13.9 13.3: SUMMARY

• The length s of a path r(t) = 〈x(t), y(t), z(t)〉 for a ≤ t ≤ b is

  

• Arc length function:

• Speed is the derivative of distance traveled with respect to time:

  

• The velocity vector v(t) = r′(t) points in the direction of motion [provided that r′(t) ≠ 0] and its magnitude υ(t) = ∥r′(t)∥ is the object’s speed.

• We say that r(s) is an arc length parametrization if ∥r′(s)∥ = 1 for all s. In this case, the length of the path for a ≤ s ≤ b is b − a.

• If r(t) is any parametrization such that r′(t) ≠ 0 for all t, then

  r₁(s) = r(g(s))

  is an arc length parametrization, where t = g(s) is the inverse of the arc length function.