$ restricted to $\partial \Omega$ be the identity mapping?

I think the answer may be no and it may have something to do with stokes' theorem and Brouwer fixed point theorem,but I can't give a proof.