Let I be the flux of G=<5e^y, 2xe^(x^2), 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 dS.(c) Compute I, the flux of G through S.

This problem involves vector calculus, specifically the concepts of curl and flux. Here's how you can solve it:

(a) Find a vector field A such that curl(A) = G.

The curl of a vector field A = <P, Q, R> is given by curl(A) = ∇ × A = <(∂R/∂y - ∂Q/∂z), (∂P/∂z - ∂R/∂x), (∂Q/∂x - ∂P/∂y)>. We want to find A such that curl(A) = G = <5e^y, 2xe^(x^2), 0>.

This gives us the system of equations:

∂R/∂y - 0 = 5e^y => R = 5ye^y + f(x,z) ∂P/∂z - ∂R/∂x = 2xe^(x^2) => P = 2xze^(x^2) + g(y,z) 0 - ∂P/∂y = 0 => P does not depend on y

From these equations, we can choose f(x,z) = g(y,z) = 0 (the simplest choice) and get A = <2xze^(x^2), 0, 5ye^y>.

(b) Calculate the circulation of A around dS.

The circulation of A around a surface S is given by the surface integral ∫∫_S curl(A) · dS. But we know that curl(A) = G, so the circulation of A around S is just the flux of G through S, which we are asked to compute in part (c).

(c) Compute I, the flux of G through S.

The flux of a vector field G through a surface S is given by the surface integral ∫∫_S G · dS. To compute this, we need to parameterize the upper hemisphere S of the unit sphere. A common parameterization is φ(u, v) = <sin(u)cos(v), sin(u)sin(v), cos(u)> for u in [0, π/2] and v in [0, 2π].

Then dS = ||φ_u × φ_v|| du dv = sin(u) du dv.

So the flux I is ∫ (from 0 to π/2) ∫ (from 0 to 2π) G(φ(u, v)) · <0, 0, 1> sin(u) du dv = ∫ (from 0 to π/2) ∫ (from 0 to 2π) 0 sin(u) du dv = 0.

So the flux of G through the upper hemisphere S of the unit sphere is 0.