Question

(a) Show that each of the vector fields F⃗ =4yi⃗ +4xj⃗ , G⃗ =3yx2+y2i⃗ +−3xx2+y2j⃗ , and H⃗ =2xx2+y2√i⃗ +2yx2+y2√j⃗ are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F⃗ , a potential function is f(x,y)= 4xy For G⃗ , a potential function is g(x,y)= 4arctan(x/y) For H⃗ , a potential function is h(x,y)= (b) Find the line integrals of F⃗ ,G⃗ ,H⃗ around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ∫CF⃗ ⋅dr⃗ = 0 ∫CG⃗ ⋅dr⃗ = ∫CH⃗ ⋅dr⃗ = 0 (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for

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