Question

Compute the flux of curl(F) through the part of the paraboloid z = x 2 + y 2 that lies below the plane z = 4 with upward-pointing unit normal vector and F = h3z,5x,−2yi.

Answer 1

Parameterize this surface (call it S) by

`\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+u^2\,\mathbf k`

with
`0\le u\le2` and
`0\le v\le2\pi`.

The normal vector to S is

`\mathbf n=(\partial\mathbf s)/(\partial u)*(\partial\mathbf s)/(\partial v)=-2u^2\cos v\,\mathbf i-2u^2\sin v\,\mathbf j+u\,\mathbf k`

Compute the curl of F :

`\\abla*\mathbf F=-2\,\mathbf i+3\,\mathbf j+5\,\mathbf k`

So the flux of curl(F) is

`\displaystyle\iint_S(\\abla*\mathbf F)\cdot\mathrm d\mathbf S=\int_0^(2\pi)\int_0^2(\\abla*\mathbf F)\cdot\mathbf n\,\mathrm du\,\mathrm dv`

`=\displaystyle\int_0^(2\pi)\int_0^2(5u+4u^2\cos v-6u^2\sin v)\,\mathrm du\,\mathrm dv=\boxed{20\pi}`

Alternatively, you can apply Stokes' theorem, which reduces the surface integral of the curl of F to the line integral of F along the intersection of the paraboloid with the plane z = 4. Parameterize this curve (call it C) by

`\mathbf r(t)=2\cos t\,\mathbf i+2\sin t\,\mathbf j+3\,\mathbf k`

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

`\displaystyle\iint_S(\\abla*\mathbf F)\cdot\mathrm d\mathbf S=\int_0^(2\pi)\mathbf F\cdot\mathrm d\mathbf r`

`=\displaystyle\int_0^(2\pi)(20\cos^2t-24\sin t)\,\mathrm dt=\boxed{20\pi}`