Let $\varphi:B^n\rightarrow B^n$ be a differential homeomorphism, $B^n$ is unit ball of $R^n$ , and $\overrightarrow{n}$ is outer normal vector of $B^n$. I feel that
$$ \frac{\partial \varphi}{\partial \overrightarrow{n}}(\partial B^n)=0 $$
In fact ,I don't know whether I'm right. And I think the differential homeomorphism from $B^n$ to $B^n$ must be rotate near the $\partial B^n$.
If I'm right ,hope for a detail proof ,If not, hope for a counterexample.
Thanks.