(7) Verify the Divergence Theorem for the vector field F = azi + yzj +32²k defined in the solid bounded by the paraboloid z = r? + y² and the plane z = 1, as shown below;

Determine the flux of the vector field F(x, y, z) = (x, y, z) across the portion of the paraboloid z = 3 - x² - y² in above the plane z = -1 with upward orientation.

Show that the vector field F(x, y, z) = (x² + y²)i – 2xyj is not conservative on R³.

(c) (d) Consider the vector field F (x, y, z) = x²i+ y²j+z²k. Let S be the surface of the region D enclosed by the cylinder x² + y² = 4 and the planes z = 0 and z = 3. Use the Divergence Theorem to calculate the flux of F across the surface S. Describe two geometric applications of Divergence Theorem.