$\mathbf{T}\=\left[\begin{array}{c}-{\sin \left(p\right)}^2\\ {\cos \left(p\right)}^2\\ \sqrt{2}\cos \left(p\right)\sin \left(p\right)\end{array}\right]$

$\mathbf{N}\=\left[\begin{array}{c}-\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ -\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ 2{\cos \left(p\right)}^2-1\end{array}\right]$

$\mathrm{\ρ}\=\frac{1}{\cos \left(p\right)\sin \left(p\right)}$

$\mathrm{\κ}\=\sqrt{2}\cos \left(p\right)\sin \left(p\right)$

Table 2.8.7(a)   Items from the Frenet formalism

$\mathrm{\rho }\prime \mathbf{T}\+\mathrm{\kappa } {\mathrm{\rho }}^2\mathbf{N}$

$\=\frac{d}{\mathrm{dp}}\left(\frac{1}{\cos \left(p\right)\sin \left(p\right)}\right) \mathbf{T}\+\left(\sqrt{2}\cos \left(p\right)\sin \left(p\right)\right) {\left(\frac{1}{\cos \left(p\right)\sin \left(p\right)}\right)}^2 \mathbf{N}$

$\=\left(-\frac{2{\cos \left(p\right)}^2-1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right) \left(\left[\begin{array}{c}-{\sin \left(p\right)}^2\\ {\cos \left(p\right)}^2\\ \sqrt{2}\cos \left(p\right)\sin \left(p\right)\end{array}\right]\right)\+\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}\left(\left[\begin{array}{c}-\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ -\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ 2{\cos \left(p\right)}^2-1\end{array}\right]\right)$

$\=\left[\begin{array}{c}\frac{2{\cos \left(p\right)}^2-1}{{\cos \left(p\right)}^2}-2\\ -\frac{2{\cos \left(p\right)}^2-1}{{\sin \left(p\right)}^2}-2\\ 0\end{array}\right]$

$\=\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$

$\mathbf{R}″·\mathbf{T}$

$\=\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]·\left[\begin{array}{c}-{\sin \left(p\right)}^2\\ {\cos \left(p\right)}^2\\ \sqrt{2}\cos \left(p\right)\sin \left(p\right)\end{array}\right]$

$\=\frac{{\sin \left(p\right)}^2}{{\cos \left(p\right)}^2}-\frac{{\cos \left(p\right)}^2}{{\sin \left(p\right)}^2}$

$\= -\frac{2{\cos \left(p\right)}^2-1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\=\mathrm{\ρ}\prime$

$\mathbf{R}″\mathbf{·}\mathbf{N}$

$\=\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]·\left[\begin{array}{c}-\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ -\sqrt{2}\cos \left(p\right)\sin \left(p\right)\\ 2{\cos \left(p\right)}^2-1\end{array}\right]$

$\=\frac{\sqrt{2}\sin \left(p\right)}{\cos \left(p\right)}\+\frac{\sqrt{2}\cos \left(p\right)}{\sin \left(p\right)}$

$\=\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}$

$\=\left(\sqrt{2}\cos \left(p\right)\sin \left(p\right)\right){\left(\frac{1}{\cos \left(p\right)\sin \left(p\right)}\right)}^2\=\mathrm{\κ} {\mathrm{\ρ}}^2$

Initialize

Loading Student:-VectorCalculus

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$\mathbf{R}\=⟨\ln \left(\cos \left(p\right)\right)\,\ln \left(\sin \left(p\right)\right)\,\sqrt{2}p⟩$$\stackrel{\text{assign}}{\to }$

Obtain $\mathrm{\ρ}\=∥\mathbf{R}\prime \left(p\right)∥$ and $\mathbf{R}″\left(p\right)$ 

$∥\frac{\ⅆ}{\ⅆ p} \mathbf{R}∥$ = $\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}$$\stackrel{\text{simplify symbolic}}{\to }$$\frac{1}{\cos \left(p\right)\sin \left(p\right)}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\rho }$

$\frac{{\ⅆ}^2}{{\ⅆ}^{}{p}^2} \mathbf{R}$ = $\left[\begin{array}{c}-1-\frac{{\sin \left(p\right)}^2}{{\cos \left(p\right)}^2}\\ -1-\frac{{\cos \left(p\right)}^2}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$$\stackrel{\text{simplify}}{\=}$$\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Temp}$$$

