1. In this problem, we explore the geometry of vector fields in space, focusing on one of the most fundamental examples in physics and engineering: the electric field generated by stationary electric charges. When a point charge q is located at the point x0 = (x0, y0, z0) in space, it produces an electric field E(x, y, z) at any other point x = (x, y, z) given by E(x, y, z) = q 40 x x0 |x x0| 3 , y y0 |x x0| 3 , z z0 |x x0| 3 , where 0 is the permittivity of free space, and |x x0| denotes the Euclidean distance from the observation point x to the source charge at x0. (a) Show that E is a conservative vector field with potential function u(x, y, z) = q 40|x x0| , that is, verify that E = u. In this formulation, the function U(x, y, z) = u(x, y, z) = q 40|x x0| is called the potential energy associated with a unit positive charge located at (x, y, z). (b) Consider two point charges +q and q placed