Answer:
check Here.
Explanation:
(a) A can be found by solving the equation curl(A) = G, which means that the curl of A in the x, y, and z directions must equal the corresponding components of G. To find A, we can use the vector identity curl(A) = del x A - del y A + del z A.
From this, we get:
del y A = 9e^y
del x A = -3x^2 ex^3
del z A = 0
So A = (f(z), g(x, y), h(x, y, z)) where f, g, h are arbitrary differentiable functions.
(b) Circulation of A around C is given by the line integral of A . ds, where C is a closed curve and ds is an infinitesimal element of C. Since we are given a specific curve 88, we need to know the parametric representation of the curve to calculate the circulation.
(c) The flux of G through S is given by the surface integral of G . dS, where dS is an infinitesimal element of the surface S. Since we are given the upper hemisphere of the unit sphere as S, we can use spherical coordinates to parametrize the surface and then use the divergence theorem to calculate the flux.
Note that the specific values of A, the circulation, and the flux are dependent on the choice of f, g, h and the representation of 88, and dS.