Consider a force in R³ defined by F(x, y, z) = (3x²y³,3 x³y² + 2y, 0). Find the potential energy U(x, y, z) such that U(0,0,0) = 0.

Use Green's theorem to calculate the work done by the force F(x, y) = xy'i + 3x?yj on a particle that is moving counterclockwise around the triangle with vertices (-3, 0), (0, 0), and (0, 3).

Suppose f(z) = u(x, y) +iv(x, y) is differentiable at zo = xo+iyo, with f'(zo) # 0. Show that the vectors Vu(xo, yo) and Vv(xo, yo) are perpendicular and have the same magnitude. Express this magnitude in terms of f'(zo).

Find the work done by the force field F in moving an object on the line segment from A to B, where F(x, y) = (2y^(3/2) )i + (3x √y)j, A(1, 1), and B(4, 4).