I came across this problem when studying Gauss Map:

Show that $\tau^2=-K$ on an asymptotic curve. Here $\tau$ is torsion of the asymptotic curve $\pmb r(u(s),v(s))$(with curvature $\kappa$ nonzero) and $K$ is total curvature.

The only thing I can figure out is that since $d\pmb{r}$ is an asymptotic direction then principal curvature $k_n=0$. Also I saw another exercise which says in such situation we have: $$\tau=\frac{1}{\sqrt {EG-F^2}} \begin{vmatrix}(\dot v)^2 & -\dot u\dot v & (\dot u)^2 \\ E & F & G \\ L & M & N \\\end{vmatrix}$$ This one I could figure out by calculation. But I can't see any connection between these two expressions.

I know this question has been asked here. But I haven't studied the theorem mentioned in that answer yet. Anyone could give another solution? Thanks in advance!


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