$\mathbf{T}\prime \left(s\right)$

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

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

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

Figure 2.4.8(a)   Graph of the curvature

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 the curvature

$\mathbf{R}$

$\stackrel{\text{curvature}}{\to }$

$\frac{1}{2}\frac{\sqrt{4{\left(\frac{1}{2}\frac{\sin \left(p\right)\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}{{\left(\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right)}^{3\/2}\cos \left(p\right)}-\frac{1}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}-\frac{{\sin \left(p\right)}^2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}{\cos \left(p\right)}^2}\right)}^2\+4{\left(-\frac{1}{2}\frac{\cos \left(p\right)\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}{{\left(\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right)}^{3\/2}\sin \left(p\right)}-\frac{1}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}-\frac{{\cos \left(p\right)}^2}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}{\sin \left(p\right)}^2}\right)}^2\+2{\cos \left(p\right)}^6{\sin \left(p\right)}^6{\left(\frac{2}{{\cos \left(p\right)}^3\sin \left(p\right)}-\frac{2}{\cos \left(p\right){\sin \left(p\right)}^3}\right)}^2}}{\sqrt{\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}}}$

$\stackrel{\text{assuming real range}}{\to }$

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

Obtain the curvature as $\mathrm{\κ}\=\parallel \mathbf{R}\prime \times \mathbf{R}″\parallel \/{\mathrm{\ρ}}^3$ 

$\mathbf{R}\prime \=\frac{\ⅆ}{\ⅆ p} \mathbf{R}$$\stackrel{\text{assign}}{\to }$

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

$\mathrm{\ρ}\=\parallel \mathbf{R}\prime \parallel$$\stackrel{\text{assign}}{\to }$

$\frac{∥\mathbf{R}\prime \times \mathbf{R}″∥}{{\mathrm{\rho }}^3}$ = $\frac{\sqrt{2}\sqrt{\frac{1}{{\sin \left(p\right)}^4{\cos \left(p\right)}^4}}}{{\left(\frac{1}{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}\right)}^{3\/2}}$$\stackrel{\text{assuming real range}}{\to }$$\cos \left(p\right)\sin \left(p\right)\sqrt{2}$

Obtain the curvature as $\mathrm{\κ}\=∥\mathbf{T}\prime \left(s\right)∥$ then graph $\mathrm{\κ}$

$\mathbf{T}\=\mathbf{R}\prime \/\mathrm{\ρ}$$\stackrel{\text{assign}}{\to }$

$∥\left(\frac{\ⅆ}{\ⅆ p} \mathbf{T}\right)\/\mathrm{\rho }∥$ = $\sqrt{2}\sqrt{{\cos \left(p\right)}^2{\sin \left(p\right)}^2}$$\stackrel{\text{assuming real range}}{\to }$$\cos \left(p\right)\sin \left(p\right)\sqrt{2}$

Display computed quantities

$\mathbf{R}\prime \,\mathbf{R}″\,\mathrm{\ρ}\,\mathbf{T}$

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

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

$\mathrm{simplify}\left(\mathrm{Curvature}\left(\mathbf{R}\,p\right)\right) assuming p\ge 0\,p\le \mathrm{\π}\/2$

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

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

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

Obtain the curvature as $\mathrm{\κ}\=\parallel \mathbf{R}\prime \times \mathbf{R}″\parallel \/{\mathrm{\ρ}}^3$ 

$\mathrm{simplify}\left(\frac{\mathrm{Norm}\left(\mathrm{CrossProduct}\left(\mathrm{diff}\left(\mathbf{R}\,p\right)\,\mathrm{diff}\left(\mathbf{R}\,p\,p\right)\right)\right)}{{\mathrm{\rho }}^3}\right) assuming p\ge 0\,p\le \mathrm{\π}\/2$

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

Obtain the curvature as $\mathrm{\κ}\=∥\mathbf{T}\prime \left(s\right)∥$ 

$\mathrm{\kappa }≔\mathrm{simplify}\left(\mathrm{Norm}\left(\frac{\mathrm{diff}\left(\mathbf{T}\,p\right)}{\mathrm{\rho }}\right)\right) assuming p\ge 0\, p\le \mathrm{\π}\/2$

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

Graph the curvature

$\mathrm{plot}\left(\mathrm{\κ}\,p\=0..\mathrm{\π}\/2\,\mathrm{size}\=\left[200\,200\right]\right)$