$\mathbf{R}″\=\mathrm{Temp}$$\stackrel{\text{assign}}{\to }$$$

Obtain T 

$\mathbf{R}$ = $\left[\begin{array}{c}\ln \left(\cos \left(p\right)\right)\\ \ln \left(\sin \left(p\right)\right)\\ \sqrt{2}p\end{array}\right]$$\stackrel{\text{tangent vector}}{\to }$$\left[\begin{array}{c}-\frac{\sin \left(p\right)}{\cos \left(p\right)}\\ \frac{\cos \left(p\right)}{\sin \left(p\right)}\\ \sqrt{2}\end{array}\right]$$\stackrel{\text{Euclidean-normalize}}{\to }$$\left[\begin{array}{c}-\frac{\sin \left(p\right)}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}\cos \left(p\right)}\\ \frac{\cos \left(p\right)}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}\sin \left(p\right)}\\ \frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\end{array}\right]$$\stackrel{\text{simplify symbolic}}{\to }$$\left[\begin{array}{c}-{\sin \left(p\right)}^2\\ {\cos \left(p\right)}^2\\ \cos \left(p\right)\sin \left(p\right)\sqrt{2}\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$T$

Obtain N 

$\mathbf{R}$ = $\left[\begin{array}{c}\ln \left(\cos \left(p\right)\right)\\ \ln \left(\sin \left(p\right)\right)\\ \sqrt{2}p\end{array}\right]$ = $\stackrel{\text{principal normal}}{\to }$$\left[\begin{array}{c}-\frac{2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\\ -\frac{2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\\ \frac{\sqrt{2}\left(2{\cos \left(p\right)}^2-1\right)}{\cos \left(p\right)\sin \left(p\right)\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\end{array}\right]$$\stackrel{\text{2-normalize}}{\to }$$\left[\begin{array}{c}-\frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\\ -\frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\\ \frac{2{\cos \left(p\right)}^2-1}{\cos \left(p\right)\sin \left(p\right)\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}\end{array}\right]$$\stackrel{\text{simplify symbolic}}{\to }$$\left[\begin{array}{c}-\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ -\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ 2{\cos \left(p\right)}^2-1\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$N$

Obtain the curvature $\mathrm{\κ}$ 

$\mathbf{R}$ = $\left[\begin{array}{c}\ln \left(\cos \left(p\right)\right)\\ \ln \left(\sin \left(p\right)\right)\\ \sqrt{2}p\end{array}\right]$ = $\stackrel{\text{curvature}}{\to }$$\frac{\sqrt{4{\left(\frac{\sin \left(p\right)\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}{2{\left(\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(p\right)}-\frac{1}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}-\frac{{\sin \left(p\right)}^2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}{\cos \left(p\right)}^2}\right)}^2+4{\left(-\frac{\cos \left(p\right)\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}{2{\left(\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(p\right)}-\frac{1}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}-\frac{{\cos \left(p\right)}^2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}{\sin \left(p\right)}^2}\right)}^2+2{\cos \left(p\right)}^6{\sin \left(p\right)}^6{\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}^2}}{2\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}$$\stackrel{\text{simplify symbolic}}{\to }$$\cos \left(p\right)\sin \left(p\right)\sqrt{2}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\kappa }$

Compute $\mathrm{\rho }\prime \mathbf{T}\+\mathrm{\kappa } {\mathrm{\rho }}^2\mathbf{N}$ and compare with $\mathbf{R}″$ 

$\left(\frac{\ⅆ}{\ⅆ p} \mathrm{\ρ}\right) \mathbf{T}\+\mathrm{\κ} {\mathrm{\ρ}}^2 \mathbf{N}$ = $\left[\begin{array}{c}-\left(\frac{1}{{\cos \left(p\right)}^2}-\frac{1}{{\sin \left(p\right)}^2}\right){\sin \left(p\right)}^2-2\\ \left(\frac{1}{{\cos \left(p\right)}^2}-\frac{1}{{\sin \left(p\right)}^2}\right){\cos \left(p\right)}^2-2\\ \left(\frac{1}{{\cos \left(p\right)}^2}-\frac{1}{{\sin \left(p\right)}^2}\right)\cos \left(p\right)\sin \left(p\right)\sqrt{2}+\frac{\sqrt{2}\left(2{\cos \left(p\right)}^2-1\right)}{\cos \left(p\right)\sin \left(p\right)}\end{array}\right]$ = $\stackrel{\text{assuming real}}{\to }$$\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$

$\mathbf{R}″$ = $\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$ 

Compare the scalar projection of $\mathbf{R}″$ on T with $\mathrm{\ρ}\prime$

$\mathbf{R}″·\mathbf{T}$ = $\frac{{\sin \left(p\right)}^2}{{\cos \left(p\right)}^2}-\frac{{\cos \left(p\right)}^2}{{\sin \left(p\right)}^2}$$\stackrel{\text{simplify}}{\=}$$\frac{-2{\cos \left(p\right)}^2+1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}$

$\frac{\ⅆ}{\ⅆ p} \mathrm{\ρ}$ = $\frac{1}{{\cos \left(p\right)}^2}-\frac{1}{{\sin \left(p\right)}^2}$$\stackrel{\text{simplify}}{\=}$$\frac{-2{\cos \left(p\right)}^2+1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}$

Compare the scalar projection of $\mathbf{R}″$ on N with $\mathrm{\κ} {\mathrm{\ρ}}^2$ 

