7.15 Differential geometry

Problem: For the surface given parametrically, find: 1. The unit vector of the normal 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 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 \( \left(u=u_{0}, v=v_{0}\right) \); 8. The Gaussian and mean curvatures of the surface; 9. The elliptic, hyperbolic and parabolic points on the given surface. \[ \left\{\begin{array}{c} x(u, v)=3 \cos u \cos v \\ y(u, v)=3 \cos u \sin v \quad u_{0}=\frac{\pi}{3}, \quad v_{0}=\frac{\pi}{4}, \quad M_{0}\left(u_{0}, v_{0}\right) . \\ z(u, v)=5 \sin u \end{array}\right. \]