Question

Use Stokes' theorem to evaluate ∬S(∇×F)⋅dS where F(x,y,z)=−3yzi+3xzj+14(x2+y2)zk and S is the part of the paraboloid z=x2+y2 that lies inside the cylinder x2+y2=1, oriented upward.

Answer 1

Stokes' theorem says the integral of the curl of $\vec F$ over $S$ is equal to the integral of $\vec F$ along the boundary of $S$, with counterclockwise orientation (when viewed from above). This boundary is the circle $x^2+y^2=1$ set in the plane $z=1$.

Parameterize this path by

$\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath+\vec k$

with $0\le t\le2\pi$. Then

$\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_{\partial S}\vec F\cdot\mathrm d\vec r$

$=\displaystyle\int_0^{2\pi}(-3\sin t\,\vec\imath+3\cos t\,\vec\jmath+14\,\vec k)\cdot(-\sin t\,\vec\imath+\cos t\,\vec\jmath)\,\mathrm dt$

$=\displaystyle3\int_0^{2\pi}\mathrm dt=\boxed{6\pi}$