$\mathbf{R}″·\mathbf{N}$ = $\frac{\sin \left(p\right)\sqrt{2}}{\cos \left(p\right)}+\frac{\cos \left(p\right)\sqrt{2}}{\sin \left(p\right)}$$\stackrel{\text{simplify}}{\=}$$\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}$$$$$

$\mathrm{\κ} {\mathrm{\ρ}}^2$ = $\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

Define R and obtain $\mathbf{R}″\,\mathbf{T}\,\mathbf{N}\,\mathrm{\ρ}\,\mathrm{\κ}$ 

$\mathbf{R}≔⟨\ln \left(\cos \left(p\right)\right)\,\ln \left(\sin \left(p\right)\right)\,\sqrt{2}p⟩\:$

$\mathbf{R}″≔\mathrm{simplify}\left(\mathrm{diff}\left(\mathbf{R}\,p\,p\right)\right)\:$

$$\begin{gathered} \mathbf{Temp}\mathbf{≔}\mathrm{TangentVector}\left(\mathbf{R}\&comma;\mathrm{normalized}\right)\&colon; \\ \mathbf{T}≔\mathrm{simplify}\left(\mathbf{Temp}\right) assuming p\&gt;0\&comma;p<\mathrm{\&pi;}\&sol;2\&colon; \end{gathered}$$

$$\begin{gathered} \mathbf{Temp}\mathbf{≔}\mathrm{PrincipalNormal}\left(\mathbf{R}\&comma;\mathrm{normalized}\right)\&colon; \\ \mathbf{N}≔\mathrm{simplify}\left(\mathbf{Temp}\right) assuming p\&gt;0\&comma;p<\mathrm{\&pi;}\&sol;2\&colon; \end{gathered}$$

$\left[\begin{array}{c}-\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ -\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ 2{\cos \left(p\right)}^2-1\end{array}\right]$

$\mathrm{\&rho;}≔\mathrm{simplify}\left(\mathrm{Norm}\left(\mathrm{diff}\left(\mathbf{R}\&comma;p\right)\right)\right) assuming p\&gt;0\&comma;p<\mathrm{\&pi;}\&sol;2\&colon;$

$\frac{1}{\cos \left(p\right)\sin \left(p\right)}$

$\mathrm{\&kappa;}≔\mathrm{simplify}\left(\mathrm{Curvature}\left(\mathbf{R}\right)\right) assuming p\&gt;0\&comma;p<\mathrm{\&pi;}\&sol;2\&colon;$

Display $\mathbf{R}″\&comma;\mathbf{T}\&comma;\mathbf{N}\&comma;\mathrm{\&rho;}\&comma;\mathrm{\&kappa;}$ 

$\mathbf{R}″\&comma;\mathbf{T}\&comma;\mathbf{N}\&comma;\mathrm{\&rho;}\&comma;\mathrm{\&kappa;}$

$\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right],\left[\begin{array}{c}-{\sin \left(p\right)}^2\\ {\cos \left(p\right)}^2\\ \cos \left(p\right)\sin \left(p\right)\sqrt{2}\end{array}\right],\left[\begin{array}{c}-\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ -\cos \left(p\right)\sin \left(p\right)\sqrt{2}\\ 2{\cos \left(p\right)}^2-1\end{array}\right],\frac{1}{\cos \left(p\right)\sin \left(p\right)},\cos \left(p\right)\sin \left(p\right)\sqrt{2}$

Obtain the right-hand side of the decomposition formula

$\mathrm{simplify}\left(\mathrm{diff}\left(\mathrm{\&rho;}\&comma;p\right) \mathbf{T}\&plus;\mathrm{\&kappa;} {\mathrm{\&rho;}}^2 \mathbf{N}\right)$

$\left[\begin{array}{c}-\frac{1}{{\cos \left(p\right)}^2}\\ -\frac{1}{{\sin \left(p\right)}^2}\\ 0\end{array}\right]$

Obtain the scalar projection of $\mathbf{R}″$ on T and compare to $\mathrm{\&rho;}\prime$ 

$\mathrm{simplify}\left(\mathrm{DotProduct}\left(\mathbf{R}″\&comma;\mathbf{T}\right)\right)$

$\frac{-2{\cos \left(p\right)}^2+1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}$

$\mathrm{simplify}\left(\mathrm{diff}\left(\mathrm{\&rho;}\&comma;p\right)\right)$

$\frac{-2{\cos \left(p\right)}^2+1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}$

Obtain the scalar projection of $\mathbf{R}″$ on N and compare to $\mathrm{\&kappa;} {\mathrm{\&rho;}}^2$ 

$\mathrm{simplify}\left(\mathrm{DotProduct}\left(\mathbf{R}″\&comma;\mathbf{N}\right)\right)$

$\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}$

$\mathrm{simplify}\left(\mathrm{\&kappa;} {\mathrm{\&rho;}}^2\right)$

$\frac{\sqrt{2}}{\cos \left(p\right)\sin \left(p\right)}$