1. Let F(x, y, z) be the vector field (x² (1 + y²z²), Q(x, y, z), ey cos(22)), where is a differentiable function of x, y, and z. Suppose div F = 2x(1 + y²z2) - 2ery sin(22) + z tan(2y + 2) + 2z2 sec²(2y + 2); and • curl F = (xey cos(22)-y tan(2y+z)-y sec² (2y+z), 2x²y²z-yery cos(22), -2x²yz²). Determine div(Q).

3. Use the divergence of the vector field to compute Y F(x, y) 2²1) ₁ + (29-2 +²8²) J = fo tan -1 tan 4 -1 (2x) dy - where C is the cardiod r= 1+ sin 0. (ry. Y 2+82, dx,

6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).