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).

Find the work done by the force field F(x, y) = (x² In(y + 5), x3 -) on a particle 3(y + 5) that moves once around the circle x? + y? = 4 oriented in the counterclockwise direction.

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).