# Compute the surface integral $\int_S (∇ × F) \cdot dS$ for $F = (x − y, x + y, ze^{xy})$ on the given surface

Let $S$ be the surface defined as $z = 4 ? 4x^2 ? y ^2$ with $z ? 0$ and oriented upward. Let $F = (x ? y, x + y, ze^{xy})$. Compute $$\int_S (? × F) \cdot dS$$.

Note that the boundary of $S$ is given by $z=0$ and $4x^2+y^2=4$. Since the surface $S$ has upward orientation, the boundary of $S$, $C=\partial S$, has counterclockwise orientation when viewed from above. Thus, a parametrization of the boundary is given by
$r(t)=(\cos t, 2\sin t, 0), 0 \leq t \leq 2\pi.$
Applying the Stokes' Theorem we have
$\iint_S \nabla \times F \cdot dS=\int_{C}F \cdot dr=\int_0^{2\pi}F(r(t))r'(t)dt$
$=\int_0^{2\pi} (\cos t -2\sin t,\cos t +2\sin t,0)\cdot (-\sin t, 2\cos t,0)dt$
$=\int_0^{2\pi}(-\cos t \sin t+2\sin^2 t+2\cos^2 t+4 \sin t \cos t)$
$=\int_0^{2\pi}2+3 \sin t \cos t dt=(2t+\frac{3}{2}\sin^2 t)|_0^{2\pi}$
$=(4\pi+\frac{3}{2}\sin^2 (2\pi))-(0+\frac{3}{2}\sin^2 (0))$
$=(4\pi+0)-(0+0)=4 \pi.$

• Savionf