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

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

$\mathrm{\ρ}\=\sqrt{2}{e}^p$

$\mathrm{\κ}\=\frac{e^{-p}}{\sqrt{2}}$

Table 2.8.2(a)   Items from the Frenet formalism

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

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

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

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

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

Initialize

Loading Student:-VectorCalculus

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

$\mathbf{R}\=⟨{\ⅇ}^p \cos \left(p\right)\,{\ⅇ}^p \sin \left(p\right)⟩$$\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{2}\sqrt{{\ⅇ}^{2p}}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{\ρ}$$$

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

Obtain T 

$\mathbf{R}$ = $\stackrel{\text{tangent vector}}{\to }$ $\stackrel{\text{Euclidean-normalize}}{\to }$ $\stackrel{\text{assuming real}}{\to }$ $\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 }$$\frac{1}{2}\frac{\sqrt{{\left(-\frac{1}{2}\frac{\sqrt{2}\left({\ⅇ}^p\cos \left(p\right)-{\ⅇ}^p\sin \left(p\right)\right)}{\sqrt{{\ⅇ}^{2p}}}-\frac{\sqrt{2}{\ⅇ}^p\sin \left(p\right)}{\sqrt{{\ⅇ}^{2p}}}\right)}^2\+{\left(-\frac{1}{2}\frac{\sqrt{2}\left({\ⅇ}^p\sin \left(p\right)\+{\ⅇ}^p\cos \left(p\right)\right)}{\sqrt{{\ⅇ}^{2p}}}\+\frac{\sqrt{2}{\ⅇ}^p\cos \left(p\right)}{\sqrt{{\ⅇ}^{2p}}}\right)}^2}\sqrt{2}}{\sqrt{{\ⅇ}^{2p}}}$$\stackrel{\text{assuming real}}{\to }$$\frac{1}{2}\sqrt{2}{\ⅇ}^{-p}$$\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 $\mathbf{R}″$ 

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

$\mathbf{R}″$ = $$

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

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

$\frac{\ⅆ}{\ⅆ p} \mathrm{\ρ}$ = $\sqrt{2}\sqrt{{\ⅇ}^{2p}}$$$$\stackrel{\text{assuming real}}{\to }$${\ⅇ}^p\sqrt{2}$

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

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

$\mathrm{\κ} {\mathrm{\ρ}}^2$ = $\sqrt{2}{\ⅇ}^{-p}{\ⅇ}^{2p}$$\stackrel{\text{simplify}}{\=}$${\ⅇ}^p\sqrt{2}$$$

Initialize

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

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

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

$\mathbf{R}≔⟨{\ⅇ}^p \cos \left(p\right)\,{\ⅇ}^p \sin \left(p\right)⟩\:$

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

$$\begin{gathered} \mathbf{Temp}\mathbf{≔}\mathrm{TangentVector}\left(\mathbf{R}\,\mathrm{normalized}\right)\: \\ \mathbf{T}≔\mathrm{simplify}\left(\mathbf{Temp}\right) assuming p\colon\colon \mathrm{real}\: \end{gathered}$$

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

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

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

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 $\mathrm{\ρ}\prime$ 

$\mathrm{simplify}\left(\mathrm{DotProduct}\left(\mathbf{R}″\,\mathbf{T}\right)\right)$ =  

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

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

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

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