Consider the vector field

1. Consider the vector field F(x, y, z) = yitejtzk. (i) Use the standard relationships between cartesian, cylindrical, and spherical variables as well as the unit vector relations given below to rewrite F as a cylindrical vector field F(p, , z) and a spherical vector field F (r, 0, 4). cos(b) - sin() 0 ep sin() cos (cp) ed 0 0 sin(0) cos($) cos(@) cos($) - sin() er sin(0) sin(6) cos(0) sin(d) cos() cos(0) - sin(@) 0 ed (ii) Compute the curl of F in its cartesian, cylindrical, and spherical forms. ep ped ez er reo rain(0)ed V X E cul = p ap VXEsph 72 sin(@) Fi pl2 F3 Fi TF2 rsin(0) F3 (iii) The figure below shows a sphere of radius 2, on which a space curve (red line) is traced out at 0 = n/4. The space curve forms the boundary of three open surfaces, defined below. SI = {(p, d, z) | p E [0, v2], $ c [0, 2x), z = V2} (black); $2 = {(r, 0, 6) |r = 2, 0 c [0, 7/4, $