Answer to Question #314546 in Vector Calculus for Jyo

Question #314546

verify stoke’s theorem when f = x ^ 2 i + y ^2 j + z ^ 2k ,s is the upper hemisphere


z =√( a ^2 − x ^2 − y ^2)


1
Expert's answer
2022-03-23T05:01:35-0400

"x=\\mathrm{a}\\sin \\varphi ,y=\\mathrm{a}\\cos \\varphi ,z=0,\\varphi :0\\rightarrow 2\\pi \\\\f=\\left( a^2\\sin ^2\\varphi ,a^2\\cos ^2\\varphi ,0 \\right) \\\\dl=\\left( \\mathrm{a}\\cos \\varphi ,-\\mathrm{a}\\sin \\varphi ,0 \\right) \\\\\\oint{fdl}=\\int_0^{2\\pi}{\\left( a^3\\sin ^2\\varphi \\cos \\varphi -a^3\\cos ^2\\varphi \\sin \\varphi \\right) d\\varphi}=0\\\\rot\\left( f \\right) =\\left| \\begin{matrix}\ti&\t\tj&\t\tk\\\\\t\\frac{\\partial}{\\partial x}&\t\t\\frac{\\partial}{\\partial y}&\t\t\\frac{\\partial}{\\partial z}\\\\\tx^2&\t\ty^2&\t\tz^2\\\\\\end{matrix} \\right|=0\\\\\\iint{rot\\left( f \\right) dS}=0\\\\Stoke\u0091s\\,\\,formula\\,\\,holds."


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