The following commands will calculate a surface integral using the Divergence Theorem of the vector field 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 over the surface defined by a hemisphere of radius 1 whose bottom is the disk 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 in the xy plane. restart:with(plots):with(linalg):with(LinearAlgebra):with(VectorCalculus):SetCoordinates(cartesian[x,y,z]); F:=<x,y,z^2>; Now we will calculate the divergence. Note the Divergence command in MAPLE will calculate the divergence of a vector field so it is necessary to convert if all we have is a general vector. divF:=Divergence(VectorField(F)); Now we create a triple integral to calculate the total flux over the hemisphere. This will be easier to do in cylindrical coordinates. Int(Int(Int(divF*r,z=0..sqrt(1-r^2)),theta=0..2*Pi),r=0..1)=int(int(int(divF*r,z=0..sqrt(1-r^2)),theta=0..2*Pi),r=0..1); We get the same thing that we got when we computed the surface integral for this surface, thus confirming the Divergence Theorem.