$S:\ \ x^2+y^2+z^2=1 \\[2ex]\ is\ the\ surface\ of\ the\ sphere\ with\ centre\ (0,0,0)\ and\ radius=\ 1. \\[2ex] \displaystyle F=yzi+zxj+xyk\ and\ dr\\[2ex] F=dxi+dyj+dzk \\[2ex] \displaystyle \therefore{}F\bullet{}dr=yzdx+xzdy+xydz \\[2ex] \displaystyle Part\ I: \\[2ex] \displaystyle Consider,\\[2ex] \displaystyle \int F\bullet{}dr=\ \int yzdx+xzdy+xydz\ \\[3ex] \displaystyle In\ the\ plane\ z=0,\\[2ex] equation\ of\ sphere\ from\ (1)\ is\ x^2+y^2=1 \\[2ex] It\ is\ a\ circle\ with\ centre\ (0,0)\ and\ radius\ =1\ ...(2)\\[2ex] Using\ parametric\ form, %eq2 \\[3ex] \displaystyle x=a\cos\theta{}=1\cos\theta{} \\[2ex] \displaystyle \therefore{}dx=-\sin\theta{}d\theta{} \\[2ex] \displaystyle y=a\sin\theta{}\ =\ 1\sin\theta{} \\[2ex] \displaystyle \therefore{}dy=\cos\theta{}d\theta{} \\[2ex] \displaystyle z=0\ \therefore{}dz=0 \\[2ex] \displaystyle For\ complete\ circle\ limits\ of\ \theta{}\ are\ 0\ to\ 2\pi{}. \\[2ex] \displaystyle \therefore{}\int F\bullet{}dr=\ \int_0^{2\pi{}}0+0+0=0\ ...(3)\\[3ex]Part\ 2: %eq3 \displaystyle F=yzi+zxj+xyk \\[2ex] \displaystyle \therefore{}\nabla{}\times{}F=\left\vert{}\begin{array}{ ccc} i & j & k \\ \frac{\partial{}}{\partial{}x} & \frac{\partial{}}{\partial{}y} & \frac{\partial{}}{\partial{}z} \\ yz & zx & xy \end{array}\right\vert{} \\[4ex] \displaystyle \ \ \ \ \ \nabla{}\times{}F=\ -i(x-x)-j(y-y)-k(z-z) \\[2ex] \displaystyle \ \ \ \ \ \nabla{}\times{}F=\ 0 \\[2ex] \displaystyle In\ xy\ plane,\ Unit\ normal\ is\ N=k\ and\ ds=dxdy \\[2ex] \displaystyle \therefore{}\ N\bullet{}\left(\ \nabla{}\times{}F\right)=k\bullet{}0=0 \\[2ex] \displaystyle \therefore{}\int N\bullet{}\left(\ \nabla{}\times{}F\right)ds=0\ ..\left(4\right) \\[3ex] From\ (3)\ and\ (4) \\[2ex] \displaystyle \int F\bullet{}dr\ \ =\int N\bullet{}\left(\ \nabla{}\times{}F\right)ds \\[2ex] \displaystyle \therefore{}\ Stokes\ theorem\ is\ verified. \\[3ex] $