Differential geometry

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Problem: For a curve given parametrically, find: 1. Vectors of the accompanying trihedron at the point \( t=t_{0} \); 2. Planes and lines of the accompanying trihedron at the point \( t=t_{0} \); 3. Tangent lines parallel to coordinate planes; 4. Contiguous planes perpendicular to the coordinate axes; 5. Curvature and torsion of a curve at the point \( t=t_{0} \)

7.12 Differential geometry

15.19 $

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.

7.13 Differential geometry

25.32 $

Problem: Find the unit vectors of the tangent, principal normal, and binormal of the curve. Compose the equations of the contiguous plane, the normal plane, the rectifiable plane at the point \( M_{0} \). \[ x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\cos 2 t . \]

7.5 Differential geometry

12.66 $

Problem: Prove that for any parametrization of the plane, the second quadratic form is identically equal to 0 .

7.19 Differential geometry

1.52 $

Problem: For a two-sheeted hyperboloid, find: 1) the first and second quadratic forms, 2) the principal curvatures. \[ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}-1=0 \text {. } \]

7.20 Differential geometry

3.04 $

Problem: Explore the nature of points on an ellipsoid.

7.21 Differential geometry

5.06 $

Problem: Explore the nature of the points of the surface given by the equation: \[ z=z(x, y)=f\left(\sqrt{x^{2}-y^{2}}\right) . \]

7.22 Differential geometry

2.53 $

Problem: Taking point a) as the definition of a geodesic surface, prove its properties in the remaining points: a) at each point, the normal to the surface is the principal normal of the line. b) at each point of the line, its geodesic curvature is 0 . c) its curvature is equal to the absolute value of the normal curvature. d) straightening plane coincides with the tangent plane to the surface.

7.23 Differential geometry

3.8 $

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