Please answer all 4 questions

(1) (4+4 points) Consider the vector field F : R3 - R3, F(x, y, z) = (sin(y) + z cos(r), r cos(y) + sin(z), y cos(z) + sin(x)). (a) Compute curl( F) (x, y, z). (b) Is F a gradient field? If so, find a potential f of F. (2) (2+2 points) Let F be as in problem (1). (a) Compute div(F)(x, y, z). (b) Is the origin a sink, source or balanced point of F? (3) (6 points) Consider an elliptic piece of wire, represented by the curve in R2. Assume that this piece of wire has constant mass density 1. Verify that its center of mass is the origin. Hint: Do not get discouraged by the fact that the length of & cannot be computed exactly. (4) (8 points) Consider the parametrization 7 : [0, 47] - R', v(t) = (t cost, t sint) of a counter-clockwise spiral C starting at the origin. What is the average distance of a point on C to the origin