find the curvature and torsion of the curve z=u,y=1+u/u,z=1-u^2/u
a)
"r'(u)=\\langle1, -\\dfrac{1}{u^2},-\\dfrac{1}{u^2}-1\\rangle"
"r''(u)=\\langle0, \\dfrac{2}{u^3}, \\dfrac{2}{u^3}\\rangle"
"|r'(u)|=\\sqrt{(1)^2+(-\\dfrac{1}{u^2})^2+(-\\dfrac{1}{u^2}-1)^2}"
"=\\dfrac{\\sqrt{2u^4+2u^2+2}}{u^2}"
"r'(u)\\times r''(u)=\\begin{vmatrix}\n i & j & k \\\\\n\\\\\n 1 & -\\dfrac{1}{u^2} & -\\dfrac{1}{u^2}-1 \\\\ \\\\\n 0 & \\dfrac{2}{u^3}&\\dfrac{2}{u^3}\n\\end{vmatrix}"
"=i\\begin{vmatrix}\n -\\dfrac{1}{u^2} & -\\dfrac{1}{u^2}-1 \\\\ \\\\\n \\dfrac{2}{u^3} & \\dfrac{2}{u^3}\n\\end{vmatrix}-j\\begin{vmatrix}\n 1 & -\\dfrac{1}{u^2}-1 \\\\ \\\\\n 0& \\dfrac{2}{u^3}\n\\end{vmatrix}"
"+k\\begin{vmatrix}\n 1 & -\\dfrac{1}{u^2} \\\\ \\\\\n 0 & \\dfrac{2}{u^3}\n\\end{vmatrix}=-\\dfrac{2}{u^3}i-\\dfrac{2}{u^3}j+\\dfrac{2}{u^3}k"
"|r'(t)\\times r''(t)|=\\sqrt{(-\\dfrac{2}{u^3})^2+(-\\dfrac{2}{u^3})^2+(\\dfrac{2}{u^3})^2}"
"=\\dfrac{2\\sqrt{3}}{u^2|u|}"
Find curvature
"=\\dfrac{\\dfrac{2\\sqrt{3}}{u^2|u|}}{(\\dfrac{\\sqrt{2u^4+2u^2+2}}{u^2})^{3}}"
"=\\dfrac{\\sqrt{6}u^2|u|}{2(u^4+u^2+1)^{3\/2}}"
"\\kappa(u)=\\dfrac{\\sqrt{6}u^2|u|}{2(u^4+u^2+1)^{3\/2}}"
b)
"(r'(u)\\times r''(u))\\cdot r'''(u)=0-\\dfrac{2}{u^3}(-\\dfrac{6}{u^4})+\\dfrac{2}{u^3}(-\\dfrac{6}{u^4})"
"=0"
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