Question

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)

Answer 1

$x=\mathrm{a}\sin \varphi ,y=\mathrm{a}\cos \varphi ,z=0,\varphi :0\rightarrow 2\pi \\f=\left( a^2\sin ^2\varphi ,a^2\cos ^2\varphi ,0 \right) \\dl=\left( \mathrm{a}\cos \varphi ,-\mathrm{a}\sin \varphi ,0 \right) \\\oint{fdl}=\int_0^{2\pi}{\left( a^3\sin ^2\varphi \cos \varphi -a^3\cos ^2\varphi \sin \varphi \right) d\varphi}=0\\rot\left( f \right) =\left| \begin{matrix} i& j& k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ x^2& y^2& z^2\\\end{matrix} \right|=0\\\iint{rot\left( f \right) dS}=0\\Stoke‘s\,\,formula\,\,holds.$