The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field $\mathbf { F } ( x , y , z ) = - 8 z \mathbf { i } + 2 y \mathbf { k }$. Find $\iint _ { S } ( \operatorname { curl } \mathbf { F } ) \cdot \mathbf { N } d S$, where $S$ is the upper surface of the cylindrical container.
A) $- 7$
B) $- 2$
C) 9
D) 0
E) 4

Question

Quizplus · Accepted Answer

The curl of $\mathbf{F}$ is $\nabla \times \mathbf{F} = 0$, since the partial derivatives of the components of $\mathbf{F}$ with respect to their respective variables do not vary. Therefore, the integral of the curl of $\mathbf{F}$ over any surface, including the upper surface of the cylinder, is $0$.

The Motion of a Liquid in a Cylindrical Container of Radius