Chapter 6.12, Problem 19MP

Given $F_1=-2yi+(z-2x)j+(y+z)k,F_2=yi+2xj:$

(a) Is $F_1$ conservative? Is $F_2$ conservative?

(b) Find the work done by $F_2$ on a particle that moves around the ellipse $x=\cos \theta$ $y=2\sin \theta$ from $\theta =0$ to $\theta =2\pi$.

(c) For any conservative force in this problem find a potential function $V$ such that $F=-\nabla V$.

(d) Find the work done by $F_1$ on a particle that moves along the straight line from $(0,1,0)$ to $(0,2,5)$.

(e) Use Green’s theorem and the result of Problem 9.7 to do Part (b) above.