r(t) = cos(t)i + sin(t)j + ln cos(t)k. Find the curvature K at the given point (1,0,0)

Question

r(t) = cos(t)i + sin(t)j + ln cos(t)k. Find the curvature K at the given point (1,0,0)

Answer 1

r'(t) = <- sint, cost, - tant>; Ir'(t)I = (sin²t + cos²t + tan²)^1/2 = IsectI; t = 0;c r'(0) = <0, 1, 0>; Ir'(0)I = 1.

T'(0) = <0, 1, 0>; IT'(0)I 1

Curvature K(0) = IT'(0)I/Ir'(0)i = 1/1 = 1