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 \((1,-2,0)\text{,}\) the line segment from \((1,-2,0)\) to \((1,5,0)\text{,}\) the quarter-circle of radius \(1\) centered at \((0,5,0)\) in the plane \(y=5\) from \((1,5,0)\) to \((0,5,1)\text{,}\) and the line segment from \((0,5,1)\) to \((0,-2,1)\text{.}\)
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.
