Local tubular neighborhood

Question

Let $M\subset\Bbb R^n$ be a $C^1$ manifold. I need to prove that for any $x\in M$, there exist $\epsilon>0$ and a neighborhood $V$ of $x$ in $M$ such that the map $$\phi:\{(v,w)\in N:\ v\in V,\ \|w\|<\epsilon\}\to\Bbb R^n,\ (v,w)\mapsto v+w,$$ is a diffeomorphism between $N_{V,\epsilon}$ and its image, where $N$ is the normal bundle of $M$.

Thanks in advance.

Answer 1

HINT: Compute $D\phi_{(x,0)}\colon T_{(x,0)}N \to \Bbb R^n$. Note that $T_{(x,0)}N \cong T_xM\oplus N_xM$.