I know what you wrote! that, was my question itself.
The chain rule, in ordinary circumstances does not require monotonic parametrization, but for the arc length ,it does, so what is the reason for this apparent difference?
->ds/dt where s is the arc length in cartesian coordinates is ((dx/dt)^2+(dy/dt)^2)^(1/2).
-> Therefore by the chain rule ds/dt = ds/dp * dp/dt, but if I substitute dx/dt=dx/dp* dp/dt and dy/dt= dy/dp* dp/dt in the formula above, I get ds/dt=ds/dp * |dp/dt|??
What is happening?
->Even by...