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 $$ \int_S (? × F) \cdot dS $$.

Answer

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
\[\iint_S \nabla \times F \cdot dS=\int_{C}F \cdot dr=\int_0^{2\pi}F(r(t))r'(t)dt\]
\[=\int_0^{2\pi} (\cos t -2\sin t,\cos t +2\sin t,0)\cdot (-\sin t, 2\cos t,0)dt\]
\[=\int_0^{2\pi}(-\cos t \sin t+2\sin^2 t+2\cos^2 t+4 \sin t \cos t)\]
\[=\int_0^{2\pi}2+3 \sin t \cos t dt=(2t+\frac{3}{2}\sin^2 t)|_0^{2\pi}\]
\[=(4\pi+\frac{3}{2}\sin^2 (2\pi))-(0+\frac{3}{2}\sin^2 (0))\]
\[=(4\pi+0)-(0+0)=4 \pi.\]

  • Please leave a comment if you have any questions.

The answer is accepted.