Answer to Question #265938 in Differential Geometry | Topology for Khizar

Question #265938

find curvature r = (4/5 Cost, 1-sint, -3/2 Cost)

1
Expert's answer
2021-11-17T00:38:11-0500

"r = [\\frac{4}{5} cos(t), (1-sin(t)), -\\frac{3}{2} cos(t)]"

Since f(t), g(t), and h(t) are the components of the vector r(t), then f, g, and h are real-valued functions called the component functions of r and we can write


"r(t)=\\frac{4}{5} cos(t) i + (1-sin(t)) j - \\frac{3}{2} cos(t) k"


"r'(t)=-\\frac{4}{5} sin(t) i - cos(t) j + \\frac{3}{2} sin(t) k"

"r''(t)=-\\frac{4}{5} cos(t) i + sin(t) j + \\frac{3}{2} cos(t) k"


So the curvature is

"\u03ba(t)=\\frac{\\vert r'(t)\u00d7 r''(t)\\vert}{{\\vert r'(t)\\vert}^{3}}"


"r'(t)\u00d7 r''(t)=\\begin{vmatrix}\n i & j & k \\\\\n-\\frac{4}{5} sint & -cost & \\frac{3}{2} sint\\\\\n-\\frac{4}{5} cost & sint & \\frac{3}{2} cost\n\\end{vmatrix}"


"r'(t)\u00d7 r''(t)=(-\\frac{3}{2}cos^{2}t -\\frac{3}{2}sin^{2}t)i -(-\\frac{12}{10}cost.sint +\\frac{12}{10}cost.sint)j +(-\\frac{4}{5}sin^{2}t -\\frac{4}{5}cos^{2}t)"


"r'(t)\u00d7 r''(t)=-\\frac{3}{2}(cos^{2}t+sin^{2}t)i -0j -\\frac{4}{5}(sin^{2}t +cos^{2}t)k"


"r'(t)\u00d7 r''(t)=-\\frac{3}{2}i +0j -\\frac{4}{5}k"

So Now

"\\vert r'(t)\u00d7 r''(t)\\vert ={[(-\\frac{3}{2})^2+(-\\frac{4}{5})^2]}^{1\/2}"


"={[\\tfrac{9}{4}+\\tfrac{16}{25}]}^{1\/2}" "={[\\frac{289}{100}]}^{1\/2}" "=\\frac{17}{10}"



"\\vert r'(t)\u00d7 r''(t)\\vert =\\frac{17}{10}"


"{\\vert r'(t)\\vert}^3={[(-\\frac{4}{5}sint)^2+(cost)^{2}+(-\\frac{3}{2}sint)^2]}^{3\/2}"


"{\\vert r'(t)\\vert}^3={[\\frac{16}{25}sin^2t+cos^2t+\\frac{9}{4}sin^2t]}^{3\/2}"


"{\\vert r'(t)\\vert}^3={[\\frac{189}{100}sin^2t+cos^2t]}^{3\/2}"

"{\\vert r'(t)\\vert}^3={[\\frac{189}{100}sin^2t+1-sin^2t]}^{3\/2}={[1+\\frac{189-100}{100}sin^2t]}^{3\/2}"


"{\\vert r'(t)\\vert}^3={[1+\\frac{89}{100}sin^2t]}^{3\/2}"


"\u03ba(t)=\\frac{\\vert r'(t)\u00d7 r''(t)\\vert}{{\\vert r'(t)\\vert}^{3}} =\\frac{\\frac{17}{10}}{{[\\frac{189}{100}sin^2t+cos^2t]}^{3\/2}}"



So the curvature is ...


"\u03ba(t)=\\frac{\\frac{17}{10}}{{[\\frac{189}{100}sin^2t+cos^2t]}^{3\/2}} =\\frac{\\frac{17}{10}}{{[1+\\frac{89}{100}sin^2t]}^{3\/2}}"


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