7. Use the divergence theorem to calculate the outward flux of F through the closed surface S. (a) ♬ = r²i + y² + 22k and S is the region cut from the solid cylinder z² + y²

Let F = -9zi+ (xe#z– 2xe**)}+ 12 k. Find f, F·dĀ, and let S be the portion of the plane 2x + 3z = 6 that lies in the first octant such that 0 < y< 4 (see figure to the right), oriented upward. Z Explain why the formula F · A cannot be used to find the flux of F through the surface S. Please be specific and use a complete sentence.

1) Find the outward flux of F across the region D where F = (6x² + 2xy, 2y + x²z, 4x² y³ ) and the region D is cut from the first octant by the cylinder x + y = 4 and the plane z = 3.

Evaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =