Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function supported in $B_r(x_0)$ with total integral one. Define the local degree of $u$ at $x_0$ as
$$\int_U \eta(u) \det(Du) dx.$$
I am trying to prove that this integral is an integer. This is an exercise in Chapter 8 of Evans' "Partial Differential Equations." The preceding exercise has us show that $\eta(u) \det(Du)$ is a null lagrangian, so the integral depends only on the restriction of $u$ to $\partial U$. So, it seems that we need to use this fact somehow, and look at a function whose boundary values agree with those of $u$. But, I do not see a way to do this. Any suggestions?