i am poor in english,please forgive my expression.
there is a torus $\ T^2\ $whose parametric equation is \begin{equation*} \overrightarrow{r}(u,v)=((a+r\cos u)\cos v,(a+r\cos u)\sin v,r\sin v)\\ 0\leq u<2\pi,0\leq v<2\pi,0<r<a \end{equation*} calculate \begin{equation*} \int_{T^2}H^2+K\ d\sigma \end{equation*} while H is the mean mean curvature and K is gauss curvature.
i want to use geodesic to cut torus into simply connected surfaces such as $dv=0$,then i can calculate the integration about $K$,while i don't know use how many geodesics,but i can't deal with the $H^2$，because i only can use$\ H=\frac{LG-2MF+NE}{2(EG-F^2)}\ ,$that was so difficult to make it so that i think i make wrong.
so are there something to simply the calculate?