7.13 Differential geometry

Problem: For the surface given parametrically, find: 1. The unit normal vector at the point \( \left(u=u_{0}, v=v_{0}\right) \); 2. The equation of the tangent plane and the normal at the point \( \left(u=u_{0}, v=v_{0}\right) \) 3. The volume of the tetrahedron formed by a tangent plane at the point \( \left(u=u_{0}, v=v_{0}\right) \) 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 \( \left(u=u_{0}, v=v_{0}\right) \); 8. Gaussian and mean curvature of the surface; 9. The elliptic, hyperbolic and parabolic points on the given surface.