Question

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + 5zj + 7yk, C is the curve of intersection of the plane x + z = 8 and the cylinder x2 + y2 = 81.

Answer 1

By Stokes' theorem,

`\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\\abla*\vec F)\cdot\mathrm d\vec S`

where
`S` is the surface with
`C` as its boundary. The curl is

`\\abla*\vec F(x,y,z)=2\,\vec\imath-x\,\vec k`

Parameterize
`S` by

`\vec\sigma(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(8-u\cos v)\,\vec k`

with
`0\le u\le9` and
`0\le v\le2\pi`. Then take the normal vector to
`S` to be

`\vec\sigma_u*\vec\sigma_v=u\,\vec\imath+u\,\vec k`

Then the line integral is equal to the surface integral,

`\displaystyle\iint_S(\\abla*\vec F)\cdot\mathrm d\vec S=\int_0^(2\pi)\int_0^9(2\,\vec\imath-u\cos v\,\vec k)\cdot(u\,\vec\imath+u\,\vec k)\,\mathrm du\,\mathrm dv`

`\displaystyle=\int_0^(2\pi)\int_0^9(2u-u^2\cos v)\,\mathrm du\,\mathrm dv=\boxed{162\pi}`