Use Stokes Theorem to evaluate S curl F Solution By Stokes T

Solution

By Stokes\' Theorem, ??s curl F · dS = ?c F · dr. Parameterize C by r(t) = <2 cos t, 2, 2 sin t> for t in [0, 2p]. Therefore, ?c F · dr = ?(t = 0 to 2p) <32 cos^2(t) sin t, sin(8 cos t sin t), 8 sin t cos t> · <-2 sin t, 0, 2 cos t> dt = ?(t = 0 to 2p) [-64 cos^2(t) sin^2(t) + 16 sin t cos^2(t)] dt = ?(t = 0 to 2p) [-16 sin^2(2t) + 16 sin t cos^2(t)] dt = ?(t = 0 to 2p) [-8 (1 - cos(4t)) + 16 sin t cos^2(t)] dt = [-8(t - sin(4t)/4) - (16/3) cos^3(t)] {for t = 0 to 2p} = -16p. I hope this helps!