Activity 13.13.3

Find the circulation of $\vF = \langle 3yz, xz, -xy\rangle$ along the curve $C$ consisting of (given in order of the orientation) the quarter-circle of radius $1$ centered at $(0,-2,0)$ in the plane $y=-2$ from $(0,-2,1)$ to $(1,-2,0)$, the line segment from $(1,-2,0)$ to $(1,5,0)$, the quarter-circle of radius $1$ centered at $(0,5,0)$ in the plane $y=5$ from $(1,5,0)$ to $(0,5,1)$, and the line segment from $(0,5,1)$ to $(0,-2,1)$.

Find the circulation of $\vG = 3z^2 \vi -(z^2 + 2x) \vj +zy \vk$ along the circle in the $xy$-plane of radius $3$ centered at the origin. Assume the counterclockwise orientation of the circle.