Let F = (5mg, 9312) be a vector field in the plane, and C the path y = 532 joining (0,0) to (1, 5) in the plane. A. Evaluate fCF - dr C] B. Does the integral in part (A) depend on the path joining (0,0) to (1, 5)? (y) Suppose F = F(x, y, z) is a gradient field with F = Vf, S is a level surface of f, and C is a curve on S. What is the value of the line integral SF . dr?The domain of f(:1:, y) is the :vy-plane, and values of f are given in the table below. Find f0 grad f - d?. where C' is (a) A line from (0,2) to (4,1). fogradf-dF=C] 'l 66 65 65 67 2 63 60 59 59 59 3 62 63 64 65 64 (b) A circle of radius 1 centered at (3, 3) traversed counterclockwise. fcgradf-dF=D Let C be the positively oriented circle x2 + y = 1. Use Green's Theorem to evaluate the line integral 20y dx + 11x dy.Let C be the positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3). Use Green's Theorem to evaluate the line integral 8yxda + 3x y dy.Suppose F(x, y) = (2x - 3y)i + 4x} and C is the counter-clockwise oriented sector of a circle centered at the origin with radius 4 and central angle 7 /3. Use Green's theorem to calculate the circulation of F around C. Circulation = (Click on graph to enlarge)