Find the work done by the vector field \(\vF(x,y,z) = 6x^2z\vi + 3y^2\vj + x\vk\) on a particle that moves from the point \((3,0,0)\) to the point \((3,0,6\pi)\) along the helix given by \(\vr(t) = \langle 3\cos(t),3\sin(t),t\rangle\text{.}\)
Let \(\vF(x,y) = \langle 0,x\rangle\text{.}\) Let \(C\) be the closed curve consisting of the top half of the circle of radius \(2\) centered at the origin and the portion of the \(x\)-axis from \((2,0)\) to \((-2,0)\text{,}\) oriented clockwise. Find the circulation of \(\vF\) around \(C\text{.}\)
