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)

the answer is -6

step-by-step explanation:

the answer is -6 because its next to the variable m