7.15 Differential geometry

Problem: For the surface given parametrically, find: 1. The unit vector of the normal at the point $(u=u_0,v=v_0)$; 2. The equation of the tangent plane and the normal at the point $(u=u_0,v=v_0)$ 3. The volume of the tetrahedron formed by the tangent plane at the point ( $u=u_0,v=v_0$ ) to the given surface and coordinate planes; 4. The normals parallel to coordinate planes; 5. The first quadratic form of the surface; 6. The second quadratic form of the surface; 7. The angle between the coordinate lines of the surface at the point $(u=u_0,v=v_0)$; 8. The Gaussian and mean curvatures of the surface; 9. The elliptic, hyperbolic and parabolic points on the given surface. $$\{\begin{array}{c}x(u,v)=3\cos u\cos v\\ y(u,v)=3\cos u\sin v{u}_0=\frac{\pi }{3},v_0=\frac{\pi }{4},M_0(u_0,v_0).\\ z(u,v)=5\sin u\end{array}$$