 Solution

For an oriented surface normal $N$ and a unit-speed curve tangent $T$, take signed geodesic curvature $k_g=(dT/ds)\cdot (T\times N)$. This convention uses boundary orientation with the region on the right; reversing to the usual left-hand boundary convention reverses both orientation and the corresponding curvature convention. With positive orientation consistent this way, the Gauss-Bonnet theorem for a region $D$ with piecewise smooth boundary isFor a disk with smooth boundary there are no corner terms and $\chi (D)=1$. Stating the signed convention is necessary to reproduce the sign in part (ii).