Prove: For a smooth curve $C$ parameterized by $r(s)$ where $s$ is arc length,$r′(s)$ satisfies $|r′(s)| = 1$.
I understand that $|r'(s)|$ should equals to one because it's the magnitude of change in distance travelled with respect to arc length. However, I am having a hard time coming up with a proof. Can someone please help?
Given another equivalent parametrization of the same curve $\tilde{r}(t),$ we have that $$ s(t)=(C)\int_O^Pds=\int_{t_0}^t\left|\frac{d\tilde{r}(\tau)}{d\tau}\right|d\tau $$ so that $$ \frac{ds(t)}{dt}=\left|\frac{d\tilde{r}(t)}{dt}\right|. $$ Moreover, given the relation between the two parametrizations $$ \tilde{r}(t)=r(s(t)) $$ dy differentiating $$ \frac{d\tilde{r}(t)}{dt}=\frac{dr(s(t))}{ds}\frac{ds(t)}{dt}=\frac{dr(s(t))}{ds}\left|\frac{d\tilde{r}(t)}{dt}\right| $$ and taking the modulus $$ \left|\frac{dr(s(t))}{ds}\right|=1 $$