In the context of tensor calculus, by using Serret-Frenet formula or otherwise,
how to prove that
$\tau^2=\displaystyle\frac{r'''^2}{k^2}-k^2-(\frac{k'}{k})^2$
where $\tau$ and $k$ represent respectively torsion and curvature.
In the context of tensor calculus, by using Serret-Frenet formula or otherwise,
how to prove that
$\tau^2=\displaystyle\frac{r'''^2}{k^2}-k^2-(\frac{k'}{k})^2$
where $\tau$ and $k$ represent respectively torsion and curvature.
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
$\displaystyle k(t)=||r''(t)||, \ \tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)$
You can substitute these in the right hand of the equation and obtain the left hand side.