Final Answer:
(a) The vector field F = ⟨4x³, -2y³, 0⟩ satisfies curl(F) = ⟨8y, 2x², 0⟩.
(b) The circulation of F around the boundary ∂S of the upper hemisphere is 0.
(c) The flux of F through the upper hemisphere S is 4π.
Step-by-step explanation:
(a) To find a vector field F such that curl(F) is equal to ⟨8y, 2x², 0⟩, we need to choose F such that its components match the given curl components. In this case, F = ⟨4x³, -2y³, 0⟩ satisfies the condition.
(b) The circulation of a vector field around a closed curve is given by Stokes' theorem as the integral of the curl over the surface enclosed by the curve. In this case, the curl of F is ⟨8y, 2x², 0⟩, and the upper hemisphere S serves as the surface. Since the curl is conservative (divergence is zero), the circulation is zero.
(c) The flux of a vector field through a surface is given by the surface integral of the field over the surface. In this case, the flux of F through the upper hemisphere S is found by integrating F · n over S, where n is the unit normal to the surface. The flux is equal to 4π, as the surface integral evaluates to the area of the upper hemisphere.
In summary, we found a suitable vector field F for the given curl, showed that the circulation around ∂S is zero due to the conservative nature of the curl, and calculated the flux through the upper hemisphere S using the appropriate surface integral, yielding a result of 4π.