I have a question on the definition of the degree of a smooth map $f$ between two oriented smooth manifolds $M$ and $N$ with $M$ compact and $N$ connected. Then the degree is defined by
$$ \mbox{deg}(f)=\sum_{y\in f^{-1}(x)}\varepsilon_x $$
for $x$ a regular value of $f$ and $\varepsilon_x$ is $1$ if $T_xf$ preserves orientation and $-1$ otherwise. It can be shown that this is independent of the choice of the regular value in this case.
Now my question concerns the identity map $f:\mathbb{B}_2 \rightarrow \mathbb{R}^2$ where $\mathbb{B}_2$ denotes the unit disk in $\mathbb{R}^2$ and which is defined by $f(x,y)=(x,y)$. Then the map is clearly not surjective hence $\mbox{deg}(f)$ should be zero. However, computing the degree using the above definition for a regular value inside the unit disk yields $\mbox{deg}(f)=1$. There is obviously something wrong somewhere but I can't wrap my head around it! If anyone could help pointing out to me where the problem lies, it would be highly appreciated!
Thanks in advance!