Let f(x, y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that lies above the plane z = 2 and is oriented upward.

Consider the closed region bounded simultaneously by the paraboloid and plane, jointly denoted . By the divergence theorem,



And since we have



the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have






Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by , we have



Parameterize by




which would give a unit normal vector of . However, the divergence theorem requires that the closed surface be oriented with outward-pointing normal vectors, which means we should instead use .

Now,





So, the flux over the paraboloid alone is