Show that each of the vector fields F₁ = (2xy, x² - y², 3xz), F₂ = (y, x, z), and F₃ = (eˣ, eʸ, eᶻ) are gradient vector fields on some domain by finding a potential function for each. For F₁, a potential function is φ(x, y, z) = x²y - y²z + C₁, for F₂, a potential function is φ(x, y, z) = xy + xz + C₂, and for F₃, a potential function is φ(x, y, z) = eˣ + eʸ + eᶻ + C₃.
Find the line integrals of F₁, F₂, and F₃ around the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise.
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 F₂. (Be sure that you are able to explain why or why not.)