(i) Give the definition of the curvature $\kappa(t)$ of a plane curve $\gamma:[a, b] \longrightarrow \mathbf{R}^{2}$. Show that, if $\gamma:[a, b] \longrightarrow \mathbf{R}^{2}$ is a simple closed curve, then

$$\int_{a}^{b} \kappa(t)\|\dot{\gamma}(t)\| d t=2 \pi$$

(ii) Give the definition of a geodesic on a parametrized surface in $\mathbf{R}^{3}$. Derive the differential equations characterizing geodesics. Show that a great circle on the unit sphere is a geodesic.