On page 6 is this exercise:
" $s(t) = \int^t_a||\alpha'(u)||du$
Show that if $\alpha$ is a regular curve, i.e., $||\alpha'|| \neq$ 0 for all $t ∈ I$, then $s(t)$ is an invertible function, i.e., it is one-to-one (Hint: compute $s'(t)$)."
I am at a loss at how to differentiate s(t). How is it performed?
Note that if $x<y$ then $s(y)-s(x) = \int_x^y \|\alpha'(t)\| dt > 0$, hence $s$ is a strictly increasing continuous function, hence invertible on its range.