(1) Find the curvature k and the length of the curve from t = 0 to t = p.
We have: k = dT/ ds And dT/ ds = dT/dt / (ds/ dt)
dT/dt = 1/13 [(-24 sin 2t)i -24 cos 2t) j
Where: (dt/ds) = 1/ 13
Simplify: dT/ ds = 24 [- sin 2t i - cos 2t j] / 169
But: dT/ ds = k N
Where: N = [- sin 2t i - cos 2t j]
k = 24/169
The length of the curve from t = 0 to t = p can be obtained from: (dt/ds) = 1/ 13
So that: ds = 13 dt (for differential in arc length)
Integrating: s = ò p o 13 dt =13 p
2. For a space curve defined by:
x = exp (t) cos (t), y = exp(t) sin (t), z = exp(t)
Find the unit vector T:
T = dR/ds = dt/ds [i dx/ dt + j dy.dt + k dz/dt]
dx/dt = et [cos(t) - sin(t)}
dy/dt = et [cos (t) + sin (t)}
dz/dt = et
Whence:
T = (dt/ds)2 (et ) 2 [(cost - sin t)2 + (cos t + sin t)2 + 1]
= (dt/ds)2 (et ) 2 [1 + 2 sin t cost - 2 sin t cost + 1 + 1] = 3(dt/ds)2 (et ) 2
dt/ds = 1/ Ö (3) et
Then: T = 1/ Ö (3) [(cos t - sin t)i + (cos t + sin t)j + k]
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