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.

Parameterize the paraboloidal part by

and the planar part by

both with and .

For the paraboloidal part, take the normal vector to be

and for the planar part,

The flux over the paraboloidal part (call it ) is

and over the planar part (call it ),

but this integral vanishes because the dot product is 0. So the total flux over (the boundary of the given region ) is

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Confirming with the divergence theorem: the divergence of the given vector field is

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

(where the integral was set up with cylindrical coordinates)