Prove a curvature identity

Question

Show that N = (dT/dt)/(abs (dT/dt)) using dT/ds = k N

Answer 1

First, $$\frac{ d\mathbf T }{ dt } = \frac{ d\mathbf T }{ ds } \, \frac{ ds }{ dt } = k' \mathbf N,$$ where $k' = k \, (ds/dt)$. But since $\|\mathbf N\|$ = 1, we have $$\left\| \frac{ d \mathbf T }{ dt } \right\| = k'.$$ So $$\mathbf N = \frac{ d \mathbf T / dt } { k'} = \frac{ d \mathbf T / dt } { \left\|d \mathbf T / dt \right\| }.$$