nt) Compute the flux of the vector field F = 5xi + 5yj through the surface S, which is the part of the surface z = 9(²+ y2) above the disk of radius 3 centered at the origin, oriented upward. flux =

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.

Let I be the flux of G = (4e³,9x³ex⁹, 0) through the upper hemisphere S of the unit sphere. (a) Find a vector field A such that curl(A) = G. (b) Calculate the circulation of A around as. (c) Compute I, the flux of G through S. (a) A = (b) √ A · ds = (c) I =

a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the flux of F = (x³, y³, z³) through any closed surface positive, negative, or zero?