For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your answer. (a) F(x, y, z) = 3yi + 3x } F(x, y, z) is ? (b) F(x, y, z) = 2xi+ 2z j F(x, y, z) is ? (c) F(x, y, 2) = 2xi + 2yj + 2z k F(x, y, z) is ? (d) F(x, y, z) = = 4z it 4y j + Vx'+2 Vez+ z2 Vx3+ 22 F(x, y, z) is ?\fUse the Fundamental Theorem of Line Integrals to calculate f0 F" - d? exaCtly, if F' = 43315; + aids}, and C' is the quarter of the unit circle in the rst quadrant, traced counterclockwise from (1, 0) to (0, 1). [CF-eh} .9: Determine whether each of the following vector fields F is path independent (conservative) or not. If it is path independent, enter a potential function for it, that is, a function f(x, y) so that Vf = F. If it is path dependent, enter NONE. (a) If F(x, y) = (x + 5)i + (2y + 3)3, then f(x, y) = (b) If F(x, y) = (3x + 6y)i + (6x + 3y)j, then f(x, y) = (c) If F(x, y) = (-2y, 2x), then f(x, y) = (d) If F(x, y) = (5x2 - y), -2xy), then f(x, y) = (e) If F(x, y) = " i + 4In(x + 3)j, then x + 3 f(x, y) =\f[1 point) Consider the: vector eld F (m, y, z} = xi + sci + zk. 3) Find a JnCtion f such that F = f and 110,0, 0) = I]. 1%: y: 3) = i, b} Use part a) to compute the work done by F on a particle moving along the curve C given by r(t) = (1 + mm + (1 + 2sin2t)j + (1 + 2sin3t)k, u 1: t g g. 91 (1 point) Consider the vector field F (x, y, z) = (4z + y)i + (3z + x)j + (3y + 4x)k. a) Find a function f such that F = Vf and f(0, 0, 0) = 0. f(x, y, z) = b) Suppose C is any curve from (0, 0, 0) to (1, 1, 1). Use part a) to compute the line integral f F . dr.If F = V(4x2 + 5y*), find f. F . dr where C is the quarter of the circle a2 + y? = 1 in the first quadrant, oriented counterclockwise. SoF . di =