Question

Calc III Tutorial: Curvature and Normal Vectors

Answer 1

\({\bf{Curvature:}}\)

κ = |dT/ds|, or in plan English, the rate at which the unit tangent vector changes its direction

we can also apply chain rule to split dT/ds into

dT/dt * dt/ds

using reciprocal of dt/ds
dT/dt * (1 / (dt/ds))
= (1/|v|) * dT/dt
to get curvature as function of t instead of s

Answer 2

\({\bf{Principal~Unit~Normal:}}\)

N = (1/κ) * (dT/ds)
basically the unit vector of dT/ds
vector N will always point towards the concave side of the curve

Answer 3

alternatively,

N = (dT/dt) / |(dT/dt)|

Answer 4

Source material is section 13.4 of Thomas' Calculus, Early Transcendentals, 14th edition by Hass, Heil, Weir, et. al.