Find the flux of the curl of field F through the shell S.
-$\mathbf { F } = 5 \mathrm { y } \mathbf { i } + 2 \mathrm { x } \mathbf { j } + \cos ( \mathrm { z } ) \mathbf { k } ; \mathrm { S } : \mathbf { r } ( \mathrm { r } , \theta ) = 2 \sin \varphi \cos \theta \mathbf { i } + 2 \sin \varphi \sin \theta \mathbf { j } + 2 \cos \varphi \mathbf { k } , 0 \leq \theta \leq 2 \pi$ and $0 \leq \varphi \leq \frac { \pi } { 2 }$
A) $- 24 \pi$
B) $24 \pi$
C) $- 12$
D) $- 12 \pi$

Question

Quizplus · Accepted Answer

The curl of $\mathbf{F}$ is $\nabla \times \mathbf{F} = (0 - 0)\mathbf{i} + (0 - \sin(z))\mathbf{j} + (2 - 5)\mathbf{k} = -\sin(z)\mathbf{j} - 3\mathbf{k}$. The surface $S$ is a hemisphere of radius 2, and its normal vector points outward. The flux of $\nabla \times \mathbf{F}$ through $S$ involves only the $k$-component since the $j$-component's contribution cancels out due to symmetry. The flux is $-3$ times the area of the hemisphere, which is $2\pi(2)^2$, giving $-12\pi$.

Find the Flux of the Curl of Field F Through