I am a little confused, is a tangent vector $T$ a unit vector? I know there is a unit tangent vector defined as
$$T = \frac{r'}{|r'|}.$$
I think the most confusing thing is $T$ is being used for both Tangent vector and unit tangent vector?
Could someone clarify this?
We know the velocity vector $v = \frac{\mathrm dr}{\mathrm dt}$ is tangent to the curve $r(t)$ so it is a tangent vector, but the vector $T$ is its direction vector. Of course, this unit tangent vector (I mean $T$) is a differentiable function of $t$ whenever $v=r'(t)$ is a differentiable function of $t$.