Let $\sigma: U \rightarrow \mathbb{R}^3$ be a regular conjugate parametrization. I would like to know a "proper" formulation of the torsion of coordinate curves of $\sigma$ with respect to the components of its first and second fundamental forms (i.e. $E,F,G,e,g$).
I naturally tried to do it by expanding $\det\,(\sigma_{uuu},\sigma_{uu},\sigma_u)$ with respect to the natural frame $\{\sigma_u,\sigma_v,N\}$ but resulted in a messy and long formula (I must add that I didn't try to simplify the obtained expression further since I thought there might be a wiser approach to it).