Et f(x, y, z) = z tan−1(y2)i + z3 ln(x2 + 5)j + zk. find the flux of f across s, the part of the paraboloid x2 + y2 + z = 11 that lies above the plane z = 2 and is oriented upward.

Since the surface is closed, and the vector field is rather complicated, you can use the divergence theorem. The flux of across is given by a surface integral, which the divergence theorem asserts is equivalent to a volume integral:



where denotes the space with boundary . We have



So in fact the flux across happens to be equal (in magnitude) to the volume encased by .



Convert to cylindrical coordinates, setting