Using stokes theorem to evaluate $\int\int (curl$ $F)n$ $ds$, $F=\langle z, -2x, xy\rangle$, S being $z=4-x^2-y^2$ above the xy plane.

Question

Typed this all out and then firgure it out, so figured I'd put it up anyway in case anyone else ever needs it since I spent the time.

Answer 1

$\int\int (curl$ $F)n$ $ds$ = $F(r(t))r'(t)dt$

$r(t)=\langle2cos(t), 2sin(t), 0\rangle$

$r'(t)=\langle-2sin(t), 2cos(t), 0\rangle$

$F(r'(t))=\langle0,-4cos(t), 4sin(t)cos(t)\rangle$

$F(r(t))*r'(t)=0+-8cos^2(t) +0$

$-8\int^{2\pi}_0(cos^2(t))dt$

$-8/2\int^{2\pi}_0(1+cos(2t))dt$

$-\frac{8}{2}[\frac{sin(2t)}{2}+t]^{2\pi}_0$

$-\frac{8}{2}[0+2\pi-0-0]$

$-8\pi$