Activity 13.4.3

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 $\vr(t) = \langle 3\cos(t),3\sin(t),t\rangle$.

Let $\vF(x,y) = \langle 0,x\rangle$. 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 $(-2,0)$, oriented clockwise. Find the circulation of $\vF$ around $C$.