(a) Using the Frenet-Serret equations, prove that dr/ds.d2r/ds2×d3r/ds3 = τ/ρ2. (b) Given that the Darboux rotation vector is given by w = τTˆ + κBˆ , use the Frenet-Serret frame (Tˆ , Nˆ , Bˆ ) to show that i. dTˆ/ds = w × Tˆ ii. dNˆ/ds = w × Nˆ .

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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3. (a) Using the Frenet-Serret equations, prove that dr/ds.d2r/ds2×d3r/ds3 = τ/ρ2.
(b) Given that the Darboux rotation vector is given by w = τTˆ + κBˆ , use the Frenet-Serret
frame (Tˆ , Nˆ , Bˆ ) to show that
i. dTˆ/ds = w × Tˆ
ii. dNˆ/ds = w × Nˆ .

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