Ok, I have a confusion involving a cord. In physics, we often make the transformation from an infinitesimal cord to an infinitesimal arc: d(cord) = d(arc). And I think it's used in some math proofs too.
For exemple, imagine a position vector \vec{r}(t) of fixed norm, rotating around the z axis (i.e. \vec{\omega} = \omega \hat{z}). Let's say the radius of the circle described by the motion of its tip/head/arrow is R. After a time \Delta t, it has rotated an angle \omega \Delta t and then they (the manuals) say that it can be seen that when \Delta t is small, ||\Delta \vec{r}|| (which is a cord), is very near the length of the arc \Delta s subtended by the angle \omega \Delta t, and thus, poof, ||d\vec{r}|| = ds.
And while this seems to be true intuitively, I have never seen a proof of this statement. And when I try to do it, here's what I get:
I start from a cercle of radius R and a cord \delta subtended by an angle \theta. I find that the length of the cord is given by
\delta = 2Rsin\left(\frac{\theta}{2}\right)
Therefor,
d\delta=Rcos\left(\frac{\theta}{2}\right)d\theta
While
ds=rd\theta
So
d\delta=cos\left(\frac{\theta}{2}\right)ds
A result indicating that even the differential version is just an approximation because only true for a principal angle \theta=0.