Problem 4. In general, for a vector field F and a smooth directed closed curve C, the integral Rc= Q F . dr, measures the circulation or rotation of F along C. a) Consider the following two vector fields in 2: F1 = (-y, 20) , F2 = (-2 , - y ). Calculate the circulation of F1 and F2 along the closed curve y(t) = (cos(t), sin(t) ), where t ranges from 0 to 27. b) The normalized circulation is defined as: lim 1 ( F . dr , ICI-0 | A| Jo where | A| is the area of the surface enclosed by the closed curve C. Fix a point (x0, yo) and let C be a smooth closed curve around (x0, yo). In the case that C shrinks to (x0, yo), we write |C|-0. For a smooth vector field F = (P(x, y), Q(x, y)), show the relation lim ICI-0 | A | Jo 1 F . dr = Qx(20, yo) - Py(20, yo). Conclude that the differential circulation of a conservative force field F at any point is zero. This is a special case of a more general relation that states the curl of the gradient vector fields is zero: V x (VU) =0