Answer to Question #265916 in Differential Geometry | Topology for Khizar

Question #265916

iii) find curvature r(t) = ( Β½ Cost, 1-sint, -3/2 cost)

1
Expert's answer
2021-11-15T16:38:53-0500
"r'(t)=\\langle -\\dfrac{1}{2}\\sin t, -\\cos t, \\dfrac{3}{2}\\sin t\\rangle"

"r''(t)=\\langle -\\dfrac{1}{2}\\cos t, \\sin t, \\dfrac{3}{2}\\cos t\\rangle"

"r'(t)\\times r''(t)=\\begin{vmatrix}\n i & j & k \\\\\n -(\\sin t)\/2 & -\\cos t & (3\\sin t)\/2\\\\\n-(\\cos t)\/2 & \\sin t & (3\\cos t)\/2\n\\end{vmatrix}"


"=i\\begin{vmatrix}\n -\\cos t & (3\\sin t)\/2 \\\\\n \\sin t & (3\\cos t)\/2\n\\end{vmatrix}"

"-j\\begin{vmatrix}\n -(\\sin t)\/2 & (3\\sin t)\/2 \\\\\n -(\\cos t)\/2 & (3\\cos t)\/2\n\\end{vmatrix}"


"+k\\begin{vmatrix}\n -(\\sin t)\/2 &-\\cos t \\\\\n -(\\cos t)\/2 & \\sin t \n\\end{vmatrix}"

"=-\\dfrac{3}{2}i+\\dfrac{3}{4}\\sin t\\cos tj-\\dfrac{1}{2}k"


"|r'(t) \\times r''(t)|=\\sqrt{\\dfrac{9}{4}+\\dfrac{9}{16}\\sin^2t\\cos^2t+\\dfrac{1}{4}}"

"=\\dfrac{1}{4}\\sqrt{40+9\\sin^2t\\cos^2t}"

"|r'(t)|=\\sqrt{\\dfrac{1}{4}\\sin^2t+\\cos^2t+\\dfrac{9}{4}\\sin^2t}"

"=\\dfrac{1}{2}\\sqrt{4+6\\sin^2t}"

"k=\\dfrac{|r'(t) \\times r''(t)|}{|r'(t)|^3}=\\dfrac{\\dfrac{1}{4}\\sqrt{40+9\\sin^2t\\cos^2t}}{\\dfrac{1}{8}(4+6\\sin^2t)\\sqrt{4+6\\sin^2t}}"

"=\\dfrac{\\sqrt{40+9\\sin^2t\\cos^2t}}{(2+3\\sin^2t)\\sqrt{4+6\\sin^2t}}"



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