Answer Happy

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F(x,y)=xy2i+x2yj (i) Show that F is a conservative field on the whole plane. (ii) Find a potential function ϕ for F satisfying ϕ(2,1)=6. (iii) Use ϕ found in Part (a) (ii) to compute the following work integral ∫C​F⋅dr, where C is some arbitrary path from (0,0) to (1,2). Let G(x,y)=(x−y)i+(x+y)j (i) Compute the integral ∫C​(x−y)dx+(x+y)dy where C is the closed path given by r=(cost)i+(sint)j,(0≤t≤2π). (ii) Based only on your answer to Part (b) (i) above and the nature of the given path C, do you believe that G is a conservative vector field? [To obtain any marks at all, briefly explain your answer.]