it is known that for smooth surfaces D with boundary $\partial D$, the Euler characteristic can be computed as $\chi(D)=2-b$, where $b$ denotes the number of boundary components. I would like to know how to prove this beautiful formula. Does someone know a readable source where I can find it?
Best wishes
For the sake of completeness I will write it down here.
Since the Euler character of a graph is $2$ and because of the invariance of the Euler characteristic, the equality $\chi =\chi (Q_r) =2-r $ follows from the following proposition:
Deleting from a triangulation $K $ an arbitrary triangle and retaining its vertices and sides decreases the Euler characteristic of the triangulation by one.
Hope it helps.