Let $\vec{F}$ be the vector field $\vec{F}\left(x,y,z\right)=\left(z,x,y\right)$. Let $\rm S$ be portion of the surface $x^{2}+y^{2}+z=1$ lying above the $\rm XY$-plane, oriented upward. Evaluate the surface integral
$$\int_{\rm S}{\rm Curl}\vec{F}\cdot\vec{{\rm d}S}$$
I know how to evaluate the surface integral directly, but I don't know how to apply Stokes' theorem to solve it. Can someone please walk me through this? Thank you for your help!
$\endgroup$ 131 Answer
$\begingroup$It's actually pretty straightforward. Let the parametrization of the circle $x^2+y^2=1$ laying on the $xy$ plane be $\gamma(t)=(\cos t,\sin t,0)$, where $t\in[0,2\pi]$. Then $\gamma'(t)=(-\sin t, \cos t,0)$ and
\begin{align} \int_{\rm S}{\rm Curl}\vec{F}\cdot\vec{{\rm d}S} =\int_{0}^{2\pi} f(\gamma(t))\cdot\gamma'(t){\rm d}t =\int_{0}^{2\pi} (0,\cos t,\sin t)\cdot(-\sin t,\cos t,0){\rm d}t =\int_{0}^{2\pi} (\cos t)^2{\rm d}t \end{align}
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