5. (33 points) Consider the vector field F = ( -y/(x^2+y^2), x/(x^2 + y^2), e^z) defined in the domain D: R³\{(0, 0, t): t E R},i.e R³ without the z-axis. (a) (5 points) Show that V x F = 0 everywhere in the domain D. (b) (5 points) Find ∮_C F. dr where C' is the unit circle on the x-y plane centered at (0,0) with counter-clockwise orientation when looking from above. (hint: use the definition of line integral) (c) (5 points) Find ∮_C F. dr where C is the boundary of the square on the x-y plane with four vertices (-1,1,0), (1,1,0), (1,-1,0), and is counter-clockwise oriented when looking from tices (-1,-1,0), above. (d) (5 points) Find ∮_C F.dr where C' is a circle on the plane z = 1 centered at (2, 2) with radius one and with counter-clockwise orientation when looking from above. (e) (5 points) Is F conservative in the domain D? Justify your answer. (f) (8 points) Does there exist a scalar function f such that F=Vf in the half space {(x, y, z): x > 0}? If yes, find f. If no, justify your conclusion.