Question

component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of y and 7 respectively. Let v,v denote the velocity functions of γ and γ respectively. (1) Prove that R 2RV. In particular, at a time t e I for which v(t) lies in the ay-plane, we have (t) 2R(t). (2) Suppose the trace of γ lies on the cylinder {(x, y, z) E R3 | 11. At a time t I for which v(t) lies in the ry-plane (so that i tangent to the "waist" of the cylinder), conclude that k(t) 31 there any upper bound for k(t) under these conditions? (3) Find an optimal lower bound for n(t) at a time t I when v makes the angle θ with the xy-plane.

Answer 1

first complete question according to HomeworkLib policy.

(x(t) ,d(リ.alt) - . I(t)三ーteethedt) 4 TIO; 2.lt) х 8㈩ observe 2 3 ㄥ 2. に+4 ince numeratorhas m posit i ve terras, D ore