Answer :
Final answer:
(a) The flux of F out of S is 0, corresponding to option A).
(b) The circulation of F around C is -49π, corresponding to option B).
Explanation:
(a) To find the flux of F out of S, we need to calculate the surface integral ∬_S F ⋅ dS over the disc S. Since the vector field F is normal to the xy-plane at every point with the component k, and S lies within the xy-plane (z = 0), the flux of F out of S is 0. Therefore, option A) ∬_S F ⋅ dS = 0.
(b) To determine the circulation of F around C, we need to evaluate the line integral ∮_C F ⋅ dr along the boundary curve C. Using Green's theorem, we can relate this line integral to the double integral over the region enclosed by C. Since C is a circle of radius 7 (x² + y² = 49), and the vector field F does not have any component tangent to the curve C, the circulation around C is -49π. Therefore, option B) circulation = -49π.
