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Consider the unit tangent vector [itex] T[/itex] unit normal vector [itex] N [/itex] and binormal vector [itex] B [/itex] parametrized in terms of arc length s.

1) Show that [tex] \frac{dT}{ds} = \kappa\,N[/tex]

I think this part is fine for me. What I did was: [tex]N(t) = \frac{T'(t)}{|T'(t)|}[/tex] and said, by the chain rule, [itex] \frac{dT}{ds} \frac{ds}{dt}= T'(t) [/itex] which simplified to [tex] N(s) = \frac{|r'(t)|}{|T'(t)|} \frac{dT}{ds} => \frac{dT}{ds} = \kappa N(s) [/tex]

Can somebody confirm this is correct?

2) Use a) to show that there exists a scalar [itex] -\tau [/itex] such that [tex] \frac{dB}{ds} = -\tau\,N [/tex]

I was given a hint to try to show that [itex] \frac{dB}{ds} . B = 0 [/itex]

I took the derivative [tex]\frac{d}{ds} B = \frac{d}{ds}(T ×N) = T ×\frac{dN}{ds}[/tex]

Therefore, [tex] (T × \frac{dN}{ds}) . B = (B ×T) . \frac{dN}{ds} = N . \frac{dN}{ds}. [/tex] Am I correct in assuming the above is equal to 0?

Many thanks.

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# Proof of the Frenet-Serret formulae

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