Verify that the divergence theorem is true for the vector field f on the region e. give the flux. f(x, y, z) = x2i + xyj + zk, e is the solid bounded by the paraboloid z = 25 − x2 − y2 and the xy-plane.

Surface integral: Parameterize the closed surface by




with and , where defines the paraboloid part and the planar part of the total surface .

We have




so we get




The second integral vanishes when computing the dot product, so we're left with the first integral which reduces to



Volume integral (divergence theorem): We have divergence



By the divergence theorem, the flux is equivalent to the volume integral



where denotes the space enclosed by the surface . Converting to cylindrical coordinates lets us write the integral as



as desired.

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