Differential geometry

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Problem: Find curvature of the line $ \rho=a(1+\cos \varphi) $.

7.1 Differential geometry

2.53 $

Problem: Compose an equation for the evolutes of the line. \[ \rho=a(1+\cos \varphi) \text {. } \]

7.2 Differential geometry

3.8 $

Problem: Compose a natural equation of a curve given parametrically: \[ x=a(\cos t+t \sin t), y=a(\sin t-t \cos t) . \]

7.3 Differential geometry

2.53 $

Problem: Find the curvature and the torsion of the curve: \[ x=\cos ^{2} t, \quad y=\sin ^{3} t, \quad z=\cos 2 t . \]

7.4 Differential geometry

5.06 $

Problem: Find the tangent planes of the surface $ z=x^{4}-2 x y^{3} $, orthogonal to the vector $ \vec{a}(-2,6,1) $.

7.6 Differential geometry

2.53 $

Problem: Find the angle between the lines $ u+v=0 $, $ u=\tan v $ at their common point on the surface \[ x=u \cos v, y=u \sin v, z=u+v . \]

7.7 Differential geometry

3.8 $

Problem: Find the envelope of the family of curves $ \left(x+C^{2}\right)^{3}=(y-C)^{2} $.

7.8 Differential geometry

2.53 $

Problem: At what $ a $ and $ b $ are the torsion and the curvature of the curve $ x=a \sinh t $, $ y=a \cosh t, z=b t $ equal?

7.9 Differential geometry

5.06 $

Problem: Find the equations of the tangent plane of the torus. \[ \begin{array}{l} x=(7+5 \cos u) \cos v, \quad y=(7+5 \cos u) \sin v, \\ z=5 \sin u, \quad \text { at the point } M(u, v) . \end{array} \]

7.10 Differential geometry

3.8 $

Problem: Find the first quadratic form of the ellipsoid $ x^{2}+2 y^{2}+4 z^{2}=1 $.

7.11 Differential geometry

5.06 $

Problem: For the curve given parametrically, find: 1. The vectors of the accompanying trihedron at the point $ t=t_{0} $; 2. The planes and lines of the accompanying trihedron at the point $ t=t_{0} $ 3. The tangent lines parallel to coordinate planes; 4. The contiguous planes perpendicular to the coordinate axes; 5. The curvature and torsion of the curve at the point $ t=t_{0} $. \[ \left\{\begin{array}{c} x(t)=2 t^{3} \\ y(t)=3 t \quad t_{0}=1 . \\ z(t)=3 t^{2} \end{array}\right. \]

7.14 Differential geometry

12.66 $

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. \[ \left\{\begin{array}{c} x(u, v)=3 \cos u \cos v \\ y(u, v)=3 \cos u \sin v \quad u_{0}=\frac{\pi}{3}, \quad v_{0}=\frac{\pi}{4}, \quad M_{0}\left(u_{0}, v_{0}\right) . \\ z(u, v)=5 \sin u \end{array}\right. \]

7.15 Differential geometry

25.32 $

Problem: Find the curvature and torsion of a curve given by the equations $ x^{2}-y^{2}+z^{2}=1 $ $ y^{2}+z-2 x=0 \quad $ at the point $ M(1 ; 1 ; 1) $.

7.16 Differential geometry

7.6 $

Problem: For the line $ \vec{\tau}(t)=t^{3} \vec{\imath}+(t+1)^{2} \vec{\jmath}+\sqrt{t^{2}+1} \vec{k} $ find at the point $ t=-2 $ : a) the unit vector of tangent $ \vec{\tau} $; b) the unit vector of binormal $ \vec{b} $; c) the unit vector of principal normal $ \vec{n} $; d) the curvature $ K $; e) the equation of the tangent; f) the equation of the normal plane.

7.17 Differential geometry

12.66 $

Problem: For the line $ y=-3 x^{3} $ find: a) the curvature, b) the radius of the curvature, $ c $ ) the center of the curvature at the point with abscissa 1.

7.18 Differential geometry

3.8 $