Remark 1. (a) Recall that the gravitational vector field that a mass M placed at the origin 0 of R3 exerts on a mass m at position (x, y, z) * 0 is given by - x FGr, M,m(x, y, z) = GMm -y - Z (202 + 92+ 2 2) 3/2' (202 + 92 + 2 2) 3/2' (2 2 + 92+ 2 2)3/2 where G is the universal gravitational constant. (b) Suppose that an electric charge Q is located at the origin 0 in R3. Coulomb's law tells us that the electric force H(x, y, z) = HEL,Q,q(x, y, z) exerted by this charge on a charge q located at a point (x, y, z) 0 (and of the same sign as Q) is 2 HEL, Q,q(x, y, z) = keQq y ( 2 2 + y2 + 2 2 ) 3/2 ' ( 2 2 + 2 2 + 2 2 ) 3 / 2 ' (ac2 + 92 + 2 2) 3/2 where ke is an experimentally found constant, called the Coulomb constant or electric force constant. Problem 2. (max. 30 = 10 + 10+ 10 points) (a) Explain why both vector fields from Remark 1 (which are force fields in this case) are examples of an "inverse square law". That is, e.g. the electric force is proportional to the reciprocal of the square of the distance between the two charges, and analogously for the gravitational force. (The justification should be essentially the same in both cases. ) (b) Show that F and H pass the screening test for conservative vector fields.(c) Show that F can be written as F = Vf (and hence that it is a conservative vector field) for the potential function GMm f ( x, y, z) = ( 2 2 + y2 + 2 2) 1/2 ' Find also a potential function for H (and conclude that this is a conservative vector field too)