Differential geometry of curves and surfaces

A running track has semicircular ends of radius $36.5$ m. Along the straight parts the curvature is $0$. What is the curvature on the bends?

Which of these surfaces have Gaussian curvature $K = 0$ at every point? (Choose all that apply.)

For the cylinder of radius $2$, $X(\theta, z) = (2\cos\theta, 2\sin\theta, z)$, one finds $E = 4$, $F = 0$, $G = 1$, $L = -2$, $M = 0$, $N = 0$. What are $K$ and $H$?

The helix $\gamma(s) = (3\cos(s/5), 3\sin(s/5), 4s/5)$ has unit speed (check: $3^2 + 4^2 = 5^2$), curvature $\kappa = a/c^2$ and torsion $\tau = b/c^2$ with $a = 3$, $b = 4$, $c = 5$. Compute $\dfrac{1}{\kappa^2 + \tau^2}$.

A triangle on a sphere of radius $R$ is bounded by three great-circle arcs and has angles $100^\circ$, $80^\circ$ and $90^\circ$. By Girard's theorem its area is $R^2$ times its angle excess (in radians). What is the angle excess, in degrees?

A closed orientable surface of genus $3$ (a pretzel with three holes) sits in $\mathbb{R}^3$. Its total curvature $\iint K\, dA$ equals $m\pi$ for which integer $m$?

Why can a flat sheet of paper be rolled into a cylinder without stretching or tearing, but not wrapped smoothly around a ball?

The Lean theorem sphere_gauss_curvature has the hypothesis hφ : cos φ ≠ 0. Why is it needed?

Write signed_curvature(dx, dy, ddx, ddy) returning the signed curvature $\kappa = (x'y'' - y'x'')/(x'^2 + y'^2)^{3/2}$ of a plane curve at a point, given the values of $x', y', x'', y''$ there. Then use it to find the curvature of the ellipse $(3\cos t, \sin t)$ at $t = 0$ and $t = \pi/2$.

Why must every torus in $\mathbb{R}^3$, however it is shaped, have points of negative Gaussian curvature?