F is the three-dimensional vector field defined by F(x,y,z) = (x,y,z). In other terms, P(x,y,z) = x, Q(x,y,z) = y, R( x,y,z) = z. Also, the domain D = [0,1]x[0,1]x[0,1] is the solid unit cube, which consists of every (x,y,z) such that 0 <= x <= 1 AND 0 <= y <= 1 AND 0<= z <= 1. The surface S of the cube D consists of six square faces, with normals pointing out of the cube.   Question: Calculate the flux of F across the surface S of the cube D.

Let F = (0, f, 1) and G = (x, 0, 1) be vector fields on R, where f = f(x, y) is a smooth function f(x, y) (that is independent of z). Find a function f such that V × (F × G) = (7x, – 7y + 2x, 8). f(x, y) =

Let f(x, y) = xy² and r(t) = (¹/1²,t³). Calculate Vf. (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) Vf= Calculate r' (t). (Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form (*, *). ) r' (t) = Use the Chain Rule for Paths to evaluateƒ(r(t)) at t = 1.5. (Use decimal notation. Give your answer to three decimal places.) d dt - ƒ(r(1.5)) = = Use the Chain Rule for Paths to evaluate ƒ(r(t)) at t = -1.7. (Use decimal notation. Give your answer to three decimal places.) d dt d f(r(-1.7)) =