Curvature and torsion via vector triple products of derivatives

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Problem 1.24.126 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Starting from the definition of the principal normal v\mathbf{v} as the unit vector in the direction of the vector dτ/dsd \tau / d s (τ\tau is the unit tangent vector, and ss is the variable arc length of a given three-times continuously differentiable curve without singular points) and the definition of the binormal β=[τ,ν]\beta = [\tau , \nu ], prove that:

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(1)

the derivative of the binormal dβ/dsd \beta / d s is collinear with the principal normal;

(2)

if dτ/ds=kν,dβ/ds=−ϰνd \tau / d s = k \nu , d \beta / d s = -\varkappa \nu, then dν/ds=−kτ+ϰβd \nu / d s = -k \tau +\varkappa \beta (in other words, prove the Frenet formulas).

Problem 1.24.127 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that for a three-times continuously differentiable curve Γ={r(s);0⩽s⩽S}\Gamma = \left\{ \mathbf{r}(s) ; 0 \leqslant s \leqslant S\right\}, where ss is the variable arc length, the relations

∣d3rds3∣3=k4+k2ϰ2+(dkds)2(drds,d2rds2)=0,(drds,d3rds3)=−k2,(d2rds2,d3rds3)=kdkds \begin{gathered} \left|\frac{d^{3} \mathbf{r}}{d s^{3}}\right|^{3} = k^{4}+k^{2} \varkappa ^{2}+\left(\frac{d k}{d s}\right)^{2} \\ \left(\frac{d \mathbf{r}}{d s}, \frac{d^{2} \mathbf{r}}{d s^{2}}\right) = 0, \quad \left(\frac{d \mathbf{r}}{d s}, \frac{d^{3} \mathbf{r}}{d s^{3}}\right) = -k^{2}, \quad \left(\frac{d^{2} \mathbf{r}}{d s^{2}}, \frac{d^{3} \mathbf{r}}{d s^{3}}\right) = k \frac{d k}{d s}\end{gathered}
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Problem 1.24.128 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Under the assumptions of the previous problem, prove that

(τ,β,dβds)=ϰ,(dβds,d2βds2,d3βds3)=ϰ5ddskϰ,(dτds,d2τds2,d3τds3)=k5ddsϰk. \left(\tau , \beta , \frac{d \mathbf{\beta }}{d s}\right) = \varkappa ,\left(\frac{d \mathbf{\beta }}{d s}, \frac{d^{2} \mathbf{\beta }}{d s^{2}}, \frac{d^{3} \mathbf{\beta }}{d s^{3}}\right) = \varkappa ^{5} \frac{d}{d s} \frac{k}{\varkappa },\left(\frac{d \tau }{d s}, \frac{d^{2} \tau }{d s^{2}}, \frac{d^{3} \tau }{d s^{3}}\right) = k^{5} \frac{d}{d s} \frac{\varkappa }{k}.
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References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.