Let \(M\subset \mathbb{R}^3\) be the elliptic paraboloid given by the equation \(z=x^2+y^2.\) We can view it as a surface for revolution via the local parametrization \(F\colon \mathbb{R}^+\times(0,2\pi)\rightarrow \mathbb{R}^3\) given by \((u,v)\mapsto(u\operatorname{cos}(v),u\operatorname{sin}(v),u^2).\)

1. Determine the geodesic curvature of the parallels, where a parallel is an arc length parametrized curve arising as the intersection of \(M\) with a plane orthogonal to the axis of rotation.

2. Determine the geodesic curvature of the meridians, where a meridian is an arc length parametrized curve arising as the intersection of \(M\) with a plane containing the axis of rotation.

Let \(M\subset \mathbb{R}^3\) be an orientable surface. Let \(\gamma\colon I\rightarrow M\) be an immersed curve parametrized by arc length.

1. Prove \(D(\dot{\gamma})\) is orthogonal to both \(\dot{\gamma}\) and \(N_{\gamma},\) where \(D\) denotes the covariant derivative and \(N_{\gamma}\) denotes the unit normal on \(M\) restricted to the image of \(\gamma.\)

2. Prove that \(D(\dot{\gamma})=\kappa_gN\times \dot{\gamma},\) where \(\kappa_g\) denotes the geodesic curvature.

3. Prove that \(\kappa^2=\kappa_n^2 +\kappa_g^2\) where \(\kappa\) denotes the curvature of \(\gamma\) and \(\kappa_n\) denotes the normal curvature of \(M\) restricted to \(\gamma.\)

Compute the surface area of the torus using the local parametrization \(F\colon (0,2\pi)\times(0,2\pi)\rightarrow \mathbb{R}^3\) given by \((u,v)\mapsto((R+r\operatorname{cos}(u))\operatorname{cos}(v),(R+r\operatorname{cos}(u))\operatorname{sin}(v),r\operatorname{sin}(u)),\) where \(r,R\in\mathbb{R}^+\) and \(R>r.\)