$\mathbf{T}\=\frac{1}{\sqrt{1\+p^2}}\left[\begin{array}{c}1\\ p\end{array}\right]$

$\mathbf{N}\=\frac{1}{\sqrt{1\+p^2}}\left[\begin{array}{c}-p\\ 1\end{array}\right]$

$\mathrm{\ρ}\=2\sqrt{1\+ p^2}$

$\mathrm{\κ}\=\frac{1}{2{\left(1\+p^2\right)}^{3\/2}}$

Table 2.8.1(a)   Items from the Frenet formalism

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

$\=\frac{d}{\mathrm{dp}}\left(2\sqrt{1\+ p^2}\right) \(\frac{1}{\sqrt{1\+p^2}}\left[\begin{array}{c}1\\ p\end{array}\right]\)\+\frac{{\left(2\sqrt{1\+ p^2}\right)}^2}{2{\left(1\+p^2\right)}^{3\/2}}\left(\frac{1}{\sqrt{1\+p^2}}\left[\begin{array}{c}-p\\ 1\end{array}\right]\right)$

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

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

$\=\left[\begin{array}{c}0\\ 2\end{array}\right]$

Initialize

Loading Student:-VectorCalculus

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

$\mathbf{R}\=⟨2 p\+3\,p^2-1⟩$$\stackrel{\text{assign}}{\to }$

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

$∥\frac{\ⅆ}{\ⅆ p} \mathbf{R}∥$ = $2\sqrt{p^2\+1}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\ρ}$$$

$\mathbf{R}″\=\frac{{\ⅆ}^2}{{\ⅆ}^{}{p}^2} \mathbf{R}$$\stackrel{\text{assign}}{\to }$

Obtain T 

$\mathbf{R}$ = $\stackrel{\text{tangent vector}}{\to }$ $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$T$$$

Obtain N 

$\mathbf{R}$ = $\stackrel{\text{principal normal}}{\to }$ $\stackrel{\text{2-normalize}}{\to }$ $\stackrel{\text{assuming real}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$N$$$

Obtain the curvature $\mathrm{\κ}$ 

$\mathbf{R}$  = $\stackrel{\text{curvature}}{\to }$$\frac{1}{2}\frac{\sqrt{\frac{p^2}{{\left(p^2\+1\right)}^3}\+{\left(-\frac{p^2}{{\left(p^2\+1\right)}^{3\/2}}\+\frac{1}{\sqrt{p^2\+1}}\right)}^2}}{\sqrt{p^2\+1}}$$\stackrel{\text{assuming real}}{\to }$$\frac{1}{2{\left(p^2\+1\right)}^{3\/2}}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\κ}$$$

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

$\left(\frac{\ⅆ}{\ⅆ p} \mathrm{\ρ}\right) \mathbf{T}\+\mathrm{\κ} {\mathrm{\ρ}}^2 \mathbf{N}$ = $\stackrel{\text{simplify}}{\=}$

$\mathbf{R}″$ =  

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

$\mathbf{R}″·\mathbf{T}$ = $\frac{2p}{\sqrt{p^2\+1}}$$$$$

$\frac{\ⅆ}{\ⅆ p} \mathrm{\ρ}$ = $\frac{2p}{\sqrt{p^2\+1}}$$$

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

$\mathbf{R}″·\mathbf{N}$ = $\frac{2}{\sqrt{p^2\+1}}$$$

$\mathrm{\κ} {\mathrm{\ρ}}^2$ = $\frac{1}{2}\frac{4p^2\+4}{{\left(p^2\+1\right)}^{3\/2}}$$\stackrel{\text{simplify}}{\=}$$\frac{2}{\sqrt{p^2\+1}}$$$

Initialize

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

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

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

$\mathbf{R}≔⟨2 p\+3\,p^2-1⟩\:$

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

$\mathbf{T}≔\mathrm{TangentVector}\left(\mathbf{R}\,\mathrm{normalized}\right)\:$

Obtain N with the PrincipalNormal command, including the parameter normalized.

$$\begin{gathered} \mathbf{temp}≔\mathrm{PrincipalNormal}\left(\mathbf{R}\,\mathrm{normalized}\right)\: \\ \mathbf{N}≔\mathrm{simplify}\left(\mathbf{temp}\right) assuming p\colon\colon \mathrm{real}\: \end{gathered}$$

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

$\mathrm{\κ}≔\mathrm{simplify}\left(\mathrm{Curvature}\left(\mathbf{R}\right)\right) assuming p\colon\colon \mathrm{real}\:$

Display $\mathbf{R}″\,\mathbf{T}\,\mathbf{N}\,\mathrm{\ρ}\,\mathrm{\κ}$ 

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

Obtain the right-hand side of the decomposition formula

$\mathrm{simplify}\left(\mathrm{diff}\left(\mathrm{\ρ}\,p\right) \mathbf{T}\+\mathrm{\κ} {\mathrm{\ρ}}^2 \mathbf{N}\right) assuming p\colon\colon \mathrm{real}$

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

$\mathrm{DotProduct}\left(\mathbf{R}″\,\mathbf{T}\right)$ = $\frac{2p}{\sqrt{p^2\+1}}$$$

$\mathrm{diff}\left(\mathrm{\ρ}\,p\right)$ = $\frac{2p}{\sqrt{p^2\+1}}$$$

Obtain the scalar projection of $\mathbf{R}″$ on N and compare to $\mathrm{\κ} {\mathrm{\ρ}}^2$ 

$\mathrm{DotProduct}\left(\mathbf{R}″\,\mathbf{N}\right)$ = $\frac{2}{\sqrt{p^2\+1}}$$$

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