Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 25H / a / Solution

For an arclength-parametrized curve define the Frenet frame by $T={\alpha }^{\prime }$, $k=|{\alpha }^{\prime \prime }|>0$, $N=T^{\prime }/k$ and $B=T\times N$. Differentiating the unit lengths gives diagonal frame-derivative coefficients zero. Also $T^{\prime }=kN$ implies $N^{\prime }\cdot T=-k$, while $B^{\prime }\cdot T=-B\cdot T^{\prime }=0$. Define the torsion of a space curve by $\tau =-B^{\prime }\cdot N$. Orthogonality then givesThese are the Frenet formulae. Differentiating $B=T\times N$ also verifies the chosen torsion sign; equivalently $\tau =\det ({\alpha }^{\prime },{\alpha }^{\prime \prime },{\alpha }^{\prime \prime \prime })/|{\alpha }^{\prime }\times {\alpha }^{\prime \prime }{|}^2$.