Arc length of a parametrised differentiable curve.

� � , 15. ArclengthDefinition 15.1. Let I be an open interval. A partition P of [a,b] ⊂I is a sequence of points a = t0 0, there is a δ> 0 such that for every partition P whose mesh size is less than δ, we have �L − l(P)� < �. There are interesting examples of curves that don’t have a length. Start with interval [0, 1] in the plane. The length is 1. Now adjust this curve, by adding in a triangular hump in the middle, so that we get four line segments of length 1/3. This is a curve of length 4/3. Now add a hump to each of the four line segments of length 1/3. The length of the resulting curve is (4/3)2 . If we keep doing this, then we get more and more complicated curves, whose length at stage n is (4/3)n . This process converges to a very pointed curve whose length is infinite (in fact this curve is a fractal). However: Proposition 15.3. If �r : I −→ Rn is a C1-function, then the curve �r([a,b]) has a length � b L = ��r�(t)� dt. a Remark 15.4. The fractal curve above is continuous but it is nowhere differentiable (the curve has too many sharp points). In general the exact formula for the arclength is only of theoretical interest. However there are some contrived examples where we can calculately the arclength precisely. Example 15.5. Let r : R −→ R2 1 � , � , be the parametrised differentiable curve given by �r(t)= a cos tˆı + a sin tj.ˆThen �r�(t)= −a sin tˆı + a cos tj,ˆand so ��r�(t)� = a. Hence the length of the curve �r([0, 2π]) is � 2π L = a dt =2πa, 0 which is indeed the circumference of a circle of radius a. Example 15.6. Let r : R −→ R3 be the parametrised differentiable curve given by �r(t)= a cos tˆj + btˆı + a sin tˆk, Then �r�(t)= −a sin tˆj + bˆı + a cos tˆk, and so ��r�(t)� = √a2 + b2 . Hence the length of the curve �r([0, 2π]) is � 2π L =(a 2 + b2)1/2 dt =2π(a 2 + b2)1/2 . 0 Example 15.7. Let r : R −→ R2 be the parametrised differentiable curve given by �r(t)= a cos tˆı + b sin tj.ˆThen �r�(t)= −a sin tˆı + b cos tj,ˆand so ��r�(t)� =(a 2 sin2 t + b2 cos 2 t)1/2 Hence the length of the curve �r([0, 2π]) is � 2π L =(a 2 sin2 t + b2 cos 2 t)1/2 dt, 0 the length of an ellipse, with major and minor axes of length a and b. 2 � � � � � � � � � � � � � � � � Definition 15.8. Let �r : 1 I −→ Rn be a parametrised differentiable curve, which is of class C. Suppose that �r�(t) is nowhere zero. Given a ∈ I, define the arclength parameter s(t), by the formula t s(t)= ��r�(τ)� dτ = length of r([a,t]) if t ≥ aa length of −r([t,a]) if t