of the quotient space R²/ given below, determine wheth under the quotient projection p: R² → R²/. (a) {[(0,0)]} (b) {[(y)]: (z,y) B₁ (0,0)} (c) {[(z.y)]: (x,y) € B₂(0,0))

Consider the following vector function: A = (6xy)x+ (3x²)ŷ + (4z)2 a) Calculate the divergence and the curl of this vector function. b) Calculate the path integral of this vector function from the origin to the point (1,1,1) using two different paths: 1) going in the direction from (0,0,0) to (1,0,0), then the y direction from (1,0,0) to (1,1,0), then the 2 direction from (1,1,0) to (1,1,1) and 2) going in a straight line from (0,0,0) to (1,1,1). c) Are the results of parts a) and b) consistent with each other? Explain why or why not.

Consider the following. C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1), starting at (0, 0) (a) Find a piecewise smooth parametrization of the path C. (t,0) (2-t,t - 1) (0,3 – t) r(t) = (b) Evaluate √(x + 4√Y) ds. 0 ≤t≤ 1 1 ≤t≤2 2 ≤ t ≤ 3

Let y = 3 H -3 -2 and u = projuy = 1 (a) Find the orthogonal projection of y onto u. A (b) Compute the distance d from y to the line through u and the origin. d=